{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nStatistical inference across multiple graphs is of vital interdisciplinary interest in domains as varied as machine learning, neuroscience, and epidemiology.\nInference on random graphs frequently depends on appropriate low-dimensional Euclidean representations of the vertices of these graphs, known as {\\em graph embeddings}, typically given by spectral decompositions of adjacency or Laplacian matrices \\citep{belkin03:_laplac, STFP-2011, chatterjee_usvt}. Nonetheless, while spectral methods for parametric inference in a single graph are well-studied, multi-sample graph inference is a nascent field. See, for example, the authors' work in \\citep{tang14:_semipar,tang14:_nonpar} as among the only principled approaches to two-sample graph testing. What is more, for inference tasks involving multiple graphs---for instance, determining whether two or more graphs on the same vertex set are similar---discerning an optimal simultaneous embedding for all graphs is a challenge: how can such an embedding be structured to both provide estimation accuracy of common parameters when the graphs are similar, but retain discriminatory power when they are different?\n\nA flexible, robust embedding procedure to achieve both goals would be of considerable utility in a range of real data applications. For instance, consider the problem of community detection in large networks. While algorithms for community detection abound, relatively few approaches exist to address community {\\em classification}; that is, to leverage graph structure to successfully establish which subcommunities appear statistically similar or different. In fact, the authors' work in \\cite{lyzinski15_HSBM} represents one of the earliest forays into statistically principled techniques for subgraph classification in hierarchical networks. Not surprisingly, the creation of a graph-statistical analogue of the classical analysis-of-variance $F$-test, in which a single test procedure would permit the comparison of graphs from multiple populations, is very much an open problem of immediate import. But even given a coherent framework for extracting graph-level differences across multiple populations of graphs, there remains the further complication of replicating this at multiple scales, by isolating---in the spirit of post-hoc tests such as Tukey's studentized range---precisely which subgraphs or vertices in a collection of vertex-matched graphs might be most similar or different. \n\nOur goal in this paper, then, is to provide a unified framework to answer the following questions:\n\\begin{enumerate}[(i)]\n\\item Given a collection of random graphs, can we develop a single statistical procedure that accurately estimates common underlying graph parameters, in the case when these parameters are equal across graphs, but also delivers meaningful power for testing when these graph parameters are distinct? \n\\item Can we develop an inference procedure that identifies sources of graph similarity or difference at scales ranging from whole-graph to subgraph to vertex? For example, can we identify particular vertices that contribute significantly to statistical differences at the whole-graph level?\n\\item Can we develop an inference procedure that scales well to large graphs, addresses graphs that are weighted, directed, or whose edge information is corrupted, and which is amenable to downstream classical statistical methodology for Euclidean data?\n\\item Does such a statistical procedure compare favorably to existing state-of-the-art techniques for joint graph estimation and testing, and does it work well on real data?\n\\end{enumerate}\nHere, we address each of these open problems with a single embedding procedure. Specifically, we describe an {\\em omnibus} embedding, in which the adjacency matrices of multiple graphs on the same vertex-matched set are jointly embedded into a single space {\\em with a distinct representation for each graph and, indeed, each vertex of each graph}. We then prove a central limit theorem for this embedding, a limit theorem similar in spirit to, but requiring a significantly more delicate probabilistic analysis than, the one proved in \\cite{athreya2013limit}. We show, in both simulated and real data, that the asymptotic normality of these embedded vertices has demonstrable utility. First, the omnibus embedding performs nearly optimally for the recovery of graph parameters when the graphs are from the same distribution, but compares favorably with state-of-the-art hypothesis testing procedures to discern whether graphs are different. Second, the simultaneous embedding into a shared space\nallows for the comparison of graphs without the need to\nperform pairwise alignments of the embeddings of different graphs.\nThird, the asymptotic normality of the omnibus embedding permits the application of a wide array of subsequent Euclidean inference techniques, most notably a multivariate analysis of variance (MANOVA) to isolate statistically significant {\\em vertices} across several graphs. Thus, the omnibus embedding provides a statistically sound analogue of a post-hoc Tukey test for multisample graph inference. We demonstrate this with an analysis of real data, comparing a collection of magnetic resonance imaging (MRI) scans of human brains to identify dissimilar graphs and then to further pinpoint specific intra-graph features that account for global graph differences. \n\nThe main theoretical results of this paper are a consistency theorem for the omnibus embedding, akin to \\cite{lyzinski13:_perfec}, and a central limit theorem, akin to \\cite{athreya2013limit}, for the distribution of any finite collection of rows of this omnibus embedding. \nWe emphasize that distributional results for spectral decompositions of random graphs are few. The classic results of\n\\cite{furedi1981eigenvalues} describe the eigenvalues of the Erd\\H{o}s-R\\'{e}nyi \nrandom graph and the work of \\cite{tao2012random} concerns distributions\nof eigenvectors of more general random matrices under moment restrictions,\nbut \\cite{athreya2013limit} and \\cite{tang_lse} are among the only references for central limit theorems for spectral decompositions of adjacency and Laplacian matrices for a class of independent-edge random graphs\nbroader than the Erd\\H{o}s-R\\'{e}nyi model.\n\nOur consistency result shows that the omnibus embedding provides consistent estimates of certain unobserved vectors, called latent positions, that are associated to vertices of the graphs.\nAt present, the best available spectral estimates\nof such latent positions involve averaging across graphs followed by\nan embedding, resulting in a single set of estimated latent positions,\nrather than in a {\\em distinct set for each graph}.\nWe find in simulations, that our omnibus-derived estimates perform competitively\nwith these existing spectral estimates of the latent positions,\nwhile still retaining graph-specific information.\nIn addition, we show that the omnibus embedding allows for a test statistic that improves on the state-of-the-art two-sample test procedure\npresented in \\cite{tang14:_semipar} for determining whether two\nrandom dot product graphs \\citep{young2007random}\nhave the same latent positions.\n\nSpecifically, \\cite{tang14:_semipar} introduces a test statistic generated by performing a Euclidean, lower-dimensional embedding of the graph adjacency matrix \\citep[see][]{STFP-2011} of each of the two networks, followed by a Procrustes alignment \\citep{gower_procrustes,dryden_mardia_shape} of the two embeddings. Addressing the nonparametric analogue of this question---whether two graphs have the same latent position {\\em distribution}---is the focus of \\cite{tang14:_nonpar}, which uses the embeddings of each graph to estimate associated density functions.\nThe Procrustes alignment required by \\cite{tang14:_nonpar} both complicates the test statistic and, empirically, weakens the power of the test (see Section \\ref{sec:expts}). Furthermore, it is unclear how to effectively adapt pairwise Procrustes alignments to tests involving more than two graphs. The omnibus embedding allows us to avoid these issues altogether by providing a multiple-graph representation that is well-suited to both latent position estimation and comparative graph inference methods.\n\nOur paper is organized as follows. In Sec.~\\ref{sec:summary_real_data}, we give an overview of our main results and present two real-data examples in which the omnibus embedding uncovers important multiscale information in collections of brain networks. In Sec.~\\ref{sec:formal_definitions}, we present background information and formal definitions. In Sec.~\\ref{sec:main_results}, we provide detailed statements of our principal theoretical results. In Sec.~\\ref{sec:expts}, we present simulation data that illustrates the power of the omnibus embedding as a tool for both estimation and testing. In our Supplementary Material, we provide detailed proofs, including a sharpening of a vertex-exchangeability argument for bounding residual terms in the difference between omnibus estimates and true graph parameter values. We conclude with a discussion of extensions and open problems on multi-sample graph inference.\n\n\\section{Summary of main results and applications to real data}\\label{sec:summary_real_data}\n\nRecall that our goal is to develop a single spectral embedding technique for multiple-graph samples that (a) estimates common graph parameters, (b) retains discriminatory power for multisample graph hypothesis testing and (c) allows for a principled approach to identifying specific vertices that drive graph similarities or differences. In this section, we give an informal description of the omnibus embedding and a pared-down statement of our central limit theorem for this embedding, keeping notation to a minimum. We then demonstrate immediate payoffs in exploratory data analysis, leaving a more detailed technical descriptions of the method and our results for later sections.\n\nTo provide a theoretically-principled paradigm for graph inference for stochastically-varying networks, we focus on a particular class of random graphs. \n\tWe define a {\\em graph} $G$ to be an ordered pair of $(V,\\mathcal{E})$ where $V$ is the {\\em vertex} or {\\em node} set, and $\\mathcal{E}$, the set of {\\em edges}, is a subset of the Cartesian product of $V \\times V$. In a graph whose vertex set has cardinality $n$, we will usually represent $V$ as $V=\\{1, 2, \\dots, n\\}$, and we say there {\\em is an edge between} $i$ and $j$ if $(i,j)\\in \\mathcal{E}$. The {\\em adjacency} matrix $\\mathbf{A}$ provides a representation of such a graph:\n\t$$\\mathbf{A}_{ij}=1 \\textrm{ if }(i,j) \\in \\mathcal{E}, \\textrm{ and }\\mathbf{A}_{ij}=0 \\textrm{ otherwise. }$$ \n\tWhere there is no danger of confusion, we will often refer to a graph $G$ and its adjacency matrix $\\mathbf{A}$\n\tinterchangeably.\n\t\n\tAny model of a stochastic network must describe the probabilistic mechanism of connections between vertices. We focus on a class of {\\em latent position} random graphs \\cite{Hoff2002,diaconis2007graph,asta_cls},in which every vertex has associated to it a (typically unobserved) {\\em latent position}, itself an object belonging to some (often Euclidean) space $\\mathcal{X}$. Probabilities of an edge between two vertices $i$ and $j$, $p_{ij}$, are a function $\\kappa(\\cdot,\\cdot): \\mathcal{X} \\times \\mathcal{X} \\rightarrow [0,1]$ (known as the {\\em link function}) of their associated latent positions $(x_i, x_j)$. Thus $p_{ij}=\\kappa(x_i, x_j)$, and given these probabilities, the entries $\\mathbf{A}_{ij}$ of the adjacency matrix $\\mathbf{A}$ are independent Bernoulli random variables with success probabilities $p_{ij}$. We consolidate these probabilities into a matrix $\\mathbf{P}=(p_{ij})$, and write $\\mathbf{A} \\sim \\mathbf{P}$ to denote this relationship. \n\t\n\tThe latent position graph model has tremendous utility in modeling natural phenomena. For example, individuals in a disease network may have a hidden vector of attributes (prior illness, high-risk occupation) that observers of disease dynamics do not see, but which nevertheless strongly influence the chance that such an individual may become ill or infect others. Because the link function is relatively unrestricted, latent position models can replicate a wide array of graph phenomena \\citep{olhede_wolfe_histogram}. \n\t\t\n\t\tIn a $d$-dimensional {\\em random dot product} graph \\citep{young2007random}, the latent space is an appropriately-constrained subspace of $\\mathbb{R}^d$, and the link function is simply the dot product of the two latent $d$-dimensional vectors. The invariance of the inner product to orthogonal transformations is a nonidentifiability in the model, so we frequently specify accuracy up to a rotation matrix $\\mathbf{W}$.\n\t\tRandom dot product graphs are often divided into two types: those in which the latent positions are fixed, and those in which the latent positions are themselves random. We will address both cases here: in our theoretical results, the latent positions $X_i \\in \\mathbb{R}^d$ for vertex $i$ are drawn independently from a common distribution $F$ on $\\mathbb{R}^d$; and our practical applications, we consider how to use an omnibus embedding to address the question of equality of potentially non-random latent positions. For the case in which the latent positions are drawn at random from some distribution $F$, an important graph inference task is the inference of properties of $F$ from an observation of the graph alone. In the graph inference setting, there is both randomness in the latent positions, and {\\em given these latent positions}, a subsequent conditional randomness in the existence of edges between vertices. \nA key to inference in such models is the initial step of consistently estimating the unobserved $X_i$'s from a spectral decomposition of $\\mathbf{A}$, and then using these estimates, denoted $\\hat{X}_i$, to infer properties of $F$. \n\n\nFor an RDPG with $n$ vertices, the $n\\times d$ matrix of latent positions $\\mathbf{X}$ is formed by taking vector $\\mathbf{X}_i$ associated to vertex $i$ to be the $i$-th row of $\\mathbf{X}$. Then $\\mathbf{P}=[p_{ij}]$, the matrix of probabilities of edges between vertices, is easily expressed as $\\mathbf{P}=\\mathbf{X}\\bX^T$. The aforementioned nonidentifiability is now transparent: if $\\mathbf{W}$ is orthogonal, then $\\mathbf{X}\\mathbf{W} \\mathbf{W}^T \\mathbf{X}^T=\\mathbf{P}$ as well, so the rotated latent positions $\\mathbf{X}\\mathbf{W}$ generate the same the matrix of probabilities.\nGiven such a model, a natural inference task is that of estimating the latent\nposition matrix $\\mathbf{X}$ up to some orthogonal transformation.\n\nBecause of the assumption that the matrix $\\mathbf{P}$ is of comparatively low rank,\nrandom dot product graphs can be analyzed with a number of tools from classical linear algebra,\nsuch as singular-value decompositions of their adjacency matrices. Nevertheless, this tractability does not compromise the utility of the model. Random dot product graphs are flexible enough to approximate a wide class of independent-edge random graphs \\citep{tang2012universally},\nincluding the stochastic block model \\citep{holland,karrer2011stochastic}.\n\nUnder mild assumptions, the adjacency matrix $\\mathbf{A}$ of a random dot product graph is a rough approximation of the matrix $\\mathbf{P}=[p_{ij}]$ of edge probabilities in the sense that the spectral norm of $\\mathbf{A}-\\mathbf{P}$ can be controlled;\nsee for example \\cite{oliveira2009concentration} and \\cite{lu13:_spect}.\nIn \\cite{STFP-2011}, \\cite{STFP-2011} and \\cite{lyzinski13:_perfec}, it is established that, under eigengap assumptions on $\\mathbf{P}$, a partial spectral decomposition of the adjacency matrix $\\mathbf{A}$, known as the {\\em adjacency spectral embedding} (ASE), allows for consistent estimation of the true, unobserved latent positions $\\mathbf{X}$. That is, if we define $\\hat{\\mathbf{X}}=\\bU_{\\bA}\\bS_{\\bA}^{1\/2}$, where $\\bS_{\\bA}$ is the diagonal matrix of the top $d$ eigenvalues of $A$, sorted by magnitude, and if $\\bU_{\\bA}$ are the associated unit eigenvectors, then the rows of this truncated eigendecomposition of $\\mathbf{A}$ are consistent estimates $\\{\\hat{\\bX}_i\\}$ of the latent positions $\\{\\mathbf{X}_i\\}$. Of course, these latent positions are often the parameters we wish to estimate. In \\cite{lyzinski15_HSBM}, it is shown that embedding the adjacency matrix and then performing a novel angle-based clustering of the rows is key to decomposing large, hierarchical networks into structurally similar subcommunities. In \\cite{athreya2013limit}, it is shown that the suitably-scaled eigenvectors of the adjacency matrix converge in distribution to a Gaussian mixture. In this paper, we prove a similar result for an omnibus matrix generated from multiple independent graphs.\n\nThe ASE provides a consistent estimate for the true latent positions in a random dot product graph up to orthogonal transformations. Hence a Procrustes distance between the adjacency spectral embedding of two graphs on the same vertex set serves as a test statistic for determining whether two random dot product graphs have the same latent positions\n\\citep{tang14:_semipar}.\nSpecifically,\nlet $\\mathbf{A}^{(1)}$ and $\\mathbf{A}^{(2)}$ be the adjacency matrices of two random dot product graphs on the same vertex set (with known vertex correspondence), and let $\\hat{\\bX}$ and $\\hat{\\bY}$ be their respective adjacency spectral embeddings.\nIf the two graphs have the same generating $\\mathbf{P}$ matrices, \nit is reasonable to surmise that the Procrustes distance\n\\begin{equation} \\label{eq:semipar:proc} \\min_{\\mathbf{W} \\in \\mathcal{O}^{d \\times d}}\\|\\hat{\\bX} -\\hat{\\bY} \\mathbf{W} \\|_F,\\end{equation}\nwhere $\\| \\cdot \\|_F$ denotes the Frobenius norm of a matrix, will be relatively small.\nIn \\cite{tang14:_semipar}, the authors show that a scaled version of the Procrustes distance in \\eqref{eq:semipar:proc} provides a valid and consistent\ntest for the equality of latent positions for a pair of random dot product\ngraphs. Unfortunately, the fact that a Procrustes minimization must be performed\nboth complicates the test statistic and compromises its power.\n\nHere, we instead consider an embedding of an {\\em omnibus matrix}, defined as follows. Given two independent $d$-dimensional RDPG adjacency matrices $\\mathbf{A}^{(1)}$ and $\\mathbf{A}^{(2)}$, on the same vertex set with known vertex correspondence, the {\\em omnibus matrix} $\\mathbf{M}$ is given by\n\\begin{equation}\\label{eq:omnibus_M}\n\\mathbf{M}=\\begin{bmatrix}\n\\mathbf{A}^{(1)} \t\t\t& \\frac{\\mathbf{A}^{(1)}+\\mathbf{A}^{(2)}}{2}\\\\\n\\frac{\\mathbf{A}^{(1)} + \\mathbf{A}^{(2)}}{2}\t& \\mathbf{A}^{(2)}\n\\end{bmatrix},\n\\end{equation}\nNote that this matrix easily extends to a sequence of graphs $\\mathbf{A}^{(1)}, \\cdots, \\mathbf{A}^{(m)}$, where the block diagonal entries are the matrices $\\mathbf{A}^{(i)}$ and the $(l,k)$-th off-diagonal block is the matrix $\\frac{\\mathbf{A}^{(k)}+ \\mathbf{A}^{(l)}}{2}$.\n\nAnalogously to our notation for the adjacency spectral embedding $\\hat{\\mathbf{X}}$, let $\\bS_{\\bM}$ represent the $d \\times d$ matrix of top $d$ eigenvalues of $\\mathbf{M}$, ordered again by magnitude, and let $\\bU_{\\bM}$ be the $mn \\times d$-dimensional matrix of associated eigenvectors. Define the {\\em omnibus embedding}, denoted $\\textrm{OMNI}(\\mathbf{M})$, by $\\bU_{\\bM} \\bS_{\\bM}^{1\/2}$. \nWe stress that $\\textrm{OMNI}(\\mathbf{M})$ produces $m$ separate points in Euclidean {\\em for each graph vertex}---effectively, one such point for each copy of the multiple graphs in our sample. This property renders the omnibus embedding useful for all manner of post-hoc inference. \n\nIf we consider the rows of the omnibus embedding as potential estimates for the latent positions, two immediate questions are arise. Are these estimates consistent, and can we describe a scaled limiting distribution for them as graph size increases? We answer both of these in the affirmative.\n\\\\\n\n\\noindent {\\bf Key result 1: The rows of the omnibus embedding provide consistent estimates for graph latent positions.} If the latent positions of the graphs $\\mathbf{A}^{(1)}, \\cdots, \\mathbf{A}^{m}$ are equal, then under mild assumptions, the rows of $\\textrm{OMNI}(\\mathbf{M})$ provide consistent estimates of their corresponding latent positions. Specifically, if $h=n(s-1)+ i$, where $1 \\leq i \\leq n$ and $1 \\leq s \\leq m$, then there exists an orthogonal matrix $\\mathbf{W}$ such that\n$$\\max_{1 \\leq i \\leq n}\\|(\\bU_{\\bM}\\bS_{\\bM}^{1\/2})_{h}-\\mathbf{W}\\mathbf{X}_i\\|<\\frac{C \\log mn}{\\sqrt{n}}$$\nwith high probability.\nThis consistency result is especially useful because it bounds the error between true and estimated latent positions for {\\em all latent positions simultaneously}. It further guarantees that the omnibus embedding competes well against the current best-performing estimator of the common latent positions $\\mathbf{X}$, which is the adjacency spectral embedding of the sample mean matrix $\\bar{\\mathbf{A}}=\\frac{\\sum_{i=1}^m \\mathbf{A}^{(i)}}{m}$ \\citep{runze_law_large_graphs}. \nThus, the omnibus embedding is not just consistent when the latent positions are equal; it is close to near-optimal, and we exhibit this clearly in simulations (see Sec.~\\ref{sec:expts}).\n\nOur second key result concerns the limiting distribution, as the graph size $n$ increases, of the rows of the omnibus embedding in the case when the graphs are independent and have the same latent positions.\n\\\\\n\n\\noindent {\\bf Key result 2: For large graphs, the scaled rows of the omnibus embedding are asymptotically normal}. Suppose that the latent positions for each graph are drawn i.i.d from a suitable distribution $F$, and that conditional on these latent positions, the adjacency matrices $\\mathbf{A}^{(1)}, \\cdots, \\mathbf{A}^{(m)}$ are independent realizations of random dot product graphs with the given latent positions. Let $\\mathbf{Z}$ represent that $mn\\times d$ dimensional matrix of latent positions for the graphs. Let $h=n(s-1) + i$, where $1 \\leq i \\leq n$ and $1 \\leq s \\leq m-1$. Then under mild assumptions, there exists a sequence of orthogonal matrices $\\mathbf{W}_n$ such that as $n \\rightarrow \\infty$, \n$$\\sqrt{n}[(\\bU_{\\bM}\\bS_{\\bM}^{1\/2})\\mathbf{W}_n-\\mathbf{Z}]_h$$ converges to a mean-zero $d$-dimensional Gaussian mixture.\nHence the omnibus embedding allows for accurate estimation when the latent positions are equal; provides multiple points for each vertex; and under mild assumptions on the structure of the latent positions, these embeddings are approximately normal for large graph sizes. Even more remarkably, \nfor testing whether two graphs have the same latent positions, we can build the omnibus embedding\nand consider only the Frobenius norm of the difference between the matrices defined by, respectively, the first $n$ and the second $n$ rows of this decomposition. This matrix difference, {\\em without any further Procrustes alignment}, also serves as a test statistic for the equality of latent positions, which brings us to our third point. \n\\\\\n\n\\noindent {\\bf Key simulation evidence: The omnibus embedding has meaningful power for two- and multi-sample graph hypothesis testing}. The omnibus embedding, without subsequent Procrustes alignments, yields an improvement in power over state-of-the-art methods in two-graph testing, as borne out by a comparison on simulated data. Combined with our earlier bounds on the $2 \\to \\infty$-norm difference between true and estimated latent positions, this demonstrates that the omnibus embedding provides estimation accuracy when the graphs are drawn from the same latent positions and improved discriminatory power when they are different.\nWhat is more, the omnibus embedding produces multiple points for each vertex and our asymptotic normality guarantees that these points are approximately normal. As a consequence, the omnibus embedding not only permits the discovery of graph-wide differences, but also the {\\em isolation of vertices} that contribute to these differences, via, for instance, a multivariate analysis of variance (MANOVA) applied to these embedded vectors. This leads us to our final point.\n\\\\\n\n\\noindent {\\bf Key real data analysis: the omnibus embedding isolates graph-wide differences and gives principled evidence for vertex significance in those differences, and works well on complicated real data}. On weighted, directed, noisily-observed real graphs, slight modifications to the omnibus embedding procedure yield genuine exploratory insights into graph structure and vertex importance, with actionable import in application domains. \n\nTo demonstrate this, we present in the next subsections detailed analyses\nof two neuroscientific data sets.\nThe first is a connectomic data set, in which we analyze a collection of paired diffusion MRI (dMRI) brain scans across 57 patients \\citep{kiar_two_truths}, a comparison of 114 graphs with 172 common vertices. \nSecond, we consider the COBRE data set \\citep{AineETAL2017},\na collection of functional MRI scans of \n$54$ schizophrenic patients and $69$ healthy controls,\nwith each scan yielding a graph on $264$ common vertices.\nIn both data sets,\nwe show how the omnibus embeddings can identify whole-graph differences\nas well as particular vertices involved in this difference.\n\n\\subsection{Discerning vertex-level difference in paired brain scans of human subjects}\n\\label{subsec:BNU1}\n\nAs a case study in the utility of our techniques, we consider data from human brain scans collected on 57 subjects at Beijing Normal University. The data, labeled ``BNU1\", is available at \\\\\n\\texttt{http:\/\/fcon\\_1000.projects.nitrc.org\/indi\/CoRR\/html\/bnu\\_1.html}.\n\nThis diffusion MRI data comprises two scans on each of 57 different patients, for a total of 114 scans. The dMRI data was converted into weighted graphs via the Neuro-Data-MRI-to-Graphs (NDMG) pipeline of \\cite{kiar_two_truths}, with vertices\nrepresenting sub-regions defined via spatial proximity and\nedges by tensor-based fiber streamlines connecting these regions. As such, by condensing vertices in a given brain region still further, neuroscientists can represent this data at different scales. We focus on data at a resolution in which the $m=114$ graphs have $n=172$ common vertices. (We point out that in \\cite{cep_two_truths}, this same data, at slightly different scale, serves as a useful illustration of different structural properties uncovered by different spectral embeddings). Our inference goals are to determine which of these graphs appear statistically similar and to elucidate which vertices might be key contributors to such difference.\n\nWe binarize the graphs via a simple thresholding operation,\nreplacing all nonzero edge weights with 1, and leaving unchanged edges of weight zero. For reference, Figure~\\ref{fig:sample_adj_mat} shows a visualization of one pair of adjacency matrices. Alternative approaches to weighted graphs, such as a replacement of weights by a rank-ordering, are also possible and have proven useful \\citep[see][]{cep_two_truths}, but we do not pursue this here.\n\n\\begin{figure}[tbh!]\n \\centering\n \\subfloat[]{\\includegraphics[width=0.4\\columnwidth]{expts\/BNU1Figs\/A1.pdf} }\n \\subfloat[]{\\includegraphics[width=0.4\\columnwidth]{expts\/BNU1Figs\/A2.pdf} }\n \\caption{Binarized adjacency matrices for two BNU-1 brain graphs}\n \\label{fig:sample_adj_mat}\n\\end{figure}\n\nWith these $m=114$ binarized, undirected adjacency matrices, we generate the $m$-fold omnibus matrix $M$, the $m$-fold analogue of the matrix in Eq.~\\eqref{eq:omnibus_M}, which is of size $(114 \\times 172)\\times(114 \\times 172)$.\nTo select a dimension for the omnibus embedding,\nwe apply the profile-likelihood method of \\cite{zhu06:_autom} to $M$.\nThis procedure performs model selection (i.e., estimates the rank of $\\mathbb{E} M$)\nby locating an elbow in the screeplot of the eigenvalues of $M$\n(shown in Figure~\\ref{fig:embedded_dim_BNU}(a))\nThis yields an estimated embedding dimension of $\\hat{d}=10$.\nAs a check, we perform the same estimation procedure\non all of the 114 graphs as well.\nReassuringly, we recover an estimated embedding dimension close to $10$\nfor almost all of them, and proceed with $\\hat{d}=10$,\nas summarized by the boxplot in Fig.~\\ref{fig:embedded_dim_BNU}(b).\n\\begin{figure}[tbh!]\n \\centering\n \\subfloat[]{\\includegraphics[width=0.4\\columnwidth]{expts\/BNU1Figs\/emb-1.pdf} }\n \\subfloat[]{\\includegraphics[width=0.4\\columnwidth]{expts\/BNU1Figs\/emb-2.pdf} }\n \\caption{(a) Eigenvalues and embedding dimension for the omnibus matrix as BNU data. Note the second elbow at $\\hat{d}=10$. (b) Box plot of the first three elbows in the scree plot identified by the profile-likelihood method of \\cite{zhu06:_autom} applied to the 114 brain graphs individually. Note that the second elbow is concentrated around $\\hat{d} = 10$.}\n \\label{fig:embedded_dim_BNU}\n\\end{figure}\nWe now construct a {\\em centered} omnibus matrix, in which we subtract the sample mean $\\bar{\\mathbf{A}}=m^{-1}\\sum_{l=1}^m \\mathbf{A}^{(l)}$ from the omnibus matrix.\nThat is, we construct an omnibus matrix\nfrom the centered graphs $\\mathbf{B}^{(l)}=\\mathbf{A}^{(l)}-\\bar{\\mathbf{A}}$\ninstead of the observed graphs $\\mathbf{A}{(l)}$.\nHaving constructed this centered omnibus matrix,\nwe embed it into $\\hat{d}=10$ dimensions,\nproducing a matrix $\\hat{Z} \\in \\mathbb{R}^{mn \\times \\hat{d}}$,\nwith $\\hat{d}=10$ columns and $mn = 114\\times172$ rows.\nObserve that $\\hat{Z}$ can be subdivided into 114 blocks each of size 172,\none for each graph.\nFor convenience, we denote these submatrices, each of size $172\\times 10$, by\n$\\hat{{\\bf X}}^{(1)}, \\cdots, \\hat{{\\bf X}}^{(114)}$.\nUnder our model assumptions, each of these submatrices is an estimate of the latent position matrix of the corresponding brain graph.\n\nBecause the omnibus embedding introduces an alignment between graphs by placing an average on the off-diagonal blocks of the omnibus matrix, we find that merely considering a Frobenius norm difference between blocks of the omnibus embedding, i.e.,\n$$\\|\\hat{{\\bf X}}^{(l)}-\\hat{{\\bf X}}^{(k)}||_F,$$\n{\\em without} any further Procrustes alignments, provides meaningful power in distinguishing between graphs with different latent positions (again, see Sec.~\\ref{sec:expts} for simulation evidence, and Sec.~\\ref{sec:conc} for theoretical discussion of why the omnibus embedding obviates the need for further subspace alignments). As a consequence, we can create a $114\\times 114$ dissimilarity matrix $\\mathbf{D}=(\\mathbf{D}_{kl}) \\in \\mathbb{R}^{114 \\times 114}$, defined as\n$$\\mathbf{D}_{kl}:=\\|\\hat{{\\bf X}}^{(l)}-\\hat{{\\bf X}}^{(k)}||_F$$\nwhich records the Frobenius norm differences of the omnibus embeddings of the $k$-th and $l$-th graph in our collection. We illustrate this dissimilarity matrix in the first panel of Fig.~\\ref{fig:omni_dissim_matrix}, and we show how this matrix can be hierarchically clustered. \n\\begin{figure}[h!]\n \\centering\n \\subfloat[]{\\includegraphics[width=0.5\\columnwidth]{expts\/BNU1Figs\/Abar-2.pdf}}\n \\subfloat[]{\\includegraphics[width=0.5\\columnwidth]{expts\/BNU1Figs\/Tmat-C3.pdf}} \n \\caption{(a) Omnibus dissimilarity matrix $\\mathbf{D}$ across 114 graphs. (b) Results of hierarchical clustering of this dissimilarity matrix. }\n \\label{fig:omni_dissim_matrix}\n\\end{figure}\n\nUsing classical multidimensional scaling \\citep{cox_MDS}, we embed this dissimilarity matrix into $2$-dimensional Euclidean space.\nThis yields a collection of 114 points in $\\mathbb{R}^2$, each one of which represents\none graph.\nWe then cluster this collection of points using Gaussian mixture modeling,\nin which we select $c=3$ clusters according to the Bayesian Information Criterion (BIC). The three resulting clusters are depicted in Fig.~\\ref{fig:GMM_clustering_BNU}(b). We remark that out of 57 subjects, only 10 subject scans are divided across clusters, suggesting that the clusters capture meaningful similarity across graphs.\n\\begin{figure}[tbh!]\n \\centering\n \\subfloat[]{\\includegraphics[width=0.5\\columnwidth]{expts\/BNU1Figs\/Abar-4.pdf}}\n \\subfloat[]{\\includegraphics[width=0.5\\columnwidth]{expts\/BNU1Figs\/Abar-5.pdf}} \n \\caption{(a) Clustering using Gaussian mixture modeling and selection of $c=3$ clusters as applied to the $2$-dimensional CMDS embedding of $\\mathbf{D}$. (b) Visualization of the embeddings and their resulting clusters. Each point represents a single graph from the BNU1 data set.}\n \\label{fig:GMM_clustering_BNU}\n\\end{figure}\n\nIf it were truly the case that all of these graphs\nhad the same latent positions,\nour main central limit theorem would ensure that for large graph sizes\nthese embedded points would be asymptotically normal.\nThus, if we consider these three clusters as identifying three distinct types\nof graphs, we can now compare embedded latent positions of individual vertices\nacross graphs to determine which vertices play the most similar or\ndifferent roles in their respective graphs.\nThe (theoretical) asymptotic normality leads us to consider a multivariate analysis of variance (MANOVA).\nFor each vertex, we consider the embedded points corresponding to that vertex\nthat arise from the graphs in Cluster 1, Cluster 2, and Cluster 3,\nrespectively.\n{\\em Because we have multiple embedded points for each vertex and these\nembedded points are asymptotically normal,} MANOVA is,\nas an exploratory tool, principled.\nMANOVA produces $p$-value for each vertex, associated to the test of equality\nof the true mean vectors for the normal distributions governing the embedded\npoints in each of the $3$ classes\nSince there are 172 vertices, we obtain 172 corresponding $p$-values, and we correct for multiple comparisons using the Bonferroni correction. The $p$-values are ordered by significance in Fig.~\\ref{fig:MANOVA_pvalue_BNU}.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{expts\/BNU1Figs\/manova-2.pdf}\n\\caption{MANOVA $p$-values, with vertices sorted by significance and adjusted for multiple comparisons. The dotted lines indicate the $p=0.05$ threshold (green) and the threshold after Bonferroni correction.}\n\\label{fig:MANOVA_pvalue_BNU}\n\\end{figure}\nWe focus on one of the two most significant vertices, Vertex 98.\nRecall that by nature of the omnibus embedding, we have multiple points\nembedded in $\\mathbb{R}^{\\hat{d}}$, each of which correspond to the $98$-th vertex\nin one of the 114 graphs. Partitioning these $114$ points according to the clusters associated to their respective graphs, we can perform nonparametric tests of difference across\nthese collections of points to further illuminate how this\nvertex differs in its behavior across the three graph clusters.\nFig.~\\ref{fig:MANOVA_most_sig_vertex_cluster_diff_BNU_dim1_and_2}\nillustrates how strikingly different are the first two principal dimensions\nof the embedded points for this vertex across the three different clusters.\nFor contrast, we examine one of the least significant vertices, Vertex 124, and reproduce the analogous plots to those in Fig.\\ref{fig:MANOVA_most_sig_vertex_cluster_diff_BNU_dim1_and_2} for its first principal dimension.\nThe results are displayed in Fig.~\\ref{fig:MANOVA_least_sig_vertex_cluster_diff_BNU}.\n\n\\begin{figure}[!h]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{expts\/BNU1Figs\/sigv-X1-bar.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{expts\/BNU1Figs\/sigv-X1-violin.pdf}}\\\\\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{expts\/BNU1Figs\/sigv-X2-bar.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{expts\/BNU1Figs\/sigv-X2-violin.pdf}}\n\\caption{(a) For the most significant vertex (Vertex 98), a histogram of the first principal dimension of embedded latent position, grouped by cluster. (b) For this same vertex, estimated mean and confidence intervals for the first dimension of the embedded position, again grouped by cluster. (c) and (d) show the analogous plots for the second principal dimension. Observe that the first principal dimension distinguishes between the first and second cluster, and the second principal dimension between the first and the third cluster.}\n\\label{fig:MANOVA_most_sig_vertex_cluster_diff_BNU_dim1_and_2}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{expts\/BNU1Figs\/sigv-least-bar.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{expts\/BNU1Figs\/sigv-least-violin.pdf}\n}\n\\caption{(a) For the least significant vertex, a histogram of the first dimension of estimated latent positions, grouped by cluster. (b) For this same vertex, estimated means and confidence intervals for the first dimension of the estimated latent position, again grouped by cluster.}\n\\label{fig:MANOVA_least_sig_vertex_cluster_diff_BNU}\n\\end{figure}\n\nThe contrast between Figs.~\\ref{fig:MANOVA_most_sig_vertex_cluster_diff_BNU_dim1_and_2} and \\ref{fig:MANOVA_least_sig_vertex_cluster_diff_BNU} is striking.\nBecause the omnibus embedding gives us multiple points for each vertex,\nwe are able to isolate vertices that are responsible for between-graph\ndifferences\nand then interface with neuroscientists to discern what physical distinctions\nmight be present at this vertex or brain location across the clusters of graphs.\n\nWe recognize, of course, some immediate concerns with our procedure.\nFirst, the fact that our clusters are determined post-hoc implies that\nthe embedded vectors are not independent samples from different populations.\nSecond, the asymptotic normality of the embedded positions is a large-sample\nresult, and applies to an arbitrary but finitely fixed collection of rows. Despite these limitations, we stress that our theoretical results supply a principled foundation on which to build a more refined analysis, and to date this is among the only approaches for the identification and\ncomparison of individual vertices and their role in\ndriving differences between (populations of) graphs.\n\n\\subsection{Identifying brain regions associated with schizophrenia}\n\\label{subsec:COBRE}\n\nWe next consider the COBRE data set \\citep{AineETAL2017},\na collection of scans of both schizophrenic and healthy patients.\nEach scan yields a graph on $n = 264$ vertices, corresponding to 264\nbrain regions of interest \\citep{PowerETAL2011},\nwith edge weights given by correlations between BOLD signals\nmeasured in those regions.\nThe data set contains scans for\n$54$ schizophrenic patients and $69$ healthy controls, for a total of\n$m = 123$ brain graphs.\n\nWe follow the general framework of our BNU1 analysis above.\nUnder the null hypothesis that all $m$ graphs share the same underlying\nlatent positions, the omnibus embedding yields for each vertex\na collection of $m$ points in $\\mathbb{R}^d$ that are\nnormally distributed about the true latent position of that vertex.\nBy applying an omnibus embedding to the $m=123$ subjects in the COBRE dataset,\nwe can therefore test,\nfor each vertex $i \\in [264]$, whether or not the healthy and\nschizophrenic populations display a difference in that vertex,\nby comparing the latent positions of vertex $i$ associated with the\nschizophrenic patients against those associated with the healthy patients.\nThat is, let $m_s = 54$ denote the number of schizophrenic patients\nand $m_h = 69$ denote the number of healthy controls,\nwith respective embeddings given by\n$$\\{ X^{(j)}_i : j =1,2,\\dots,m_s \\}, \\textrm{ and } \\{ Y^{(j)}_i : j=1,2,\\dots,m_h \\}$$\nWe can test whether the samples\n$$\\{ X^{(1)}_i, X^{(2)}_i, \\dots, X^{(m_h)} \\} \\subseteq \\mathbb{R}^d\n\\textrm{ and } \\{ Y^{(1)}_i, Y^{(2)}_i, \\dots, Y^{(m_s)} \\} \\subseteq \\mathbb{R}^d$$\nappear to come from the same distribution.\nBy Theorem~\\ref{thm:main},\nif all $m$ subjects' graphs are drawn from the same\nunderlying RDPG, then it is natural to test the hypothesis that both\n the $X_i^(j)$ and then $Y_i^(j), 1 \\leq j \\leq m_k$ are drawn from the same normal distribution.\nWe use Hotelling's $t^2$ test \\citep{Hotelling1931,Anderson2003} (and we remark that experiments applying a permutation test for this same\npurpose yield broadly similar results).\nWe note that while in the BNU1 data example in Section~\\ref{subsec:BNU1},\nwe required a clustering to discover collections of similarly-behaving\nnetworks, the COBRE data set already has two populations of interest\nin the form of the healthy and schizophrenic patients.\n\nWe begin by building the omnibus matrix of $m=123$ brain graphs,\neach on $n=264$ vertices.\nIn contrast to the BNU1 data presented above,\nhere we work with the weighted graph obtained from scans,\nrather than binarizing them.\nWe apply a three-dimensional omnibus embedding to these $m$ graphs,\nyielding $123$ points in $\\mathbb{R}^3$ for each of the $n=264$ brain regions\nfor a total of $32472$ points.\nFor each vertex $i \\in \\{1,2,\\dots,264\\}$,\nthere are $m=123$ points in $\\mathbb{R}^3$ each corresponding\nto vertex $i$ in one of the brain graphs.\n$54$ of these $123$ points correspond to the estimated\nlatent position of the $i$-th vertex in the schizophrenic patients,\nwhile the remaining $69$ points correspond to the estimated latent position\nof the $i$-the vertex in the healthy patients.\nFor each vertex $i$, we apply Hotelling's $t^2$ test to assess whether or not\nthe healthy and schizophrenic estimated latent positions appear to come\nfrom different populations.\nThus, for each of the $264$ regions of interest, we obtain a $p$-value\nthat captures the extent to which the estimated latent positions of the\nhealthy and schizophrenic patients appear to differ in their distributions.\n\nFigure~\\ref{fig:pval_by_parcel} summarizes the result of the procedure\njust described. Using the Power parcellation \\citep{PowerETAL2011},\nwe group the 264 brain regions into larger {\\em parcels},\nwhich capture what are believed by\nneuroscientists to correspond to functional subnetworks of the brain.\nFor example, a parcel called the\n{\\em default mode network} is associated with wakeful, undirected thought\n(i.e., mind wandering), and is implicated in\nschizophrenia \\citep{WhitfieldGabrieliETAL2009,FoxETAL2015}.\nWe collect, for each of the 14 Power parcels, the $p$-values associated with\nall of the brain regions (i.e., vertices) in that parcel, and display in\nFigure~\\ref{fig:pval_by_parcel} a histogram of those $p$-values.\nUnder this setup,\nparcels in which the populations are largely the same will have histograms\nthat appear more or less flat,\nwhile parcels in which schizophrenic patients\ndisplay different behavior from their healthy counterparts will result\nin left-skewed histograms.\nObserving Figure~\\ref{fig:pval_by_parcel}, we see strong visual evidence\nthat the default mode, the sensory\/somatomotor hand\nand the uncertain parcels are affected by schizophrenia.\n\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=\\columnwidth]{expts\/hotelling\/pval_by_parcel.pdf}\n \\caption{Histograms of the distribution of $p$-values within each parcel.\n Each histogram corresponds to one of the fourteen parcels in the\n\tPower parcellation \\citep{PowerETAL2011},\n\tand shows the distribution of the $p$-values obtained from\n\tapplying the Hotelling $t^2$ test to the omnibus embeddings\n\tof the brain regions in that parcel.\n\tWe see that certain parcels (most notably the default mode,\n\tsensory\/somatomotor hand, and uncertain parcels)\n\tclearly display non-uniform $p$-value distributions, suggesting that\n\tthese parcels differ in schizophrenic patients compared to\n\ttheir healthy counterparts.\n }\n \\label{fig:pval_by_parcel}\n\\end{figure}\n\nHere again we see the utility of the omnibus embedding.\nThanks to the alignment of the embeddings across all $123$ graphs in the\nsample, we obtain, after comparatively little processing, a concise summary of\nwhich vertices differ in their behavior across the two populations of\ninterest. Further, this information can be summarized into an simple display of information---in this case, summarizing which Power parcels are likely involved in schizophrenia---that is interpretable by neuroscientists and other domain specialists.\n\n\\section{Background, notation, and definitions}\\label{sec:formal_definitions}\nWe now turn toward a more thorough exploration of the theoretical\nresults alluded to above. We begin by establishing notation\nand a few definitions that will prove useful in the sequel.\n\\subsection{Notation and Definitions}\nFor a positive integer $n$, we let $[n] = \\{1,2,\\dots,n\\}$,\nand denote the identity, zero and all-ones matrices by, respectively,\n$\\mathbf{I}$, $\\mathbf{0}$ and $\\mathbf{J}$.\nFor an $n \\times n$ matrix $\\mathbf{H}$, we let $\\lambda_i(\\mathbf{H})$ denote\nthe $i$-th largest eigenvalue of $\\mathbf{H}$ and \nwe let $\\sigma_i(\\mathbf{H})$ denote the $i$-th singular value of $\\mathbf{H}$.\nWe use $\\otimes$ to denote the Kronecker product.\nFor a vector $\\mathbf{v}$, we let $\\| \\mathbf{v} \\|$ denote the Euclidean norm of $\\mathbf{v}$.\nFor a matrix $\\mathbf{H} \\in \\mathbb{R}^{n_1 \\times n_2}$,\nwe denote by $\\mathbf{H}_{\\cdot j}$\nthe column vector formed by the $j$-th column of $\\mathbf{H}$,\nand let $\\mathbf{H}_{i \\cdot}$ denote the row vector\nformed by the $i$-th row of $\\mathbf{H}$.\nFor ease of notation, we let $\\mathbf{H}_i \\in \\mathbb{R}^{n_2}$\ndenote the \\emph{column} vector formed by transposing the $i$-th row\nof $\\mathbf{H}$. That is, $\\mathbf{H}_i = (\\mathbf{H}_{i \\cdot})^T$.\nWe let $\\| \\mathbf{H} \\|$ denote the spectral norm of $\\mathbf{H}$,\n$\\| \\mathbf{H} \\|_F$ denote the Frobenius norm of $\\mathbf{H}$\nand $\\|\\mathbf{H}\\|_{2 \\rightarrow \\infty}$ denote the maximum of the Euclidean norms\nof the rows of $\\mathbf{H}$, i.e., $\\|\\mathbf{H}\\|_{2 \\rightarrow \\infty}=\\max_{i} \\| \\mathbf{H}_i \\|$.\nWhere there is no danger of confusion, we will often refer to a graph $G$ and its adjacency matrix $\\mathbf{A}$\ninterchangeably.\nThroughout, we will use $C > 0$ to denote a constant, not depending on $n$,\nwhose value may vary from one line to another.\nFor an event $E$, we denote its complement by $E^c$.\nGiven a sequence of events $\\{ E_n \\}$,\nwe say that $E_n$ occurs with high probability,\nand write $E_n \\text{ w.h.p. }$,\nif $\\Pr[ E_n^c ] \\le Cn^{-2}$ for $n$ sufficiently large.\nWe note that $E_n$ w.h.p. implies, by the Borel-Cantelli Lemma,\nthat with probability $1$ there exists an $n_0$ such that\n$E_n$ holds for all $n \\ge n_0$.\n\nOur focus here is on $d$-dimensional random dot product graphs,\nfor which the edge connection probabilities arise as inner products between\nvectors, called latent positions, that are associated to the vertices.\nTherefore, we define an {\\em an inner product distribution} as a probability distribution over a suitable subset of $\\mathbb{R}^d$, as follows:\n\\begin{definition}\n\\label{def:innerprod}\n\\emph{($d$-dimensional Inner Product Distribution)}\nLet $F$ be a probability distribution on $\\mathbb{R}^d$.\nWe say that $F$ is a\n\\emph{$d$-dimensional inner product distribution}\non $\\mathbb{R}^d$ if for all $\\mathbf{x},\\mathbf{y} \\in \\operatorname{supp} F$, we have $\\mathbf{x}^T \\mathbf{y} \\in [0,1]$.\n\\end{definition}\n\\begin{definition}\\label{def:RDPG}\n\\emph{(Random Dot Product Graph)}\nLet $F$ be a $d$-dimensional inner product distribution\nwith $\\mathbf{X}_1,\\mathbf{X}_2,\\dots,\\mathbf{X}_n \\stackrel{\\text{i.i.d.}}{\\sim} F$, collected in the rows of the matrix\n$\\mathbf{X}=[\\mathbf{X}_1, \\mathbf{X}_2, \\dots, \\mathbf{X}_n]^T \\in \\mathbb{R}^{n \\times d}$.\nSuppose $\\mathbf{A}$ is a random adjacency matrix given by\n\\begin{equation} \\label{eq:rdpg}\n\\Pr[\\mathbf{A}|\\mathbf{X}]=\n\\prod_{i 0$ denote the eigengap of\n\\begin{equation} \\label{eq:def:Delta}\n\\mathbf{\\Delta} = \\mathbb{E} \\mathbf{X}_1 \\mathbf{X}_1^T \\in \\mathbb{R}^{d \\times d},\n\\end{equation}\nthe second moment matrix of $\\mathbf{X}_1 \\sim F$.\nThat is, $\\delta = \\lambda_d( \\mathbf{\\Delta} ) > 0 = \\lambda_{d+1}( \\mathbf{\\Delta} )$.\nWe note that $\\mathbf{\\Delta}$ can be chosen diagonal without loss of generality\nafter a suitable change of basis \\citep{athreya2013limit}.\nWe assume further that $\\mathbf{\\Delta}$ is such that its diagonal entries are\nin nonincreasing order, so that\n$\\mathbf{\\Delta}_{1,1} \\ge \\mathbf{\\Delta}_{2,2} \\ge \\dots \\ge \\mathbf{\\Delta}_{d,d} = \\delta.$\nWe assume that the matrix $\\mathbf{\\Delta}$ is constant in $n$,\nso that $d$ and $\\delta$ are constants, while the number of graphs\n$m$ is allowed to grow with $n$.\nWe leave for future work the exploration of the case where the\nmodel parameters are allowed to vary with the number of vertices $n$.\n\nSince we rely on spectral decompositions, we begin with a straightforward one:\nthe spectral decomposition of the positive semidefinite matrix $\\mathbf{P}=\\mathbf{X}\\bX^T$.\n\\begin{definition}\\emph{(Spectral Decomposition of $\\mathbf{P}$)}\nSince $\\mathbf{P}$ is symmetric and positive semidefinite, let\n$\\mathbf{P}= \\bU_{\\bP} \\bS_{\\bP} \\bU_{\\bP}^T$ denote its spectral decomposition,\nwith $\\bU_{\\bP} \\in \\mathbb{R}^{n \\times d}$ having orthonormal columns\nand $\\bS_{\\bP} \\in \\mathbb{R}^{d \\times d}$ diagonal\nwith nonincreasing entries\n$(\\bS_{\\bP})_{1,1}\\ge (\\bS_{\\bP})_{2,2} \\ge \\cdots \\ge (\\bS_{\\bP})_{d,d} > 0$.\n\\end{definition}\n\nWe note that while $\\mathbf{P} = \\mathbf{X} \\mathbf{X}^T$ is not observed,\nexisting spectral norm bounds\n\\citep[e.g.,][]{oliveira2009concentration,lu13:_spect}\nestablish that if $\\mathbf{A} \\sim \\operatorname{Bernoulli}(\\mathbf{P})$,\nthe spectral norm of $\\mathbf{A}-\\mathbf{P}$ is comparatively small.\nAs a result, we regard $\\mathbf{A}$ as a noisy version of $\\mathbf{P}$,\nand we begin our inference procedures with a spectral decomposition of $\\mathbf{A}$.\n\\begin{definition}\\emph{\\citep[Adjacency Spectral Embedding;][]{STFP-2011}}\nLet $\\mathbf{A} \\in \\mathbb{R}^{n \\times n}$ be the\nadjacency matrix of an undirected $d$-dimensional random dot product graph.\nThe $d$-dimensional \\emph{adjacency spectral embedding} (ASE) of $\\mathbf{A}$\nis a spectral decomposition of $\\mathbf{A}$ based on its top $d$ eigenvalues,\nobtained by\n$\\operatorname{ASE}(\\mathbf{A},d) = \\bU_{\\bA} \\bS_{\\bA}^{1\/2}$, where $\\bS_{\\bA} \\in \\mathbb{R}^{d \\times d}$ is a diagonal\nmatrix whose entries are the top eigenvalues of $\\mathbf{A}$ (in nonincreasing order)\nand $\\bU_{\\bA} \\in \\mathbb{R}^{n \\times d}$ is the matrix whose columns are the\northonormal eigenvectors corresponding to the eigenvalues in $\\bS_{\\bA}$.\n\\end{definition}\n\n\\begin{remark}\\label{remark:nonneg_eig}\nWe observe that without any additional assumptions, the top $d$ eigenvalues\nof $\\mathbf{A}$ are not guaranteed to be nonnegative.\nHowever, under our eigengap assumptions on $\\mathbf{\\Delta}$,\nthe i.i.d.-ness of the latent positions ensures that for large $n$,\nthe eigenvalues of $\\mathbf{A}$ will be nonnegative with high probability\n(see Observation~\\ref{obs:deltaLB} in the Supplementary Material).\n\\end{remark}\n\nGiven a set of $m$ adjacency matrices distributed as\n$$(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},\\mathbf{X}) \\sim \\operatorname{JRDPG}(F,n,m)$$\nfor distribution $F$ on $\\mathbb{R}^d$,\na natural inference task is to recover the $n$ latent positions\n$\\mathbf{X}_1,\\mathbf{X}_2,\\dots,\\mathbf{X}_n \\in \\mathbb{R}^d$ shared by the vertices of the $m$ graphs.\nTo estimate the underlying latent positions from these $m$ graphs, \\cite{runze_law_large_graphs} provides justification for the estimate\n$\\bar{\\bX} = \\operatorname{ASE}( \\bar{\\bA}, d )$, where $\\bar{\\bA}$ is the sample mean of the\nadjacency matrices $\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)}$.\nHowever, $\\bar{\\bX}$ is ill-suited to any task that requires\ncomparing latent positions across the $m$ graphs,\nsince the $\\bar{\\bX}$ estimate collapses the $m$ graphs into a single\nset of $n$ latent positions.\nThis motivates the \\emph{omnibus embedding},\nwhich still yields a single spectral decomposition, but with a separate $d$-dimensional representation for each of the $m$ graphs.\nThis makes the omnibus embedding useful for {\\em simultaneous}\ninference across all $m$ observed graphs.\n\\begin{definition}\\emph{(Omnibus embedding)}\nLet $\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)} \\in \\mathbb{R}^{n \\times n}$\nbe (possibly weighted) adjacency matrices\nof a collection of $m$ undirected graphs.\nWe define the $mn$-by-$mn$ omnibus matrix\nof $\\mathbf{A}^{(1)}, \\mathbf{A}^{(2)}, \\dots, \\mathbf{A}^{(m)}$ by\n\\begin{equation} \\label{eq:omnidef}\n\\mathbf{M} =\n\\begin{bmatrix}\n\\mathbf{A}^{(1)} & \\frac{1}{2}(\\mathbf{A}^{(1)} + \\mathbf{A}^{(2)}) & \\dots & \\frac{1}{2}(\\mathbf{A}^{(1)} + \\mathbf{A}^{(m)}) \\\\\n\\frac{1}{2}(\\mathbf{A}^{(2)} + \\mathbf{A}^{(1)}) & \\mathbf{A}^{(2)} & \\dots & \\frac{1}{2}(\\mathbf{A}^{(2)} + \\mathbf{A}^{(m)}) \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\frac{1}{2}(\\mathbf{A}^{(m)} + \\mathbf{A}^{(1)}) & \\frac{1}{2}(\\mathbf{A}^{(m)} + \\mathbf{A}^{(2)})\n& \\dots & \\mathbf{A}^{(m)} \\end{bmatrix},\n\\end{equation}\nand the $d$-dimensional \\emph{omnibus embedding} of\n$\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)}$\nis the adjacency spectral embedding of $\\mathbf{M}$:\n$$ \\operatorname{OMNI}(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},d)= \\operatorname{ASE}( \\mathbf{M}, d ). $$\n\\end{definition}\nIf $(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},\\mathbf{X}) \\sim \\operatorname{JRDPG}(F,n,m)$,\nthen the omnibus embedding provides a natural approach to \nestimating $\\mathbf{X}$ \\emph{without} collapsing the $m$ graphs into a single\nrepresentation as with $\\bar{\\bX} = \\operatorname{ASE}(\\bar{\\bA},d)$.\nUnder the JRDPG, the omnibus matrix has expected value\n$$ \\mathbb{E} \\mathbf{M} = \\tilde{\\bP} = \\mathbf{J}_m \\otimes \\mathbf{P} = \\bU_{\\Ptilde} \\bS_{\\Ptilde} \\bU_{\\Ptilde}^T $$\nfor $\\bU_{\\Ptilde} \\in \\mathbb{R}^{mn \\times d}$ having $d$ orthonormal columns\nand $\\bS_{\\Ptilde} \\in \\mathbb{R}^{d \\times d}$ diagonal.\nSince $\\mathbf{M}$ is a reasonable estimate for $\\tilde{\\bP} = \\mathbb{E} \\mathbf{M}$\n\\citep[see, for example,][]{oliveira2009concentration},\nthe matrix $\\hat{\\bZ} = \\operatorname{OMNI}(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},d)$\nis a natural estimate of the $mn$ latent positions\ncollected in the matrix\n$\\mathbf{Z} = [\\mathbf{X}^T \\mathbf{X}^T \\dots \\mathbf{X}^T]^T \\in \\mathbb{R}^{mn \\times d}$.\nHere again, as in Remark~\\ref{rem:nonid}, $\\hat{\\bZ}$ only recovers\nthe true latent positions $\\mathbf{Z}$ up to an orthogonal rotation.\nThe matrix\n\\begin{equation} \\label{eq:Zstruct}\n\\bZ^* = \\begin{bmatrix} \\bX^* \\\\ \\bX^* \\\\ \\vdots \\\\ \\bX^* \\end{bmatrix}\n = \\bU_{\\Ptilde} \\bS_{\\Ptilde}^{1\/2} \\in \\mathbb{R}^{mn \\times d},\n\\end{equation}\nprovides a reasonable canonical choice of latent positions,\nso that $\\mathbf{Z} = \\bZ^* \\mathbf{W}$ for some suitably-chosen orthogonal matrix\n$\\mathbf{W} \\in \\mathbb{R}^{d \\times d}$, and our main theorem shows that we can recover\n$\\mathbf{Z}$ (up to orthogonal rotation) by recovering $\\bZ^*$.\n\n\n\n\\section{Main Results}\\label{sec:main_results}\n\nIn this section, we give theoretical results on the\nconsistency and asymptotic distribution of the estimated latent positions based on the omnibus matrix $\\mathbf{M}$.\nIn the next section,\nwe demonstrate from simulations that the omnibus embedding can be successfully leveraged for subsequent inference, specifically two-sample testing.\n\nLemma~\\ref{lem:omni2toinf} shows that the omnibus embedding\nprovides uniformly consistent estimates of the true latent positions,\nup to an orthogonal transformation,\nroughly analogous to Lemma 5 in \\cite{lyzinski13:_perfec}.\nLemma~\\ref{lem:omni2toinf}\nshows consistency of the omnibus embedding under the $2 \\rightarrow \\infty$ norm,\nimplying that all $mn$ of the estimated latent positions \nare near their corresponding true positions.\nWe recall that the orthogonal transformation $\\tilde{\\bW}$\nin the statement of the lemma is necessary since,\nas discussed in Remark~\\ref{rem:nonid},\n$\\mathbf{P} = \\mathbf{X} \\mathbf{X}^T = (\\mathbf{X} \\mathbf{W})(\\mathbf{X} \\mathbf{W})^T$\nfor any orthogonal $\\mathbf{W} \\in \\mathbb{R}^{d \\times d}$.\n\\begin{lemma}\\label{lem:omni2toinf}\n\tWith $\\tilde{\\bP}$, $\\mathbf{M}$, $\\bU_{\\bM}$, and $\\bU_{\\Ptilde}$ defined as above, there exists\n\tan orthogonal matrix $\\tilde{\\bW} \\in \\mathbb{R}^{d \\times d}$ such that\n\twith high probability,\n\t\\begin{equation}\\label{eq:omni2toinf_actualbound}\n\t\\|\\bU_{\\bM} \\bS_{\\bM}^{1\/2}-\\bU_{\\Ptilde} \\bS_{\\Ptilde}^{1\/2} \\tilde{\\bW} \\|_{2 \\rightarrow \\infty}\n\t\\le \\frac{Cm^{1\/2} \\log mn }{\\sqrt{n}} .\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThis result is proved in the supplemental material.\n\\end{proof}\n\nAs noted earlier, our central limit theorem for the omnibus embedding is analogous to a similar result proved in \\cite{athreya2013limit}, but with the crucial difference that we no longer require that the second moment matrix have distinct eigenvalues. As in \\cite{athreya2013limit}, our proof here depends on writing the difference between a row of the omnibus embedding and its corresponding latent position as a pair of summands: the first, to which a classical Central Limit Theorem can be applied, and the second, essentially a combination of residual terms, which converges to zero. The weakening of the assumption of distinct eigenvalues necessitates significant changes in how to bound the residual terms. In fact, \\cite{athreya2013limit} adapts a result of \\cite{bickel_sarkar_2013}---the latter of which depends on the assumption of distinct eigenvalues---to control these terms. Here, we resort to somewhat different methodology: we prove instead that analogous bounds to those in \\cite{lyzinski15_HSBM,tang_lse} hold for the estimated latent positions\nbased on the omnibus matrix $\\mathbf{M}$, and this enables us to establish that here, too, the rows of the omnibus embedding are also approximately normally distributed. Further, en route to this limiting result, we compute the explicit variance of the omnibus matrix, and show that as $m$, the number of graphs embedded, increases, this contributes to a reduction in the variance of the estimated latent positions.\n\n\\begin{theorem} \\label{thm:main}\nLet $(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},\\mathbf{X}) \\sim \\operatorname{JRDPG}(F,n,m)$ for some\n$d$-dimensional inner product distribution $F$ and let $\\mathbf{M}$ denote\nthe omnibus matrix as in \\eqref{eq:omnidef}. Let\n$\\mathbf{Z} = \\bZ^* \\mathbf{W}$ with $\\bZ^*$ as defined in Equation~\\eqref{eq:Zstruct},\nwith estimate\n$\\hat{\\bZ} = \\operatorname{OMNI}(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},d)$.\nLet $h = m(s-1) + i$ for $i \\in [n],s \\in [m]$, so that $\\hat{\\bZ}_h$\ndenotes the estimated latent position\nof the $i$-th vertex in the $s$-th graph $\\mathbf{A}^{(s)}$.\nThat is, $\\hat{\\bZ}_h$ is the\ncolumn vector formed by transposing the $h$-th row of the matrix\n$\\hat{\\bZ} = \\bU_{\\bM} \\bS_{\\bM}^{1\/2} = \\operatorname{OMNI}(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},d)$.\nLet $\\Phi(\\mathbf{x},\\mathbf{\\Sigma})$ denote the cdf of a (multivariate)\nGaussian with mean zero and covariance matrix $\\mathbf{\\Sigma}$,\nevaluated at $\\mathbf{x} \\in \\mathbb{R}^d$.\nThere exists a sequence of orthogonal $d$-by-$d$ matrices\n$( \\Wtilde_n )_{n=1}^\\infty$ such that for all $\\mathbf{x} \\in \\mathbb{R}^d$,\n$$ \\lim_{n \\rightarrow \\infty}\n \\Pr\\left[ n^{1\/2} \\left( \\hat{\\bZ} \\Wtilde_n - \\mathbf{Z} \\right)_h\n \\le \\mathbf{x} \\right]\n= \\int_{\\operatorname{supp} F} \\Phi\\left(\\mathbf{x}, \\mathbf{\\Sigma}(\\mathbf{y}) \\right) dF(\\mathbf{y}), $$\nwhere\n$\\mathbf{\\Sigma}(\\mathbf{y}) = (m+3)\\mathbf{\\Delta}^{-1} \\tilde{\\bSigma}(\\mathbf{y}) \\mathbf{\\Delta}^{-1}\/(4m), $\n$\\mathbf{\\Delta}$ is as defined in \\eqref{eq:def:Delta} and\n$$\\tilde{\\bSigma}(\\mathbf{y})\n= \\mathbb{E}\\left[ (\\mathbf{y}^T \\mathbf{X}_1 - ( \\mathbf{y}^T \\mathbf{X}_1)^2 ) \\mathbf{X}_1 \\mathbf{X}_1^T \\right].$$\n\\end{theorem}\n\\begin{proof}\nThis result is proved in the supplemental material.\n\\end{proof}\n\n\\section{Experimental results}\n\\label{sec:expts}\n\nIn this section, we present experiments on synthetic data\nexploring the efficacy of the omnibus embedding described above.\nWe consider both estimation of latent positions and two-sample graph testing.\n\\subsection{Recovery of Latent Positions}\nPerhaps the most ubiquitous estimation problem for RDPG data is that of\nestimating the latent positions (i.e., the rows of the matrix $\\mathbf{X}$); consequently, we begin by exploring how well the omnibus embedding recovers the latent\npositions of a given random dot product graph.\nIf one wishes merely to estimate the latent positions $\\mathbf{X}$\nof a set of $m$ graphs\n$(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},\\mathbf{X}) \\sim \\operatorname{JRDPG}(F,n,m)$,\nthe estimate $\\bar{\\bX} = \\operatorname{ASE}( \\sum_{i=1}^m \\mathbf{A}^{(i)}\/m, d )$,\nthe embedding of the sample mean of the adjacency matrices\nperforms well asymptotically \\citep{runze_law_large_graphs}.\nIndeed, all else equal,\nthe embedding $\\bar{\\bX}$ is preferable to the omnibus embedding\nif only because it requires an eigendecomposition\nof an $n$-by-$n$ matrix rather\nthan the much larger $mn$-by-$mn$ omnibus matrix.\n\n\\begin{figure}[t!]\n \\centering\n \n \\includegraphics[width=0.6\\columnwidth]{expts\/fullARE2v1\/sqerr_by_vxs.pdf}\n \\caption{Mean squared error (MSE) in recovery of latent positions (up to rotation) in a 2-graph joint RDPG model as a function of the number of vertices. The figure shows the performance of ASE applied to a single graph (red), ASE embedding of the mean graph (gold), the Procrustes-based pairwise embedding (blue), the omnibus embedding (green) and the mean omnibus embedding (purple). Each point is the mean of 50 trials, with error bars indicating two times the standard error. We see that the mean omnibus embedding (OMNIbar) achieves performance competitive with that of the optimal embedding $\\operatorname{ASE}(\\bar{\\bA},d)$, while the Procrustes alignment estimation is notably inferior to the other two-graph techniques for graphs of size between 80 and 200 vertices (and we note that the gap appears to persist at larger graph sizes, though it shrinks).}\n \\label{fig:compareMSE}\n\\end{figure}\n\nOf course, the omnibus embedding can still be used to\nto estimate the latent positions, potentially at the cost of\nincreased variance.\nFigure \\ref{fig:compareMSE} compares the mean-squared error of various\ntechniques for estimating the latent positions for a random dot product graph.\nThe figure plots the (empirical) mean squared error in recovering the\nlatent positions of a $3$-dimensional\nJRDPG as a function of the number of vertices $n$.\nEach point in the plot is the empirical mean of 50 independent trials.\nIn each trial, the vertex latent positions are drawn i.i.d. from a\nDirichlet with parameter $[1,\\,1,\\,1]^T \\in \\mathbb{R}^{3}$.\nHaving generated a random set of latent positions, we generate two graphs,\n$\\mathbf{A}^{(1)},\\mathbf{A}^{(2)} \\in \\mathbb{R}^{n \\times n}$ independently,\nbased on this set of latent positions.\nThus, we have $(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\mathbf{X}) \\sim \\operatorname{JRDPG}(F,n,2)$,\nwhere $F = \\operatorname{Dir}([1,\\,1,\\,1]^T)$ is a Dirichlet with parameter\n$[1,\\,1,\\,1]^T \\in \\mathbb{R}^3$, and $n$ varies.\nThe lines correspond to\n\\begin{enumerate}\n\\item {\\bf ASE1 (red)}: we embed only one of the two observed graphs,\n and use only the ASE of that graph to estimate the latent positions\n in $\\mathbf{X}$. That is, we consider $\\operatorname{ASE}(\\mathbf{A}^{(1)})$ as our estimate\n of $\\mathbf{X}$, ignoring entirely the information present in\n $\\mathbf{A}^{(2)}$. This condition serves as a baseline for how much\n additional information is provided by the second graph $\\mathbf{A}^{(2)}$.\n\\item {\\bf Abar (gold)}: we embed the average of the two graphs,\n $\\bar{\\bA} = (\\mathbf{A}^{(1)} + \\mathbf{A}^{(2)})\/2$ as $\\hat{\\bX} = \\operatorname{ASE}( \\bar{\\bA}, 3 )$.\n As discussed in, for example, \\cite{runze_law_large_graphs},\n this is the lowest-variance estimate of the latent positions $\\mathbf{X}$.\n\\item {\\bf OMNI (green)}: We apply the omnibus embedding to obtain\n $\\hat{\\bZ} = \\operatorname{ASE}(\\mathbf{M},3)$,\n where $\\mathbf{M}$ is as in Equation~\\eqref{eq:omnidef}.\n We then use only the first $n$ rows of\n $\\hat{\\bZ} \\in \\mathbb{R}^{2n \\times d}$ as our estimate of $\\mathbf{X}$.\n Thus, this embedding takes advantage of the information available\n in both graphs $\\mathbf{A}^{(1)}$ and $\\mathbf{A}^{(2)}$, but does not\n use both graphs equally, since the first rows of $\\hat{\\bZ}$ are based\n primarily on the information contained in $\\mathbf{A}^{(1)}$.\n\\item {\\bf OMNIbar (purple)}: We again apply the omnibus embedding to obtain\n estimated latent positions\n $\\hat{\\bZ} = \\operatorname{ASE}(\\mathbf{M},3)$, but this time we use all available\n information by averaging the first $n$ rows and the second $n$ rows\n of $\\hat{\\bZ}$.\n\\item {\\bf PROCbar (blue)}: We separately embed the graphs\n $\\mathbf{A}^{(1)}$ and $\\mathbf{A}^{(2)}$, obtaining two separate estimates of the\n latent positions in $\\mathbb{R}^3$.\n We then align these two sets of estimated latent positions\n via Procrustes alignment, and average the aligned embeddings to obtain\n our final estimate of the latent positions.\n\\end{enumerate}\nFirst, let us note that ASE applied to a single graph (red)\nlags all other methods.\nThis is expected, since all other methods assessed in\nFigure~\\ref{fig:compareMSE} use information from both observed graphs\n$\\mathbf{A}^{(1)}$ and $\\mathbf{A}^{(2)}$ rather than only $\\mathbf{A}^{(1)}$.\nWe see that all other methods perform essentially equally well\non graphs of 50 vertices or fewer.\nGiven the dearth of signal in these smaller graphs,\nwe do not expect any method to\nrecover the latent positions accurately.\n\nCrucially, however, we see that the OMNIbar estimate (purple) performs nearly\nidentically to the Abar estimate (gold), the natural choice among spectral methods for the estimation latent positions\n\\citep[for more on the efficiency of Abar, see][]{runze_law_large_graphs}.\nThe Procrustes estimate (in blue)\nprovides a two-graph analogue of ASE (red):\nit combines two ASE estimates via Procrustes alignment,\nbut does not enforce an {\\em a priori} alignment of the estimated latent positions\nin the manner of the omnibus embedding does (we discuss this enforced alignment in \\ref{sec:conc} as well.)\nAs predicted by the results in \\cite{lyzinski13:_perfec} and \\cite{tang14:_semipar},\nthe Procrustes estimate is competitive with the Abar (gold)\nestimate for suitably large graphs.\nThe OMNI estimate (in green) serves, in a sense, as an in-between\nmethod, in that it uses information available from both graphs,\nbut in contrast to Procrustes (blue), OMNIbar (purple)\nand Abar (gold), it does not make complete use of the information\navailable in the second graph.\nFor this reason, it is noteworthy that the OMNI estimate\noutperforms the Procrustes estimate for graphs of 80-100 vertices.\nThat is, for certain graph sizes,\nthe omnibus estimate appears to more optimally leverage the information in both graphs\nthan the Procrustes estimate does,\ndespite the fact that the information in the second graph has comparatively little\ninfluence on the OMNI embedding.\n\n\\subsection{Two-graph Hypothesis Testing}\n\nWe now turn to the matter of using the omnibus embedding for testing\nthe semiparametric hypothesis that two observed graphs are drawn from the\nsame underlying latent positions.\nSuppose we have a set of points\n$\\mathbf{X}_1,\\mathbf{X}_2,\\dots,\\mathbf{X}_n,\\mathbf{Y}_1,\\mathbf{Y}_2,\\dots,\\mathbf{Y}_n \\in \\mathbb{R}^d$.\nLet the graph $G_1$ with adjacency matrix $\\mathbf{A}^{(1)}$ have edges distributed\nindependently as\n$ \\mathbf{A}^{(1)}_{ij} \\sim \\operatorname{Bernoulli}( \\mathbf{X}_i^T \\mathbf{X}_j )$.\nSimilarly, let $G_2$ have adjacency matrix $\\mathbf{A}^{(2)}$ with edges\ndistributed independently as\n$ \\mathbf{A}^{(2)}_{ij} \\sim \\operatorname{Bernoulli}( \\mathbf{Y}_i^T \\mathbf{Y}_j )$.\nAs discussed previously,\nwhile $\\bar{\\bA} = (\\mathbf{A}^{(1)}+\\mathbf{A}^{(2)})\/2$ may be optimal for estimation of latent\npositions, it is not clear how to use the embedding $\\operatorname{ASE}(\\bar{\\bA},d)$ to\ntest the following hypothesis:\n\\begin{equation} \\label{eq:H0}\n H_0 : \\mathbf{X}_i = \\mathbf{Y}_i \\enspace \\forall i \\,\\in [n].\n\\end{equation}\nOn the other hand,\nthe omnibus embedding provides a natural test of\nthe null hypothesis \\eqref{eq:H0}\nby comparing the first $n$ and last $n$ embeddings of the omnibus matrix\n$$ \\mathbf{M} = \\begin{bmatrix} \\mathbf{A}^{(1)} & (\\mathbf{A}^{(1)} + \\mathbf{A}^{(2)})\/2 \\\\\n (\\mathbf{A}^{(1)} + \\mathbf{A}^{(2)})\/2 & \\mathbf{A}^{(2)}\n \\end{bmatrix}. $$\nIntuitively, when $H_0$ holds,\nthe distributional result in Theorem~\\ref{thm:main} holds,\nand the $i$-th and $(n+i)$-th rows of $\\operatorname{OMNI}(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},d)$\nare equidistributed (though they are not independent).\nOn the other hand, when $H_0$ fails to hold, there exists at least one\n$i \\in [n]$ for which the $i$-th and $(n+i)$-th rows of $\\mathbf{M}$ are \\emph{not}\nidentically distributed, and thus the corresponding embeddings are\nalso distributionally distinct.\nThis suggests a test that compares the first $n$ rows of\n$\\operatorname{OMNI}(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},d)$\nagainst the last $n$ rows (see below for details).\nHere, we empirically explore the power this test against its\nProcrustes-based alternative from \\cite{tang14:_semipar}.\n\nOur setup is as follows.\nWe draw $\\mathbf{X}_1,\\mathbf{X}_2,\\dots,\\mathbf{X}_n \\in \\mathbb{R}^3$ i.i.d. according to a\nDirichlet distribution $F$ with parameter\n$\\vec{\\balpha} = [1, 1, 1]^T$.\nAssembling these $n$ points into a matrix\n$ \\mathbf{X} = [\\mathbf{X}_1 \\mathbf{X}_2 \\dots \\mathbf{X}_n]^T \\in \\mathbb{R}^{n \\times 3}, $\nwe can generate a graph $G_1$ with adjacency matrix $\\mathbf{A}^{(1)}$\nwith entries\n$ \\mathbf{A}^{(1)}_{ij} \\sim \\operatorname{Bernoulli}( (\\mathbf{X} \\mathbf{X}^T)_{ij} )$.\nThus, $(\\mathbf{A}^{(1)},\\mathbf{X}) \\sim \\operatorname{RDPG}(F,n)$.\nWe generate a second graph $G_2$ by first\ndrawing random points $\\mathbf{Z}_1,\\mathbf{Z}_2,\\dots,\\mathbf{Z}_n \\stackrel{\\text{i.i.d.}}{\\sim} F$.\nSelecting a set of indices $I \\subset [n]$ of size $k < n$ uniformly at\nrandom from among all such $\\binom{n}{k}$ sets,\nwe let $G_2$ have latent positions\n$$ \\mathbf{Y}_i = \\begin{cases} \\mathbf{Z}_i & \\mbox{ if } i \\in I \\\\\n \\mathbf{X}_i & \\mbox{ otherwise. } \\end{cases} $$\nAssembling these points into a matrix\n$\\mathbf{Y} = [\\mathbf{Y}_1, \\mathbf{Y}_2, \\dots, \\mathbf{Y}_n]^T \\in \\mathbb{R}^{n \\times 3}, $\nwe generate graph $G_2$ with adjacency matrix $\\mathbf{A}^{(2)}$\nwith edges generated independently according to\n$ \\mathbf{A}^{(2)}_{ij} \\sim \\operatorname{Bernoulli}( (\\mathbf{Y} \\mathbf{Y}^T)_{ij} ).$\nThe task is then to test the hypothesis\n\\begin{equation} \\label{eq:ptsnull}\n H_0 : \\mathbf{X} = \\mathbf{Y}.\n\\end{equation}\nTo test this hypothesis, we consider two different tests, one based on\na Procrustes alignment of the adjacency spectral embeddings of $G_1$ and $G_2$\n\\citep{tang14:_semipar}\nand the other based on the omnibus embedding.\nBoth approaches are based on estimates of the latent positions\nof the two graphs.\nIn both cases we use a test statistic of the form\n$ T = \\sum_{i=1}^n \\| \\hat{\\bX}_i - \\hat{\\bY}_i \\|_F^2, $\nand accept or reject based on a Monte Carlo estimate of the\ncritical value of $T$ under the null hypothesis,\nin which $\\mathbf{X}_i = \\mathbf{Y}_i$ for all $i \\in [n]$.\nIn each trial, we use $500$ Monte Carlo iterates to estimate the\ndistribution of $T$.\n\nWe note that in the experiments presented here,\nwe assume that the latent positions\n$\\mathbf{X}_1,\\mathbf{X}_2,\\dots,\\mathbf{X}_n$ of graph $G_1$ are known for sampling purposes,\nso that the matrix $\\mathbf{P} = \\mathbb{E} \\mathbf{A}^{(1)}$ is known exactly, rather than\nestimated from the observed adjacency matrix $\\mathbf{A}^{(1)}$.\nThis allows us to sample from the true null distribution.\nAs proved in \\cite{lyzinski13:_perfec},\nthe estimated latent positions $\\hat{\\bX}_1 = \\operatorname{ASE}(\\mathbf{A}^{(1)})$\nand $\\hat{\\bX}_2 = \\operatorname{ASE}( \\mathbf{A}^{(2)} )$ recover the true latent positions\n$\\mathbf{X}_1$ and $\\mathbf{X}_2$ (up to rotation) to arbitrary accuracy\nin $(2,\\infty)$-norm for suitably large $n$~\\citep{lyzinski13:_perfec}.\nWithout using this known matrix $\\mathbf{P}$, we would require that our matrices\nhave tens of thousands of vertices before the variance associated with\nestimating the latent positions would no longer overwhelm the signal present\nin the few altered latent positions.\n\nThree major factors influence the complexity of testing\nthe null hypothesis in Equation \\eqref{eq:ptsnull}:\nthe number of vertices $n$,\nthe number of changed latent positions $k = |I|$,\nand the distances $\\|\\mathbf{X}_i - \\mathbf{Y}_i\\|_F$ between the latent positions.\nThe three plots in Figure \\ref{fig:trueP:power} illustrate\nthe first two of these three factors.\nThese three plots show the power of two different approaches to testing\nthe null hypothesis \\eqref{eq:ptsnull} for different sized graphs\nand for different values of $k$, the number of altered latent positions.\nIn all three conditions, both methods improve as the number of vertices\nincreases, as expected, especially since we do not require\nestimation of the underlying expected matrix $\\mathbf{P}$ for Monte Carlo\nestimation of the null distribution of the test statistic.\nWe see that when only one vertex is changed, neither method has power much above $0.25$.\nHowever, in the case of $k = 5$ and $k = 10$, is it clear that the\nomnibus-based test achieves higher power than the Procrustes-based\ntest, especially in the range of 30 to 250 vertices.\n\n\\begin{figure}[t!]\n \\centering\n \n \n \n \\subfloat[]{ \\includegraphics[width=0.5\\columnwidth]{expts\/truePonediff\/onediff_power_by_vxs.pdf} }\n \\subfloat[]{ \\includegraphics[width=0.5\\columnwidth]{expts\/truePfivediff\/fivediff_power_by_vxs.pdf} }\\\\\n \\subfloat[]{ \\includegraphics[width=0.5\\columnwidth]{expts\/truePtendiff\/tendiff_power_by_vxs.pdf} }\n \\caption{Power of the ASE-based (blue) and omnibus-based (green)\n tests to detect when the two graphs being testing differ in\n (a) one, (b) five, and (c) ten of their latent positions.\n Each point is the proportion of 1000 trials for which the given\n technique correctly rejected the null hypothesis,\n and error bars denote two standard errors of this empirical mean\n in either direction. }\n \\label{fig:trueP:power}\n\\end{figure}\n\nFigure~\\ref{fig:driftpow} shows the effect of the difference between the latent position matrices under null and alternative.\nWe consider a $3$-dimensional RDPG on $n$ vertices,\nin which one latent position, $i \\in [n]$,\nis fixed to be equal to $\\mathbf{x}_i = (0.8, 0.1, 0.1)^T$\nand the remaining latent positions are drawn i.i.d. from a Dirichlet\nwith parameter $\\vec{\\balpha} = (1,1,1)^T$.\nWe collect these latent positions in the rows\nof the matrix $\\mathbf{X} \\in \\mathbb{R}^{n \\times 3}$.\nTo produce the latent positions $\\mathbf{Y} \\in \\mathbb{R}^{n \\times 3}$ of the second graph,\nwe use the same latent positions in $\\mathbf{X}$, but we alter the $i$-th position\nto be $\\mathbf{Y}_i = (1-\\lambda)\\mathbf{x}_i + \\lambda (0.1,0.1,0.8)^T$ for\n$\\lambda \\in [0,1]$ a ``drift'' parameter, controlling how much the\nlatent position changes between the two graphs.\nIntuitively, correctly rejecting $H_0 : \\mathbf{X} = \\mathbf{Y}$ is easier\nfor larger values of $\\lambda$; the greater the gap between latent position matrices under null and alternative, the more easily our test procedure should discriminate between them.\nFigure~\\ref{fig:driftpow} shows how the size of the drift parameter\ninfluences the power.\nWe see that for $n=30$ vertices (top left), neither the omnibus\nnor Procrustes test has power appreciably better than\napproximately $0.05$, largely in agreement with the\nwhat we observed in Figure~\\ref{fig:trueP:power}.\nSimilarly, when $n=200$ vertices (bottom right),\nboth methods perform approximately equally\n(though omnibus does appear to consistently outperform Procrustes testing).\nThe case of $n=50$ and $n=100$\nvertices (upper right and bottom left, respectively), though, offers a fascinating instance in which the omnibus test\nconsistently outperforms the Procrustes test.\nParticularly interesting to note is the $n=50$ case (top right),\nin which we see that performance of the Procrustes test is more or less\nflat as a function of drift parameter $\\lambda$, while the omnibus\nembedding clearly improves as $\\lambda$ increases, with performance\nclimbing well above that of Procrustes for $\\lambda > 0.8$.\n\n\\begin{figure}[t!]\n \\centering\n \n \n \n \n \\subfloat[]{ \\includegraphics[width=0.5\\columnwidth]{expts\/trueP_1drift\/pow_by_lambda_nvx30.pdf} }\n \\subfloat[]{ \\includegraphics[width=0.5\\columnwidth]{expts\/trueP_1drift\/pow_by_lambda_nvx50.pdf} } \\\\\n \\subfloat[]{ \\includegraphics[width=0.5\\columnwidth]{expts\/trueP_1drift\/pow_by_lambda_nvx100.pdf} }\n \\subfloat[]{ \\includegraphics[width=0.5\\columnwidth]{expts\/trueP_1drift\/pow_by_lambda_nvx200.pdf} }\n \\caption{Power of the ASE-based (blue) and omnibus-based (green)\n tests to detect when the two graphs being testing differ in\n their latent positions.\n Subplots show power as a function of the drift parameter\n $\\lambda$ for (a) $n=30$, (b) $n=50$, (c) $n=100$ and (d) $n=200$\n vertices.\n Each point is the proportion of 500 trials for which the given\n technique correctly rejected the null hypothesis,\n and error bars denote two standard errors of this empirical mean.}\n \\label{fig:driftpow}\n\\end{figure}\n\n\\section{Discussion and Conclusion}\\label{sec:conc}\n\nThe omnibus embedding is a simple, scalable procedure for the simultaneous embedding of multiple graphs on the same vertex set, the output of which are multiple points in Euclidean space for each graph vertex. For a wide class of latent position random graphs, this embedding generates accurate estimates of latent positions and supplies empirical power for distinguishing when graphs are statistically different. Our consistency results in the $2 \\to \\infty$ norm for the omnibus-derived estimates are competitive with state-of-the-art spectral approaches to latent position estimation, and our distributional results for the asymptotic normality of the rows of the omnibus embedding render principled the application of classical Euclidean inference techniques, such as analyses of variance, for the comparison of multiple population of graphs and the identification of drivers of graph similarity or difference at multiple scales, from whole graphs to subcommunities to vertices. We illustrate the utility of the omnibus embedding in data analyses of two different collections of noisy, weighted brain scans, and we uncover new insights into brain regions and vertices that are responsible for graph-level differences in two distinct data sets. \n\nFurther, we quantify the impact of multiple graphs on the variance of the rows of the embedding, specifically in relation to the variance given in \\cite{athreya2013limit}.\nThis result shows that as the number of graphs, $m$, grows, a significant reduction in the variance is achievable.\nExperimental data suggest that the omnibus embedding is\ncompetitive with state-of-the-art, multiple-graph spectral estimation of latent positions, and we surmise that the variance of the rows in the omnibus embedding is close to optimal for latent position estimators derived from the adjacency spectral embedding.\nThat is, the variance of the omnibus embedding is asymptotically equal to the variance obtained by first averaging the $m$ graphs to get $\\bar{\\bA}$\n(which corresponds, in essence, to the maximum likelihood estimate for $\\mathbf{P}$),\nand then performing an adjacency spectral embedding of $\\bar{\\bA}$.\nLet $\\hat{\\bZ}_i$ correspond to the $i$-th row of\n$\\hat{\\bZ} = \\operatorname{OMNI}(\\mathbf{A}^{(1)},\\mathbf{A}^{(2)},\\dots,\\mathbf{A}^{(m)},d)$, and let\n$\\bar{\\bX}_i$ denote the $i$-th row of the adjacency spectral embedding of $\\bar{\\bA}$.\nLet $\\bar{\\hat{\\bZ}}_i$ denote the average value of the $m$ rows of $\\hat{\\bZ}$\ncorresponding to the $i$-th vertex,\nnamely, the average of the $m$ vectors corresponding to the $i$-th vertex in the omnibus embedding.\nWe conjecture that averaging the rows of the omnibus embedding accounts for all of the reduction in variance when one compares a single row of the omnibus embedding and a single row of $\\bar{\\bX}$.\nFigure~\\ref{fig:compareMSE} provides weak evidence in favor of this\nconjecture, since it illustrates that the MSE of both the omnibus-\nand Procrustes-based estimates of the latent position estimates\nare very close to that of the estimate based on the mean adjacency matrix.\n\\begin{conjecture}\\emph{(Decomposition of Variance)}\nWith notation as above, for large $n$,\n$$ \\operatorname{Var}\\left( \\sqrt{n}\\bar{\\bX}_i \\right)\n\\approx \\operatorname{Var} \\left( \\sqrt{n}\\bar{\\hat{\\bZ}}_i \\right)\n<\\operatorname{Var}\\left( \\sqrt{n}\\hat{\\bZ}_i \\right). $$\n\\end{conjecture}\n\nWe have also demonstrated that the omnibus embedding can be profitably deployed\nfor two-sample semiparametric hypothesis testing of graph-valued data.\nOur omnibus embedding provides a natural mechanism for the\nsimultaneous embedding of multiple graphs into a single vector space.\nThis eliminates the need for multiple Procrustes alignments,\nwhich were required in previously-explored\napproaches to multiple-graph testing \\citep{tang14:_semipar}.\nIn the two-graph hypothesis testing framework of \\cite{tang14:_semipar},\neach graph is embedded separately.\nUnder the assumption of equality of latent positions\n(i.e., under $H_0$ in Equation ~\\eqref{eq:H0}),\nwe note that embedding the first graph estimates the true latent positions\n$\\mathbf{X}$ up to a unitary transformation in $\\mathbb{R}^{d \\times d}$.\nCall this estimate $\\hat{\\bX}_1$.\nSimilarly, $\\hat{\\bX}_2$, the estimates based on the second graph,\nestimates $\\mathbf{X}$ only up to some {\\em potentially different} unitary rotation,\ni.e., $\\hat{\\bX}_2 \\approx \\mathbf{X} \\mathbf{W}^*$ for some unitary $\\mathbf{W}^*$.\nProcrustes alignment is thus required to discover the rotation aligning $\\hat{\\bX}_1$ with $\\hat{\\bX}_2$.\nIn \\cite{tang14:_semipar}, it was shown that this Procrustes alignment,\ngiven by\n\\begin{equation} \\label{eq:procmin}\n \\min_{\\mathbf{W} \\in \\mathcal{O}_d} \\| \\hat{\\bX}_1 - \\hat{\\bX}_2 \\mathbf{W} \\|_F,\n\\end{equation}\nconverges under the null hypothesis.\nThe effect of this Procrustes alignment on subsequent inference\nis ill-understood. At the very least, it has the potential to\nintroduce variance, and our simulations in Section \\ref{sec:expts}\nsuggest that it negatively impacts performance in both estimation\nand testing settings.\nFurthermore, when the matrix $\\mathbf{P} = \\mathbf{X} \\mathbf{X}^T$\ndoes not have distinct eigenvalues\n(i.e., is not uniquely diagonalizable), this Procrustes step is unavoidable,\nsince the difference $\\|\\hat{\\bX}_1 - \\hat{\\bX}_2\\|_F$ need not converge at all.\n\nIn contrast, our omnibus embedding builds an alignment of the graphs\ninto its very structure. To see this, consider, for simplicity, the $m=2$ case.\nLet $\\mathbf{X} \\in \\mathbb{R}^{n \\times d}$ be the matrix whose rows are the latent positions\nof both graphs $G_1$ and $G_2$, and let $\\mathbf{M} \\in \\mathbb{R}^{2n \\times 2n}$ be their\nomnibus matrix.\nThen\n\\begin{equation*}\n\\mathbb{E} \\mathbf{M} = \\tilde{\\bP} = \\begin{bmatrix} \\mathbf{P} & \\mathbf{P} \\\\ \\mathbf{P} & \\mathbf{P} \\end{bmatrix}\n = \\begin{bmatrix} \\mathbf{X} \\\\ \\mathbf{X} \\end{bmatrix}\n \\begin{bmatrix} \\mathbf{X} \\\\ \\mathbf{X} \\end{bmatrix}^T.\n\\end{equation*}\nSuppose now that we wish to factorize $\\tilde{\\bP}$ as\n$$ \\tilde{\\bP} = \\begin{bmatrix} \\mathbf{X} \\\\ \\mathbf{X} \\mathbf{W}^* \\end{bmatrix}\n \\begin{bmatrix} \\mathbf{X} \\\\ \\mathbf{X} \\mathbf{W}^* \\end{bmatrix}^T\n = \\begin{bmatrix} \\mathbf{P} & \\mathbf{X} (\\mathbf{W}^*)^T \\mathbf{X}^T \\\\\n \\mathbf{X} \\mathbf{W}^* \\mathbf{X}^T & \\mathbf{P} \\end{bmatrix}. $$\nThat is, we want to consider graphs $G_1$ and $G_2$ as being generated from\nthe same latent positions,\nbut in one case, say, under a \\emph{different} rotation.\nThis possibility necessitates the Procrustes alignment in the case of\nseparately-embedded graphs.\nIn the case of the omnibus matrix,\nthe structure of the $\\tilde{\\bP}$ matrix implies that $\\mathbf{W}^*=\\mathbf{I}_d$.\nThus, in contrast to the Procrustes alignment,\nthe omnibus matrix incorporates an alignment {\\em a priori}.\nSimulations show that the omnibus embedding\noutperforms the Procrustes-based test for equality of latent positions,\nespecially in the case of moderately-sized graphs.\n\nTo further illustrate the utility of this omnibus embedding, consider the case of testing whether three different random dot product graphs have the same generating latent positions. The omnibus embedding gives us a {\\em single} canonical representation of all three graphs: Let $\\hat{\\bX}^O_1$, $\\hat{\\bX}^O_2$, and $\\hat{\\bX}^O_3$ be the estimates for the three latent position matrices generated from the omnibus embedding.\nTo test whether any two of these random graphs have the same generating latent positions, we merely have to compare the Frobenius norms of their differences, as opposed to computing three separate Procrustes alignments.\nIn the latter case, in effect, we do not have a canonical choice of coordinates in which to compare our graphs simultaneously.\n\nIn our analysis of BNU1 data, we ``center\" our omnibus matrix, by first considering $\\mathbf{B}^{(i)}=\\mathbf{A}^{(i)}-\\bar{\\mathbf{A}}$ and then performing an omnibus embedding on the $\\mathbf{B}^{(i)}$ matrices. While our theorems are written for the uncentered case, the analysis of the centered version proceeds along similiar lines. We find that in many practical settings, centering meaningfully improves our ability to detect differences across graphs, and we offer the following conjectures as to why. First, we surmise that centering allows us to better assess covariance structure between estimated latent positions, and thereby improve clustering in a dissimilarity matrix. Second, centering can mitigate the effect of degree heterogeneity across graphs. Third, centering can dampen the potentially noisy impact of common subgraphs, if they exist, to more clearly address graph difference.\n\nInvestigating the impact of centering, both for theory and practice, is ongoing, and it is a prominent open problem in the analysis of the omnibus embedding. Of course, other open problems abound, such as an analysis of the omnibus embedding when the $m$ graphs are correlated, are weighted, or are corrupted by occlusion or noise; a closer examination of the impact of the Procrustes alignment on power; the development of an analogue to a Tukey test for determining which graphs differ when we test equality of multiple graphs; the comparative efficiency of the omnibus embedding relative to other spectral estimates; and finally, results for the omnibus embedding under the alternative, when the graph distributions are unequal. The elegance of the omnibus embedding, especially its anchoring in a long and robust history of spectral inference procedures, makes it an ideal point of departure for multiple graph inference, and the richness of the open problems it inspires suggests that the omnibus embedding will remain a key part of the graph statistician's arsenal.\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\nUnveiling the star formation histories of elliptical galaxies is key\nto our understanding of galaxy formation. Being able to resolve their\nseemingly homogenous distribution is hampered by the fact that their\nlight is dominated by old, i.e. low-mass stars, which do not evolve\nsignificantly even over cosmological times. Furthermore, the presence\nof small amounts of young stars as recently discovered in NUV studies\n\\citep{fs00,yi05,kav07} reveals a complex history of star formation\nthat requires proper estimates of mass-weighted ages, in contrast with\nthe luminosity-weighted ages that simple stellar populations (SSPs)\ncan only achieve. The majority of papers dealing with age estimates of\nthe stellar populations of elliptical galaxies rely on such SSPs\n\\citep[see e.g. ][]{kd98,sct00,tm05}, and it is only recently that special\nemphasis has been made on the need to go beyond simple populations\n\\citep{fyi04,serra07,idi07}\n\nDating the (old) stellar populations of elliptical galaxies has been\nfraught with difficulties, the most prominent being the\nage-metallicity degeneracy, whereby the photo-spectroscopic properties\nof a galaxy of a given age and metallicity can be replicated by a\nyounger or older galaxy at a suitably higher or lower metallicity,\nrespectively. To some extent, this problem has been overcome by the\nmeasurement of pairs of absorption line indices \\citep{wo94}, one\nwhose change in equivalent width (EW) is dominated by the average\nmetallicity of the population and the other dominated by the average\nage of the population. Typical metal-sensitive line strengths are the\nMg feature at 5170\\AA\\ , the iron lines around 5300\\AA , or a\ncombination such as [MgFe] \\citep{gon93}. Balmer lines are more\nage-sensitive and are often combined with metal-sensitive lines to break the\ndegeneracy. However, measurements of EWs of Balmer lines can be\naffected by the age-metallicity degeneracy because of the presence of\nnearby absorption lines. Such is the case of H$\\gamma$, with the\nprominent G band at 4300\\AA\\ or the CN bands in the vicinity of the\nH$\\delta$ line. This paper is partly motivated by the need to define a\nmethod to estimate EWs that minimise the sensitivity of metallicity on\nBalmer lines by a proper estimate of the continuum.\n\nEven at relatively younger ages (a few Gyr) where such uncertainties\nare reduced, considerable degeneracies still remain. The confirmation\nof recent star formation (RSF) occuring in early type galaxies\n\\citep{yi05,kav07} has raised the problem that the existence of a\nyoung population can considerably affect the parameters derived\nthrough SSP analysis. \\citet{sct00} and later \\citet{serra07} showed\nthat even relatively small mass fractions ($\\sim$1\\%) of young stars \ncan distort age and metallicity estimates and in moderate cases\n($\\sim$10\\%) completely overshadow the older population. In addition,\nthe age and mass fraction of any younger sub-population will also be\ndegenerate, with larger mass fraction of relatively older sub\npopulations having the same effect as smaller fractions of younger\nones.\n\n\\citet{schiavon04} noticed that using different Balmer lines\n(H$\\beta$, H$\\gamma$ and H$\\delta$) to estimate the age gives slightly\ndifferent results, which was suggested to show that the galaxy had\nundergone recent star formation. Contrary to this, \\citet{tm04} find\nthat H$\\gamma$ and H$\\delta$ equivalent widths are more affected by a\nnon-solar $\\alpha$\/Fe ratio on higher order Balmer lines. This effect\nis due to the increase of metal lines at bluer wavelengths, thereby\ndistorting the continuum as measured by a side-band method. This was\nexpanded by \\cite{serra07} who remodelled synthetic 2-burst models\nusing H$\\beta$ and H$\\gamma_A$ and achieved a result consistent with\nboth papers, concluding that a mistmatch between the three Balmer line\nestimates could possibly reveal underlying younger populations.\n\nMoving forward with such analysis -- beyond simple populations and\nluminosity-weighted parameters -- requires improvements on both the\nH$\\gamma$ and H$\\delta$ measurements. Balmer line equivalent widths\nsuffer from the effects of the metallicity due to the presence of such\nlines in the spectral region used to determine the continuum\n\\citep{wo97,tm04,pro07}. Although considerable work has already been\ndone in this area \\citep[see e.g. ][]{rose94,jw95,va99,yam06}, we test\na completely different approach.\n\nIn this paper we present a comprehensive analysis of the stellar\npopulations of 14 Virgo cluster ellipticals using several models to\ndescribe the star formation history, exploring the discrepancies found\nbetween simple and composite stellar populations. In an attempt to\ncombat the problems discussed above, we introduce a new method for the\nmeasurement of equivalent widths using a high percentile running\n``median'' to describe the continuum. The properties of this new method\nare exploited by using various age- and metallicity-sensitive spectral\nfeatures.\n\n\\begin{figure*}\n\\begin{minipage}{18cm}\n\\begin{center}\n\\includegraphics[width=3.3in]{BFPS_f1a.eps}\n\\includegraphics[width=3.3in]{BFPS_f1b.eps}\n\\caption{Dependence of the age-metallicity gradient -- ($\\Delta$\nAge\/Age)(\/$\\Delta$ Z\/Z) -- on the choice of BMC parameters, namely the\nlevel at which the 'boosted median' is taken (\\emph{left}) or the size\nof the kernel (\\emph{right}). On the left panel a number of kernel\nsizes is shown as labelled: $\\Delta\\lambda$=50\\AA\\ (dashed); 100\\AA\\\n(solid) and 200\\AA\\ (dotted). On the right panel a number of\nconfidence levels are considered: 50\\% (dotted); 70\\% (long dashed);\n90\\% (solid) and 95\\% (short dashed). The grey horizontal line in\nboth panels is the estimate from the standard side-band method (SB), and the\ngrey vertical shaded area marks our choice of 'boosted median' parameters.}\n\\label{fig:DADZ}\n\\end{center}\n\\end{minipage}\n\\end{figure*}\n\n\\section{The sample}\n\nWe use a sample of 14 elliptical galaxies in the Virgo cluster, for\nwhich moderate resolution spectroscopy is available at high\nsignal-to-noise ratio \\citep[S\/N\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 100\n\\AA$^{-1}$,][]{yam06,yam08}. Eight galaxies were observed with FOCAS\nat the 8m Subaru telescope; the other six were observed with ISIS at\nthe 4.2m William Herschel Telescope (WHT). Observations from Subaru\nspan the spectral range $\\lambda\\simeq 3800-5800$\\AA\\ , whereas the\nspectra taken at the WHT span a narrower window, namely $\\lambda\\simeq\n4000-5500$\\AA\\ . The resolution (FWHM) of both data sets is similar:\n2\\AA\\ (Subaru) and 2.4\\AA (WHT). \n We refer the interested reader to \\citet{yam06} for\ndetails about the data reduction process. We compare those spectra\nwith composites of the R$\\sim 2000$ synthetic models of\n\\citet{bc03}, updated to the 2007 version \\citep{cb07}. \nWe resampled the observed spectra from the original\n0.3\\AA\\ to 1\\AA\\ per pixel, performing an average of the spectra over\na 1.5\\AA\\ window, in order to have a sampling more consistent with the\nactual resolution.\n\nIn this paper we use two alternative sets of information, either the full\nspectral energy distribution or targeted absorption lines. For the\nformer, we consider a spectral window\naround the 4000\\AA\\ break, which is a strong age indicator (albeit\nwith a significant degeneracy in metallicity, especially for\nevolved stellar populations). In order to minimise the effect of\nan error in the flux calibration, we do not choose the full spectral\nrange of the spectra, restricting the analysis to 3800--4500\\AA\\ for the\nSubaru spectra, and 4000--4500\\AA\\ for the WHT spectra.\n\nThe second method focuses on a reduced number of spectral\nlines. Following the traditional approach \\citep[see\ne.g.][]{kd98,sct00,tm05,sbla06}, we use a set of age-sensitive and\nmetallicity-sensitive lines. In the next section we describe in detail\nthe indices targeted by our analysis and describe a new algorithm that\nimproves on the ``standard'' method to determine the continuum in\ngalaxy spectra.\n\n\n\n\n\n\\section{Measuring Equivalent Widths}\n\nWe focus on a reduced set of absorption lines originally defined in\nthe Lick\/IDS system \\citep{lick} and extensions thereof\n\\citep{wo97}. As age-sensitive lines we use the Balmer lines H$\\beta$,\nH$\\gamma$, H$\\delta$, the G-band (G4300) and the 4000\\AA\\ break\n(D4000). We use the standard definition of [MgFe] \\citep{gon93}, as a\nmetal-sensitive tracer, which is a reliable proxy of overall\nmetallicity, with a very mild dependence on [$\\alpha$\/Fe] abundance\nratio \\citep{tm03}.\n\nThe standard method to determine the equivalent widths of galaxy\nspectra relies on the definition of a blue and a red side-band to\ndetermine the continuum. A linear fit to the average flux in the blue \nand red side-bands is used to track the continuum in the line\n\\citep[see e.g.][]{sct00}. This method, although easy to implement,\nhas an important drawback as neighbouring lines can make a significant\ncontribution to the flux in the blue and red passbands, introducing\nunwanted age\/metallicity effects. For instance, the H$\\gamma$ index\n\\citep{wo97} is defined with the blue side-band located close to the\nprominent G-band, around 4300\\AA. This definition causes\nnon-physical negative values of the H$\\gamma$ line in \\emph{absorption}, \nas the depression caused by the G-band makes the flux in\nthe $H\\gamma$ line (wrongly) appear in \\emph{emission}. This has not\nprevented the community from using this line as a sensitive age tracer, as\nlong as models and data are treated in the same way. \\cite{va99}\ndefined new measurements of this line in order to reduce the\nmetallicity degeneracy mainly introduced by the choice of the side\nbands. They avoid this by selecting specific regions\nless affected by the metal absorption lines.\n\n\nH$\\delta$ is another Balmer line which has been recently considered to\nbe affected by neighbouring metal lines -- most notably the CN molecular bands --\nwhich reduce the age sensitivity of the index \\citep{pro07}.\n\nIn this paper we present an alternative method to determine the equivalent\nwidths. Our method does not rely on the definition of blue and red side-bands\nand minimizes the contamination from neighbouring lines. This \nmethod is simple to apply and we propose it for future studies of stellar\npopulations in galaxies\\footnote{A C-programme that computes EWs using our\nproposed BMC method from an ASCII version of an SED can be obtained from\nus (\\emph{ferreras@star.ucl.ac.uk}).}.\n\n\n\\subsection{The Boosted Median Continuum (BMC)}\n\nOur measure of equivalent width follows the standard procedure\ncomparing observed flux in the line and the corresponding\n``interpolated'' continuum in the same wavelength range. For an\nequivalent width measured in \\AA:\n\n\\begin{equation}\nEW = \\int_{\\lambda_1}^{\\lambda_2} \\Big[ 1-\\frac{\\Phi(\\lambda)}{\\Phi_C(\\lambda)}\\Big]\nd\\lambda,\n\\end{equation} \n\n\\noindent\nwhere $\\lambda_1$ and $\\lambda_2$ define the wavelength range of the\nline, $\\Phi(\\lambda)$ is the observed flux, and $\\Phi_C(\\lambda)$ is\nthe flux from the continuum. Rather than defining the continuum as a\nlinear fit between a blue and a red side-band, we propose the\n``boosted median'' of the flux, defined at each wavelength as the 90th\npercentile of the flux values within a 100\\AA\\ window.\n\nThis method is defined by two parameters, namely the choice of\npercentile (90\\% in our case) and the size of the kernel\n($\\Delta\\lambda=$100\\AA). The kernel size needs to be large enough to\navoid the small scale variations of the spectra, but small enough to\navoid distortions from large scale structure of the spectra such as\nthe breaks at 4000\\AA\\ and 4300\\AA, or flux calibration errors\n\\footnote{Applying a flux calibration distortion of ~5\\% --\nconsistent with that found in current surveys (SDSS) -- \ncauses negligible effects on the EW measured with the BMC method (below 1\\%).}.\nThe choice of percentile also suffers a similar balancing act, since\nit should be high enough to select the true continuum but at a value\nthat would avoid it becoming dominated by noise. In order to determine\nthe optimal choice, a range of values for these two parameters was\nstudied on a number of simple stellar populations taken from the\nmodels of \\citet{bc03} including the effect of velocity dispersion and\nnoise. Out of the simulations, we adopted the 90th percentile of the\nflux within a 100\\AA\\ window. However, this choice is not critical,\ngiven the robustness of the method in which the continuum is selected\n(i.e. a median). The advantage of averaging over a large enough\nwavelength range is that the effect of strong metallic lines in the\nvicinity of the index is limited. Furthermore, this pseudo-continuum\nis found to be less susceptible to noise (see below).\n\n\\begin{table*}\n\\caption{Equivalent Widths of Virgo Elliptical galaxies using a 90\\% \nBoosted Median Continuum (see text for details). \nAll values given in \\AA\\ and measured at the observed $\\sigma$, with the 1$\\sigma$\nuncertanities in brackets below each measurement.}\n\\label{tab:EWs}\n\\begin{tabular}{lrccccccccc}\n\\hline\\hline\nGalaxy & $\\sigma^1$ & H$\\beta_{20}$ & H$\\gamma_{20}$ & H$\\delta_{20}$ & \nMgb$_{20}$ & Fe5270$_{20}$ & Fe5335$_{20}$ & G4300$_{20}$ & D4000 & [MgFe]$_{20}$\\\\\n\\hline\nNGC 4239 & 82 & 2.993 & 2.097 & 2.356 & 2.972 & 2.864 & 1.673 & 6.130 & 1.441 & 2.596 \\\\\n & & (0.051) & (0.064) & (0.056) & (0.044) & (0.046) & (0.060) & (0.046) & (0.006) & (0.032) \\\\\nNGC 4339 & 142 & 2.666 & 1.488 & 1.442 & 3.807 & 3.163 & 1.772 & 6.642 & 1.560 & 3.065 \\\\\n & & (0.047) & (0.054) & (0.082) & (0.045) & (0.037) & (0.050) & (0.042) & (0.005) & (0.024) \\\\\nNGC 4365 & 245 & 2.191 & 0.732 & 1.195 & 3.819 & 2.971 & 1.615 & 6.289 & --- & 2.959 \\\\\n & & (0.026) & (0.042) & (0.070) & (0.023) & (0.028) & (0.031) & (0.021) & (---) & (0.015) \\\\\nNGC 4387 & 105 & 2.431 & 1.698 & 1.974 & 3.614 & 3.237 & 1.703 & 7.317 & --- & 2.988 \\\\\n & & (0.044) & (0.066) & (0.068) & (0.039) & (0.057) & (0.061) & (0.050) & (---) & (0.024) \\\\\nNGC 4458 & 104 & 2.464 & 1.622 & 1.676 & 3.520 & 2.719 & 1.670 & 6.570 & 1.504 & 2.779 \\\\\n & & (0.039) & (0.053) & (0.055) & (0.037) & (0.042) & (0.047) & (0.041) & (0.006) & (0.025) \\\\\nNGC 4464 & 135 & 2.302 & 1.547 & 1.523 & 3.648 & 2.913 & 1.521 & 6.813 & --- & 2.844 \\\\\n & & (0.038) & (0.053) & (0.070) & (0.035) & (0.034) & (0.036) & (0.037) & (---) & (0.023) \\\\\nNGC 4467 & 75 & 2.605 & 1.797 & 1.875 & 3.839 & 3.209 & 1.747 & 6.760 & 1.458 & 3.085 \\\\\n & & (0.050) & (0.063) & (0.065) & (0.044) & (0.052) & (0.056) & (0.049) & (0.005) & (0.026) \\\\\nNGC 4472 & 306 & 2.225 & 0.793 & 1.149 & 3.650 & 2.719 & 1.535 & 5.742 & 1.525 & 2.787 \\\\\n & & (0.031) & (0.069) & (0.081) & (0.039) & (0.039) & (0.042) & (0.027) & (0.005) & (0.021) \\\\\nNGC 4473 & 180 & 2.410 & 1.233 & 1.592 & 3.990 & 3.212 & 1.808 & 6.918 & --- & 3.164 \\\\\n & & (0.028) & (0.044) & (0.060) & (0.026) & (0.023) & (0.041) & (0.020) & (---) & (0.021) \\\\\nNGC 4478 & 132 & 2.627 & 1.653 & 1.855 & 3.719 & 3.378 & 1.838 & 6.764 & --- & 3.114 \\\\\n & & (0.042) & (0.051) & (0.069) & (0.036) & (0.039) & (0.040) & (0.039) & (---) & (0.023) \\\\\nNGC 4489 & 73 & 3.204 & 2.084 & 2.328 & 3.217 & 3.248 & 1.903 & 6.455 & 1.520 & 2.878 \\\\\n & & (0.047) & (0.054) & (0.075) & (0.046) & (0.048) & (0.045) & (0.050) & (0.005) & (0.028) \\\\\nNGC 4551 & 105 & 2.663 & 1.717 & 1.736 & 3.907 & 3.416 & 2.027 & 6.828 & 1.531 & 3.261 \\\\\n & & (0.046) & (0.058) & (0.056) & (0.034) & (0.038) & (0.044) & (0.036) & (0.005) & (0.024) \\\\\nNGC 4621 & 230 & 2.284 & 0.964 & 1.312 & 4.072 & 3.014 & 1.678 & 6.250 & --- & 3.091 \\\\\n & & (0.016) & (0.024) & (0.040) & (0.012) & (0.017) & (0.018) & (0.011) & (---) & (0.009) \\\\\nNGC 4697 & 168 & 2.397 & 1.289 & 1.661 & 3.806 & 3.198 & 1.832 & 6.603 & 1.524 & 3.094 \\\\\n & & (0.029) & (0.037) & (0.071) & (0.027) & (0.026) & (0.033) & (0.021) & (0.004) & (0.017) \\\\\n\\hline\n\\end{tabular}\n\n{$^1$ Velocity dispersions given in km\/s, from \\citet{yam06}.\\hfill}\n\\end{table*}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f2.eps}\n\\caption{Comparison between the EWs measured by the side-band method \n(SB; horizontal)\nand our proposed 'Boosted Median Continuum' (BMC; vertical) for H$\\beta$,\nH$\\gamma_F$, H$\\delta_F$, G4300, Fe5270, Fe5335, Mgb, D4000 and\n[MgFe]. The error bars are shown at the 1$\\sigma$ level. Notice only 8\ngalaxies (observed with Subaru) have a measurement of the 4000\\AA\\\nbreak. For the remaining six galaxies our data does not extend bluer\nthan 4000\\AA .}\n\\label{fig:EWSBBMC}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:DADZ} motivates our choice of parameters. We show the\nage-metallicity sensitivity -- ($\\Delta$Age\/Age)\/($\\Delta$Z\/Z) -- for\nsimple stellar populations measured at 12~Gyr and solar\nmetallicity. We do not follow the same definition as in\n\\citet{wo94}. Instead, we change the age from the reference value by\n2~Gyr and find the change in metallicity required from the fiducial SSP\nthat gives the same variation in the EW. The grey horizontal bar is\nthe value determined from the standard side-band method. Smaller\nvalues of the gradient imply a better disentanglement of the\nage-metallicity degeneracy. In the left (right) panels the horizontal\naxis explores a range of confidence levels (kernel sizes). The lines\ncorrespond to various choices of kernel size (left) and confidence\nlevel (right) as labelled. H$\\beta$ (\\emph{top}) behaves quite\nrobustly with respect to the choice of BMC parameters. H$\\gamma$\n(\\emph{middle}) shows that the presence of large scale features such\nas the break found around the G band at 4300\\AA\\ can affect the\nestimate if a large kernel size is chosen (200\\AA, dotted line,\n\\emph{left}). Finally, H$\\delta$ (\\emph{bottom}) shows that too small\na kernel size ($\\Delta\\lambda=$50\\AA, dashed line, \\emph{ left}) or\ntoo high a confidence level (95\\%, short dashed line, \\emph{ right})\ncan affect the estimate. In this case the effect is caused by nearby\nfeatures that will contaminate the estimate of the pseudo-continuum.\nOur choice of 100\\AA\\ kernel size and 90\\% level is thereby justified\nby the need to avoid both short and long-scale features in the\nSEDs. Furthermore, lower confidence levels should be avoided as they\nwill give flux values closer to an average that will not reflect the\ntrue continuum given the presence of numerous absorption lines. Too\nhigh values of the level will make the measurement more prone to\nhigher uncertainties at low signal-to-noise ratios. Monte Carlo\nsimulations of noise show that for our choice of BMC parameters the\nerrors for the BMC EWs are always smaller than those from the\nstandard side-band method.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f3.eps}\n\\caption{Comparison of the H$\\gamma_F$ and H$\\delta_F$ equivalent\nwidth measured with the side-band (SB) and the Boosted Median method\n(BMC). \nIn order to eliminate the dependence of the EWs on\nvelocity dispersion, we select those galaxies with $\\sigma\\leq 150$~km\/s \nand smooth them to this maximum velocity dispersion. \\emph{Top}:\nWe select NGC 4458 \\& NGC 4467, which show similar EWs in\nthe SB method but have differing H$\\delta$ EWs when using BMC. In the\nrightmost panels we plot the spectra from both galaxies. Overplotted\nare the pseudo-continuum of the side band (slanted straight line) and\nthe BMC (running along the top of the SED). The identical EWs\naccording to the SB method are due to the decreased flux in the red\npassband of NGC 4467, most likely caused by increased CN absorption,\nwhich hides the stronger intrinsic absorption in H$\\delta$. The\nspectra are normalised to an average value of 1 across the blue\npassband of H$\\delta_F$ to highlight the effect of CN absorption.\n\\emph{Bottom}: We highlight H$\\gamma$ in NGC 4387 \\& NGC 4467,\nfor which the SB method indicates a considerable difference in EW\nwhile the BMC identifies almost none. Looking at the spectra (right) of\nboth galaxies we can see that the difference in EW from the SB method\nis caused by an increase in the depth of the G4300 feature,\nartificially lowering the pseudo-continuum. The spectra are\nnormalised to 1 at 4365\\AA .}\n\\label{fig:Balmer1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f4.eps}\n\\caption{As in figure~\\ref{fig:Balmer1} we compare the H$\\gamma_F$\nand H$\\delta_F$ equivalent widths measured with the side-band and the\nBoosted Median method. In order to eliminate the dependence of the EWs\nand spectra on velocity dispersion, we select -- in contrast to\nfigure~\\ref{fig:Balmer1} -- those galaxies with $\\sigma\\leq 200$ km\/s.\nThe SEDs are smoothed to this maximum velocity dispersion.\n\\emph{Top}: The galaxies NGC 4473 \\& NGC 4478 are selected since they\nshow differing H$\\delta$ SB EWs but similar BMC measured EWs. The\nspectra (right) identifies the cause of the discrepancy. The spectra\nof NGC 4473 redwards of the H$\\delta$ line is severely affected by CN\nabsorption, lowering the average flux in this side band and distorting\nthe derived pseudo-continuum. The BMC recovers a significantly more\nrobust value. The spectra are normalised to an average value of 1\nacross the blue passband of H$\\delta_F$ to highlight the effect of CN\nabsorption. \\emph{Bottom}: NGC 4458 \\& NGC 4478 show\ndiscrepant SB EWs. This difference\ncan be seen on the right, where the spectra of both galaxies are\ndisplayed. The BMC correctly identifies the larger value of H$\\gamma$\nin NGC 4478, which is hidden in the SB method due to the increased absorption\nof G4300 feature located in the blue pass band of H$\\gamma_F$. The\nspectra are normalised to 1 at 4365\\AA.}\n\\label{fig:Balmer2}\n\\end{center}\n\\end{figure}\n\nHence, the method for generating the BMC is fairly simple. At every\nwavelength the 90th percentile from all flux measurements within\n50\\AA\\ on either side is assigned as the continuum at that wavelength.\nWith the continuum thus defined, for each line strength we only have\nto define the central wavelength and the spectral window over which\nthe line is measured. Given that our method maps quite robustly the\nunderlying continuum, we decided to fix a 20\\AA\\ width for all line\nstrengths considered in this paper -- hereafter, our BMC-based\nequivalent widths are labelled with a $20$ subindex.\n\n\\begin{table*}\n\\caption{Range of parameters explored in this paper (see text\nfor a description of each model).}\n\\label{tab:params}\n\\begin{tabular}{lccl}\n\\hline\\hline\nModel\/Param & MIN & MAX & Comments\\\\\n\\hline\nSSP & & & 2 params\\\\\nAge & 3 & 13 & Gyr\\\\\n$\\log(Z\/Z_\\odot)$ & -1.5 & +0.3 & Metallicity\\\\\n\\hline\nEXP & & & 3 params\\\\\n$\\log\\tau$ (Gyr) & -1 & +0.9 & Exp. Timescale\\\\\n$z_F$ & 0.1 & 5 & Formation epoch\\\\\n$\\log(Z\/Z_\\odot)$ & -1.5 & +0.3 & Metallicity\\\\\n\\hline\n2BST & & & 4 params\\\\\nt$_{\\rm O}$ & 3 & 13 & Old (Gyr)\\\\\nt$_{\\rm Y}$ & 0.1 & 3 & Young (Gyr)\\\\\nf$_{\\rm Y}$ & 0.0 & 0.5 & Mass fraction\\\\\n$\\log(Z\/Z_\\odot)$ & -1.5 & +0.3 & Metallicity\\\\\n\\hline\nCXP & & & 3 params\\\\\n$\\log\\tau_1$ (Gyr) & -2 & +0.5 & SF Timescale\\\\\n$\\log\\tau_2$ (Gyr) & -2 & +0.5 & Enrichment Timescale\\\\\n$z_F$ & 0.1 & 5 & Formation epoch\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\nThis is motivated by the fact that for some definitions of the line\nstrengths, any 'contaminating' lines falling within the central\nbandpass can have a stronger effect on the BMC method compared to the\nstandard side-band method. A clear example is found in the $H\\beta$\nline, where the effect of FeI(~4871\\AA) which sits on the shoulder of\nthe Balmer line within the standard definition of the central passband\n(width 28.75\\AA\\ ) clearly distorts the measurement of the equivalent\nwidth. This is slightly 'compensated' in the side-band method through\nthe presence in the blue side-band of another significant Fe line at\n4891\\AA. This is clearly not the case for the BMC method, in which the\npseudo-continuum is effectively independent of the values of both Fe\nabsortion lines. Hence, in the BMC method it is desirable to choose a\ncentral passband which only targets the line of interest. \nNevertheless, the method is versatile enough to define wider central\nbandpasses.\n\nThus, to avoid this problem we define a central passband wavelength as\nnarrow as possible within the usual spectral resolutions targeted in\nunresolved stellar populations. We follow the 'F' type passbands as\nused for the Balmer indices \\citep{wo97}, defining a 20\\AA\\ window\ncentered on the line of interest for all indices\nconsidered\\footnote{Our BMC-based EWS are labelled with a '20'\nsubindex, e.g. H$\\beta_{20}$}. In the case of the metal line strengths -- \nwhich usually involve clusters of lines -- the major feature is chosen\nas the center of the new 20\\AA\\ index. In the case of Mgb$_{20}$, this\nis centered between the MgI doublet at 5167\/5172\\AA. Fe5270$_{20}$ is\ndefined by the FeI line at 5270\\AA\\ and Fe5335$_{20}$ uses the FeI\nline at 5328\\AA. The central passband of the G-band is 4300\\AA. Such\na simple approach is nevertheless very versatile in its definition of\nany line strength. The new $D4000$ break feature uses the definition\ngiven by \\citet{bal99}, with the difference that it is the ratio of\nthe continuum flux obtained by the BMC method within those wavelengths\nthat is used here.\n\n\nTable~\\ref{tab:EWs} shows the (BMC-measured) equivalent widths of the\nVirgo elliptical galaxies targeted in this paper. The measurements are\nobtained directly from the observed spectra, presented in\n\\citet{yam06} and have not been corrected with respect to velocity\ndispersion. Analogously to the standard method, one could either\ncorrect the observed EWs for the effect of velocity dispersion, or use\nthe observed EWs in the modelling. For the latter, one must then\nmeasure the model EWs on spectra with the same resolution and velocity\ndispersion as the targeted galaxy. We follow this approach in the\npaper. We note at this point that the effect of velocity\ndispersion on the EWs of Balmer lines measured with the BMC method is\nincreased slightly since the method uses spectral information to\nbetter define the pseudo-continuum, which is destroyed by an\nincreasing $\\sigma$. This effect of course affects any other method which\nmeasures line strengths from unresolved spectra\n\\citep[e.g.][]{va99}. The numbers in brackets correspond to the\n1-$\\sigma$ uncertainty, obtained from a Monte Carlo simulation that\ngenerates 500 realizations of each SED, adding noise corresponding to\nthe SNR of the observations. Notice that 6 of the galaxies do not have\na measured D4000 as they were observed over a spectral range that does\nnot include the blue passband used in the definition of D4000.\n\n\\renewcommand{\\arraystretch}{1.1}\n\\begin{table*}\n\\begin{center}\n\\begin{minipage}{18cm}\n\\caption{Ages and metallicities of Virgo elliptical galaxies\naccording to the four models used in this paper. Error bars\nquoted at the 90\\% confidence level.}\n\\label{tab:Ages}\n\\begin{tabular}{lrrc|rrc|rrc|rrc}\n\\hline\\hline\nGalaxy & \\multicolumn{3}{c}{SSP} &\\multicolumn{3}{c}{2BST} &\\multicolumn{3}{c}{EXP} &\\multicolumn{3}{c}{CXP}\\\\\nNGC & Age(Gyr) & log(Z\/Z$_\\odot$) & $\\chi_r^2$ & Age(Gyr) & log(Z\/Z$_\\odot$) & $\\chi_r^2$ & Age(Gyr) & log(Z\/Z$_\\odot$) & $\\chi_r^2$ & Age(Gyr) & log(Z\/Z$_\\odot$) & $\\chi_r^2$ \\\\\n\\hline\n4239 & $ 5.0_{ -0.6}^{ +0.6}$ & $-0.13_{-0.07}^{+0.07}$ & 0.29 &\n $ 6.6_{ -1.7}^{ +2.6}$ & $-0.14_{-0.08}^{+0.08}$ & 0.27 &\n $ 5.3_{ -0.8}^{ +1.0}$ & $-0.14_{-0.07}^{+0.07}$ & 0.27 &\n $ 5.5_{ -0.9}^{ +0.6}$ & $+0.06_{-0.07}^{+0.04}$ & 0.83\\\\\n\n4339 & $ 8.2_{ -1.6}^{ +1.6}$ & $+0.18_{-0.10}^{+0.06}$ & 1.60 &\n $ 8.5_{ -1.9}^{ +1.7}$ & $+0.19_{-0.10}^{+0.06}$ & 1.58 &\n $ 8.8_{ -1.7}^{ +1.5}$ & $+0.16_{-0.09}^{+0.07}$ & 1.61 &\n $ 9.6_{ -1.7}^{ +1.5}$ & $+0.25_{-0.05}^{+0.03}$ & 1.23\\\\\n\n4365 & $11.6_{ -0.6}^{ +1.0}$ & $+0.26_{-0.04}^{+0.03}$ & 2.30 &\n $11.7_{ -0.6}^{ +0.9}$ & $+0.26_{-0.04}^{+0.03}$ & 2.30 &\n $11.4_{ -0.4}^{ +0.4}$ & $+0.27_{-0.02}^{+0.02}$ & 2.76 &\n $12.1_{ -0.3}^{ +0.1}$ & $+0.30_{-0.02}^{-0.00}$ & 3.11\\\\\n\n4387 & $12.5_{ -0.9}^{ +0.4}$ & $-0.09_{-0.05}^{+0.03}$ & 2.36 &\n $12.4_{ -1.1}^{ +0.5}$ & $-0.09_{-0.05}^{+0.03}$ & 2.36 &\n $11.4_{ -1.2}^{ +0.5}$ & $-0.06_{-0.03}^{+0.10}$ & 2.96 &\n $11.4_{ -0.9}^{ +0.7}$ & $+0.15_{-0.05}^{+0.02}$ & 3.15\\\\\n\n4458 & $10.7_{ -0.8}^{ +0.9}$ & $-0.22_{-0.04}^{+0.04}$ & 4.34 &\n $10.7_{ -0.9}^{ +1.0}$ & $-0.22_{-0.04}^{+0.04}$ & 4.34 &\n $10.7_{ -0.7}^{ +0.7}$ & $-0.21_{-0.05}^{+0.04}$ & 5.06 &\n $11.6_{ -1.1}^{ +0.8}$ & $-0.01_{-0.07}^{+0.06}$ & 1.90\\\\\n\n4464 & $12.7_{ -0.5}^{ +0.2}$ & $-0.14_{-0.03}^{+0.02}$ & 3.87 &\n $12.7_{ -0.4}^{ +0.2}$ & $-0.14_{-0.03}^{+0.02}$ & 3.87 &\n $11.9_{ -0.4}^{ +0.2}$ & $-0.13_{-0.02}^{+0.02}$ & 4.85 &\n $12.0_{ -0.6}^{ +0.4}$ & $+0.09_{-0.03}^{+0.01}$ & 3.68\\\\\n\n4467 & $ 6.4_{ -1.1}^{ +1.3}$ & $+0.16_{-0.07}^{+0.06}$ & 2.79 &\n $ 7.6_{ -1.8}^{ +2.6}$ & $+0.19_{-0.07}^{+0.07}$ & 2.78 &\n $ 6.8_{ -1.3}^{ +1.7}$ & $+0.15_{-0.09}^{+0.06}$ & 3.33 &\n $10.0_{ -1.9}^{ +1.8}$ & $+0.21_{-0.06}^{+0.04}$ & 0.67\\\\\n\n4472 & $ 9.2_{ -0.5}^{ +0.6}$ & $+0.27_{-0.02}^{+0.02}$ & 2.55 &\n $ 9.4_{ -0.5}^{ +0.4}$ & $+0.27_{-0.02}^{+0.02}$ & 2.55 &\n $ 9.8_{ -0.7}^{ +0.8}$ & $+0.27_{-0.02}^{+0.02}$ & 2.46 &\n $11.3_{ -0.8}^{ +0.6}$ & $+0.29_{-0.03}^{+0.00}$ & 2.19\\\\\n\n4473 & $12.5_{ -0.8}^{ +0.4}$ & $+0.23_{-0.03}^{+0.05}$ & 1.65 &\n $12.4_{ -0.7}^{ +0.5}$ & $+0.23_{-0.03}^{+0.05}$ & 1.65 &\n $11.6_{ -0.5}^{ +0.4}$ & $+0.27_{-0.02}^{+0.02}$ & 2.59 &\n $12.0_{ -0.3}^{ +0.1}$ & $+0.30_{-0.02}^{-0.00}$ & 5.56\\\\\n\n4478 & $ 7.2_{ -1.3}^{ +1.3}$ & $+0.22_{-0.07}^{+0.05}$ & 0.12 &\n $ 7.6_{ -1.6}^{ +2.1}$ & $+0.23_{-0.06}^{+0.04}$ & 0.12 &\n $ 7.7_{ -1.5}^{ +1.6}$ & $+0.21_{-0.07}^{+0.06}$ & 0.10 &\n $ 8.6_{ -1.3}^{ +1.1}$ & $+0.27_{-0.03}^{+0.02}$ & 0.03\\\\\n\n4489 & $ 4.6_{ -0.8}^{ +0.6}$ & $+0.09_{-0.08}^{+0.08}$ & 2.82 &\n $ 5.6_{ -1.6}^{ +2.3}$ & $+0.10_{-0.09}^{+0.09}$ & 2.69 &\n $ 5.0_{ -0.9}^{ +0.9}$ & $+0.06_{-0.07}^{+0.09}$ & 2.84 &\n $ 4.3_{ -0.8}^{ +0.5}$ & $+0.25_{-0.05}^{+0.03}$ & 3.83\\\\\n\n4551 & $ 6.1_{ -0.5}^{ +1.0}$ & $+0.26_{-0.04}^{+0.03}$ & 3.13 &\n $10.0_{ -2.0}^{ +1.2}$ & $+0.26_{-0.03}^{+0.02}$ & 2.50 &\n $ 7.3_{ -1.4}^{ +2.1}$ & $+0.26_{-0.05}^{+0.03}$ & 2.53 &\n $ 9.8_{ -2.2}^{ +2.0}$ & $+0.28_{-0.03}^{+0.01}$ & 2.29\\\\\n\n4621 & $10.5_{ -0.3}^{ +0.2}$ & $+0.27_{-0.02}^{+0.02}$ & 10.73 & \n $10.6_{ -0.3}^{ +0.3}$ & $+0.27_{-0.02}^{+0.02}$ & 10.73 &\n $10.9_{ -0.2}^{ +0.2}$ & $+0.27_{-0.02}^{+0.02}$ & 10.67 &\n $12.2_{ -0.5}^{ +0.2}$ & $+0.30_{-0.02}^{-0.00}$ & 8.56\\\\\n\n4697 & $ 8.1_{ -0.4}^{ +0.6}$ & $+0.27_{-0.02}^{+0.02}$ & 1.65 &\n $ 8.1_{ -0.5}^{ +0.7}$ & $+0.27_{-0.02}^{+0.02}$ & 1.65 &\n $ 8.3_{ -0.6}^{ +1.0}$ & $+0.26_{-0.04}^{+0.03}$ & 1.80 &\n $11.0_{ -1.3}^{ +1.1}$ & $+0.27_{-0.03}^{+0.01}$ & 0.24\\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\end{center}\n\\end{table*}\n\\normalsize\n\n\n\\subsection{Comparison with the side-band method}\n\nFigure ~\\ref{fig:EWSBBMC} shows a comparison of the line\nstrengths measured on the same spectra using the side-band (horizontal\naxes) and the BMC methods (vertical axes). Notice the departure from a\nsimple linear relationship, mainly caused by the different way\nneighbouring lines affect the estimate of the pseudo-continuum. In\nfigures \\ref{fig:Balmer1}~and~\\ref{fig:Balmer2} we illustrate in more\ndetail the difference in the measurement of the equivalent width of\nH$\\gamma$ and H$\\delta$, looking more closely at the effect of nearby\nlines on the indices. In order to eliminate the dependence of the EWs\non velocity dispersion we classify the sample into two subsets,\naccording to velocity dispersion. Figure~\\ref{fig:Balmer1} considers\nonly those galaxies with $\\sigma\\leq 150$~km\/s and\nfigure~\\ref{fig:Balmer2} focuses on galaxies with $\\sigma\\leq\n200$~km\/s. In both cases the galaxies are smoothed to the maximum\nvelocity dispersion of the subsample, in order to make a consistent\ncomparison. We note that by comparing spectra smoothed to some maximum\n$\\sigma$ one would obtain a false sense of agreement between the\nmethods, since the effect of smoothing removes information from the\nspectra.\n\nThe left panels of figures\n\\ref{fig:Balmer1}~and~\\ref{fig:Balmer2} compare the EWs of the\ngalaxies in each subset (open black circles), but we focus on two\ngalaxies in each case (solid squares). The galaxies are chosen\nbecause they have a similar value of the EW using one method, and a\nsignificantly different value using the other method. The spectral\nregions targeted in those galaxies is shown in the rightmost\npanels. The straight short line intersecting the spectra is the\nside-band pseudo-continuum and runs between the centres of the\nflanking side band regions The continuous line running along the top\nof the spectra is the BMC continuum. These are shown in grey and black\nfor each galaxy, as labelled. The spectra are normalised to allow us\nto overplot both SEDs. In the case of H$\\gamma$ we follow\n~\\cite{va99} and normalise at the peak of 4365\\AA. For H$\\delta$ we\nnormalise to an average flux value of 1 within the wavelengths of the\nblue side-band, to highlight the effect of CN absorption towards the\nred side of the line.\n\nIn figure~\\ref{fig:Balmer1} we show those galaxies with\n$\\sigma\\leq 150$~km\/s. In the top panels, we focus on NGC 4458\nand NGC 4467, since these galaxies give a similar side-band H$\\delta$\nEW but very different BMC-based measurements. Looking in detail at\nthe SEDs on the right, the red side-band\nof NGC4467 is more affected by increased absorption, mainly caused by\nthe CN features which, as reported in \\cite{yam06}, are stronger in\nNGC4467 (CN$_1$=0.110~mag), than in NGC4458 (CN$_1$=0.074~mag). Notice\nthat this does not strongly affect the BMC pseudo-continuum. In\naddition, the H$\\delta$ absorption observed in NGC 4467 is\npossibly underestimated relative to NGC 4458. The iron abundance\nreported by \\cite{yam06} is higher in NGC 4467 by $\\sim$0.1~dex, an\neffect that would be hidden by the normalisation over the iron lines\naround $\\sim$4070\\AA. In the bottom panels of\nfigure~\\ref{fig:Balmer1} we investigate the H$\\gamma$ lines in NGC\n4387 and NGC 4467 which have a similar EW using the BMC method,\nwhereas they differ by $\\sim$0.5~\\AA\\ in the side-band method. The\nSEDs reveal that these two galaxies do indeed have similar Balmer\nabsorption, but the side-band method is 'tricked' when comparing the\nflux with the continuum (straight line). The increased G4300\nabsorption in NGC 4387 pushes the pseudo-continuum down, which reduces\nthe EW value incorrectly. This is consistent with the measured \nabsorption at this resolution of NGC4467 (G4300$_{20}$ = 6.5~\\AA)\ncompared to that in NGC4387 (G4300$_{20}$ = 7.2~\\AA).\n\nShown in figure~\\ref{fig:Balmer2} is the subsample with\nhigher velocity dispersion, \ncomparing galaxies at $\\sigma\\leq 200$km\/s. The top panel\nshows the considerable discrepancy between the side band and BMC\nmethods on measuring H$\\delta$. Since the over-abundance of CN and the\nresulting absorption tends to increase with the mass of the galaxy\n\\citep[see e.g. ][] {sbla03,tol09}, the inclusion of the more massive\nNGC 4473 serves to show how significant the effect can be. The\n$\\sim$0.6\\AA\\ difference in H$\\delta$ given by the side band method can\nbe seen to be generated by the increase in absorption in the red\nside-band region $\\sim$4120\\AA, of NGC4473. We note again that the CN\nabsorption, identified by \\cite{yam06} is higher in NGC 4473\n(CN$_1$=0.138~mag), than in NGC 4478 (CN$_1$=0.077~mag). The bottom\npanels display a more subtle situation in which the effect of the\nsurrounding metal lines is to reverse the relative values of the\nH$\\gamma$ EWs. The increased absorption, again mainly coming from the\nG4300 feature, forces the pseudo-continuum to a lower value, resulting in a \nsmall EW. The effect, while small, serves to indicate how\nsensitive the side band method can be to the surrounding spectral features.\n\nTherefore, figures~\\ref{fig:Balmer1}~and~\\ref{fig:Balmer2} show that \nour proposed method is more resilient to the effects of neighbouring \nabsorption lines. To further illustrate this point, figure~\\ref{fig:Idx} \ncompares the age-metallicity degeneracy of typical Balmer (\\emph{left}) and metal\nlines (\\emph{right}) measured either in the standard way or using the\nBMC pseudo-continuum. The shaded areas indicate the difference in the\nEWs with respect to a reference stellar population: on the left it is\nthe population with the same age at solar metallicity, and on the\nright the reference is a population with the same metallicity and\n10~Gyr old, both references marked by the vertical lines. Black and\ngrey shading correspond to the BMC and the side-band methods,\nrespectively. The panels on the left explore the\nage-sensitive Balmer indices for a range of metallicities and the\npanels on the right show metal indices for a range of ages. Ideally, a\nperfect observable would result in a horizontal line (i.e. zero\nmetallicity dependence of an age-sensitive line and vice-versa). The\nmodels span a wide range of ages and metallicities as shown in the\ncaption. This figure shows that BMC-based measurements of EWs are less\nsubject to the age-metallicity degeneracy than the side-band\nmethods. This result is especially dramatic for H$\\gamma$ and\nH$\\delta$, for which the age-metallicity degeneracy drops from\n$\\Delta$EW$\/\\Delta\\log(Z\/Z_\\odot)= -4.2$ (side-band) to $-1.9$ (BMC)\nin H$\\gamma$ or from $-3.9$ (side-band) to $-1.7$ (BMC) in H$\\delta$\n(values measured at the fiducial 10~Gyr, solar metallicity SSP).\nFurthermore, the shaded regions of the metal-line indices on the\nrightmost panels are much wider for the BMC method (black), showing\nthat at a fixed age, the BMC method spans a wider range of EW, thereby\nbeing more sensitive to metallicity (see caption for details).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f5.eps}\n\\caption{Dependence of Balmer lines on metallicity (\\emph{left})\nand ``metal'' lines on age (\\emph{right}). The standard side-band\nmethod (SB; grey) and our proposed Boosted Median Continuum (BMC; black)\nare shown for a range of metallicities and ages as shown. On the\nleft, the shaded areas correspond to an age range $[6,12]$~Gyr. On the\nright, the shaded areas span a range of metallicities: $-0.3<\\log Z\/Z_\\odot <+0.3$.\nThe vertical axis is the difference between the equivalent width of the\nline for a given metallicity (left) or age (right) and the value at\na reference point given by the vertical line (i.e. solar metallicity \non the left and 10 Gyr on the right).}\n\\label{fig:Idx}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f6.eps}\n\\caption{The uncertainties caused by an error in the\nestimate of velocity dispersion are illustrated by a \ncomparison of the difference between the EW measured\nat a fiducial value ($\\sigma=$100km\/s) and a range\nof velocity dispersions (horizontal axis). Various\nline strengths are considered, as labelled, for the\nstandard SB method ({\\sl top}) and our proposed BMC\nmethod ({\\sl bottom}).}\n\\label{fig:sigerr}\n\\end{center}\n\\end{figure}\n\nIn order to quantify the effect of an uncertainty in the velocity\ndispersion on the EWs, we compare in figure~\\ref{fig:sigerr} the\nchange in EW between a fiducial value ($\\sigma=$100km\/s) and a range\nof velocity dispersions (horizontal axis). The SB (BMC) values are\nshown in the top (bottom) panels for a number of absorption lines, as\nlabelled. One can see that the effect on BMC-measured EWs can be\ncorrected much in the same as with the standard SB method. In\naddition, for most cases the correction is smaller for BMC estimates.\n\nFurthermore, we have compared the effect of noise on our proposed BMC\npseudo-continuum and found that for a wide range of signal-to-noise\nratios (from 10 to 100 per \\AA\\ ) the uncertainty in the EW of all\nlines is always $\\sim 0.3$~dex smaller than those obtained with the\nside-band method. \nHowever due to the lower dynamical range, as the \nBMC is forced to stay above the spectra, the S\/N requirements are \nsimilar to other methods.\nWe follow \\citet{optoHB} and define the S\/N at which the method can \ndistinguish $\\pm$2.5 Gyr at 10 Gyr. For $H\\gamma$ and $H\\delta$ \nwe get a S\/N of $\\sim 50$, and for $H\\beta$ the required S\/N is $\\sim 60$.\n\nFigure~\\ref{fig:grid} shows the equivalent widths of several lines for\na grid of SSPs corresponding to a velocity dispersion of 100 km\/s\n(black) or 300 km\/s (grey). The measured EWs (without any correction\nfor velocity dispersion) are shown as black dots, along with \na characteristic error bar which mostly comes from the systematics (the\nobserved uncertainties are much smaller). We emphasize that our\nfitting method generates grids corresponding to the measured velocity\ndispersion for each galaxy. One can see that BMC EWs (left panels)\nappear less degenerate than the grids using a side-band method (right\npanels).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f7.eps}\n\\caption{Comparison of grids of SSP models using the standard\nside-band method (SB, \\emph{right}) and our proposed Boosted Median\nContinuum (BMC, \\emph{left}). A systematic error bar is included in\neach panel (the observational error bars are much smaller). We show\nthe grids for two different velocity dispersions 100km\/s (black) and\n300km\/s (grey). The solid lines connect the SSPs with ages (from top\nto bottom) 2 to 14 Gyr in steps of 2 Gyr. The dashed lines connect\nthe SSPs with metallicities (from left to right) [m\/H]=$-$1 to $+$0.2 in\nsteps of 0.2~dex. }\n\\label{fig:grid}\n\\end{center}\n\\end{figure}\n\n\\section{Modelling the SFH of elliptical galaxies}\n\nThe properties of the unresolved stellar populations of our sample are\nconstrained by comparing the targeted equivalent widths with four sets\nof generic models that describe the star formation history in terms of\na reduced number of parameters. It is our goal to assess the\nconsistency of different sets of models in fitting \\emph{independently}\nthe different spectral lines targeted as well as the full SED. The\nmajority of studies in the literature \\citep[see\ne.g. ][]{kd98,sct00,cald03,tm05} have compared measurements of EWs\nwith simple stellar populations (i.e. a single age and\nmetallicity). While those models are probably valid for the\npopulations found in globular clusters, it is imperative to go beyond\nsimple stellar populations in galaxies, whose star formation histories\ngenerate complex distributions of age and metallicity. \\cite{fyi04}\nand \\citet{pas05} showed that composite models of stellar populations\ncould result in significant differences on the average ages and\nmetallicities of galaxies. More recently, \\citet{sbla06} and\n\\citet{serra07} have explored this issue through the comparison of two\nage indicators, both concluding that composite populations are needed\nto consistently model the populations in early-type galaxies.\n\n\\begin{figure*}\n\\begin{minipage}{18cm}\n\\includegraphics[width=3.2in]{BFPS_f8a.eps}\n\\includegraphics[width=3.2in]{BFPS_f8b.eps}\n\\caption{Best fit age and metallicity values for our sample,\nusing SSPs (\\emph{left}) or EXP models (\\emph{right}). The\nerror bars shown the 68\\% confidence levels and the\nshaded regions give the age and metallicity estimates\nof \\citet{yam06} using the H$\\gamma_\\sigma$ vs. [MgFe]\ndiagram. The diamond gives the fit to the age and\nmetallicity using \\emph{only} the spectral\nenergy distribution (i.e. no line strengths are used for\nthis data point).}\n\\label{fig:maps1}\n\\end{minipage}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}{18cm}\n\\includegraphics[width=3.2in]{BFPS_f9a.eps}\n\\includegraphics[width=3.2in]{BFPS_f9b.eps}\n\\caption{Best fit age and metallicity values for our sample,\nusing 2-Burst (\\emph{left}) or CXP models (\\emph{right}). The\nerror bars shown the 68\\% confidence levels and the\nshaded regions give the age and metallicity estimates\nof \\citet{yam06} using the H$\\gamma_\\sigma$ vs. [MgFe]\ndiagram. The diamond gives the fit to the age and\nmetallicity using \\emph{only} the spectral\nenergy distribution (i.e. no line strengths are used for\nthis data point).}\n\\label{fig:maps2}\n\\end{minipage}\n\\end{figure*}\n\nIn this paper, except for the first case (namely Simple Stellar\nPopulations), the models generate a distribution of ages and\/or\nmetallicities that are used to combine the population synthesis models\nof \\citet{bc03}, assuming a \\citet{chab03} Initial Mass Function. We\nuse the updated 2007 version \\citep{cb07}. The resulting synthetic\nspectral energy distribution is smoothed to the same resolution and\nvelocity dispersion of the galaxy. A correction for Galactic reddening\nis applied, assuming the \\citet{Fitz99} law (this is mostly done for\nthe comparison of the full SEDs, as EWs are not affected). Finally,\nthe EWs are computed and compared with the observations using a\nstandard maximum likelihood method. The high S\/N of the observed\nspectra imply that our error budget is mainly dominated by the\nsystematics of the population synthesis models. It is not obvious how\nto incorporate problems such as sparsely populated stellar parameter\nspace or systematics within the models, into the analysis. As a\ncompromise we estimate the uncertainties associated with the models by\ngenerating 500 Monte Carlo simulations of gaussian noise at the level\nof S\/N $\\sim$50, that of the stellar library at the core of the\nBC03\/CB07 models \\citep[STELIB][]{stelib}. These are added in quadrature\nwith those generated for each galaxy index. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f10.eps}\n\\caption{Likelihood distribution of the marginalised average stellar ages for some\nof the Virgo cluster elliptical galaxies from our sample. The\ndistribution is shown for the four models as labelled. The grey solid \nline corresponds to the fit of the SED. The lines correspond \nto the age distribution according to [MgFe] plus either H$\\beta$ (black,solid), \nH$\\gamma$ (black,dashed) or H$\\delta$ (black,dotted),\nrespectively.}\n\\label{fig:AgeHist}\n\\end{center}\n\\end{figure}\n\n\nThe four sets of models considered in this paper are listed below,\nwith the range of parameters shown in table~\\ref{tab:params}.\n\n\\begin{enumerate}\n\\item {Simple Stellar Populations (SSP):} This has been the most\npopular method used in the analysis of the ages and metallicities of\nearly-type galaxies. The advantage lies in it simplicity: SSPs are the\nbuilding blocks of all population synthesis models, and they are\ncarefully calibrated against realistic SSPs (i.e. globular\nclusters). An SSP is primarily defined by an age and a metallicity\nalthough it is also dependent on the abundance pattern and the IMF.\nThe drawback of this method is that the parameters obtained are\ninherently \\emph{luminosity weighted}, such that a small amount of\nyoung stars can have a significant effect on the age extracted with\nthis method. Furthermore, SSPs are not expected to model galaxy\npopulations, which have an extended range of ages and\nmetallicities. By using SSPs to model early-type galaxies one makes\nthe assumption that the stellar populations have a narrow age\ndistribution compared to stellar evolution timescales.\n \n\\item {Exponential SFH (EXP):} \nA more physical scenario should consider an extended period of star\nformation. The EXP models (also called $\\tau$ models in the\nliterature) model the star formation rate as an exponentially\ndecaying function of time with timescale $\\tau$, started at an epoch\ngiven by a formation redshift $z_F$. The metallicity is assumed to be\nfixed throughout the SFH and is also left as a free parameter.\n\n\\item {Two Burst Formation (2BST):} \nRecent rest-frame NUV observations have revealed the presence of\nresidual star formation in early-type galaxies \\citep[see\ne.g.][]{sct00,fs00,yi05,kav07}. The presence of small amounts of\nyoung stars can significantly affect the derived SSP model parameters\n\\citep{serra07}. This model describes this mechanism with two simple\nstellar populations. Four parameters are left free: the age of the\nold (t$_O$) and the young components (t$_Y$), the mass fraction in\nyoung stars (f$_Y$), and the metallicity of the populations ($Z$,\nassumed to be the same for the old and the young components).\n\n\\item {Chemically Enriched Exponential (CXP):} \nAll the models considered above fix the metallicity throughout the\nSFH. More realistically, a model should incorporate in a consistent\nway the buildup of metallicity caused by previous generations of\nstars. These chemical enrichment models \\citep[see e.g.][]{bp99,fs03}\ninclude the stellar yields from intermediate and massive stars and\nresult in a distribution of metallicities as well as ages. The aim of\nthis paper is to explore simple models that can be easily\nimplemented. Hence, instead of applying a detailed model of chemical\nenrichment, we define in a purely phenomenological way a model that\nmimics those. We keep the same SFH as in the EXP models (described by\na star formation timescale $\\tau_1$ and a formation redshift\n$z_F$). Furthermore, the metallicity is assumed to increase in step with \nthe cumulative distribution of the star formation rate, namely:\n\n\\begin{equation}\nZ(t) = Z_1 + (Z_2 - Z_1) \\Big[ 1 - \\exp( - \\Delta t\/\\tau_2)\\Big],\n\\end{equation}\n\n\\noindent\nwhere $\\Delta t = t-t(z_F)$ and $\\tau_2$ is the timescale\ncorresponding to the buildup of metallicity. As a first-order\napproach it is valid to assume that metallicity increases with the\nstellar mass of the system (roughly, the cumulative distribution of\nthe star formation rate). More accurately, this timescale will depend\non the star formation efficiency or on the fraction of gas ejected in\noutflows \\citep[see e.g. ][]{fs03}. The upper and lower values of\nmetallicity are fixed to $Z_1=Z_\\odot \/10$ and $Z_2=2Z_\\odot$,\nalthough the average metallicity will be controlled by the timescale\n$\\tau_2$.\n\\end{enumerate}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f11.eps}\n\\caption{Comparison between the ages presented here and those from\nprevious studies. Y06, TF02 and C03 indicate \\citep{yam06},\n\\citep{tf02}, and \\citep{cald03}, respectively. The figure illustrates\nthe difference in the average ages when including two types of\ncomposite populations: a 2-burst model (2BST; top) or a smooth star formation\nhistory including chemical enrichment (CXP; bottom).\nThe slanted solid line\nis a 1:1 correspondence. The vertical dashed line represents the age\nof the Universe for our cosmology, and is used as a prior in the\nanalysis (see text for details).}\n\\label{fig:Ages}\n\\end{center}\n\\end{figure}\n\nWe assume a standard $\\Lambda$CDM cosmology ($\\Omega_m$=0.3; H$_0$= 70\nkm\/s\/Mpc) to constrain the maximum ages of the stellar populations to\nthe age of the Universe (13.7~Gyr). One could allow the fitting\nalgorithm to stray into older ages to explore wider volumes of\nparameter space, but this paper takes the cosmological constraints on\nthe age of the Universe at face value \\citep{wmap}. Furthermore, population\nsynthesis models are poorly calibrated for populations older than\nGalactic globular clusters. We note that previous work on the same\nspectra \\citep{yam06} probed the population synthesis models of\n\\citet{va99} all the way to their oldest available ages ($\\sim$20~Gyr),\nusing the isochrones of \\citet{bert94}.\nAlthough in only two cases, NGC4365 at 20~Gyr and NGC4464 at 18.5~Gyr, \ndid they find cosmologically contradictory ages, we still choose to avoid \nthis by limiting the maximum age.\n\n\n\\begin{table*}\n\\caption{Best fit parameters for the 2BST and CXP models}\n\\label{tab:BFit}\n\\begin{tabular}{lrllllll}\n\\hline\\hline\nGalaxy & M$_{\\rm s}^1$ & \\multicolumn{3}{c}{2BST} & \\multicolumn{3}{c}{CXP}\\\\\nNGC & $\\times 10^{10}$M$_\\odot$ & t$_O$(Gyr) & t$_Y$(Gyr) & f$_Y$ & $\\tau_1$(Gyr) & $\\tau_2$(Gyr) & z$_F$\\\\\n\\hline\n4239 & 0.9(0.7)& $ 7.9_{ -2.7}^{ +2.7}$ & $ 2.6_{ -0.3}^{ +0.3}$ & $0.08_{-0.06}^{+0.12}$ \n& $0.2_{ -0.2}^{ +0.5}$ & $0.2_{ -0.1}^{ +0.5}$ & $0.6_{ -0.1}^{ +0.2}$\\\\\n4339 & 4.6(4.6)& $ 8.8_{ -2.0}^{ +1.8}$ & $ 1.8_{ -1.3}^{ +1.1}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.01$\n& $0.4_{ -0.2}^{ +0.4}$ & $0.0_{ -0.0}^{ +0.1}$ & $1.7_{ -0.5}^{ +0.8}$\\\\\n4365 & 38.0(36.3)&$11.9_{ -0.9}^{ +0.7}$ & $ 1.5_{ -1.1}^{ +1.0}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.01$\n& $0.8_{ -0.2}^{ +0.2}$ & $2.9_{ -2.0}^{ +2.0}$ & $3.0_{ -0.2}^{ +0.2}$\\\\\n4387 & 3.2(2.8)& $12.4_{ -0.8}^{ +0.5}$ & $ 1.6_{ -1.1}^{ +1.0}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.01$\n& $0.7_{ -0.3}^{ +0.4}$ & $0.3_{ -0.1}^{ +0.2}$ & $2.6_{ -0.5}^{ +0.5}$\\\\\n4458 & 3.1(2.2)&$10.9_{ -1.0}^{ +0.7}$ & $ 1.6_{ -1.1}^{ +1.0}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.02$\n& $0.2_{ -0.1}^{ +0.4}$ & $0.2_{ -0.2}^{ +0.6}$ & $2.5_{ -0.4}^{ +0.6}$\\\\\n4464 & 1.8(1.4)& $12.7_{ -0.5}^{ +0.2}$ & $ 1.6_{ -1.1}^{ +1.0}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.01$\n& $0.6_{ -0.2}^{ +0.3}$ & $0.4_{ -0.2}^{ +0.2}$ & $2.8_{ -0.4}^{ +0.3}$\\\\\n4467 & 0.4(0.4)& $ 8.6_{ -2.6}^{ +2.6}$ & $ 2.4_{ -1.1}^{ +0.5}$ & $0.03_{-0.03}^{+0.11}$\n& $0.2_{ -0.1}^{ +0.1}$ & $0.1_{ -0.1}^{ +0.1}$ & $1.7_{ -0.6}^{ +0.8}$\\\\\n4472 & 97.7(102.3)& $ 9.5_{ -0.8}^{ +0.4}$ & $ 1.6_{ -1.1}^{ +1.0}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.02$\n& $0.9_{ -0.4}^{ +0.6}$ & $0.2_{ -0.2}^{ +0.2}$ & $2.8_{ -0.4}^{ +0.3}$\\\\\n4473 & 20.4(20.4)& $12.5_{ -0.8}^{ +0.4}$ & $ 1.5_{ -1.1}^{ +1.0}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.01$\n& $1.0_{ -0.1}^{ +0.2}$ & $1.6_{ -1.1}^{ +1.1}$ & $3.0_{ -0.2}^{ +0.2}$\\\\\n4478 & 4.7(5.6)& $ 8.3_{ -2.2}^{ +2.1}$ & $ 1.9_{ -1.3}^{ +1.0}$ & $0.02_{-0.02}^{+0.05}$ \n& $0.6_{ -0.3}^{ +0.5}$ & $0.1_{ -0.1}^{ +0.1}$ & $1.4_{ -0.3}^{ +0.6}$\\\\\n4489 & 1.2(1.2)&$ 6.9_{ -2.5}^{ +2.4}$ & $ 2.6_{ -0.4}^{ +0.3}$ & $0.13_{-0.10}^{+0.18}$ \n& $0.3_{ -0.1}^{ +0.3}$ & $0.1_{ -0.1}^{ +0.1}$ & $0.4_{ -0.1}^{ +0.1}$\\\\\n4551 & 4.0(2.8)& $10.7_{ -2.0}^{ +1.7}$ & $ 2.7_{ -0.2}^{ +0.2}$ & $0.04_{-0.03}^{+0.06}$ \n& $0.3_{ -0.1}^{ +1.1}$ & $0.2_{ -0.1}^{ +0.1}$ & $1.7_{ -0.6}^{ +0.9}$\\\\\n4621 & 33.9(33.1)& $10.7_{ -0.4}^{ +0.1}$ & $ 1.5_{ -1.1}^{ +1.0}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.02$\n& $1.0_{ -0.3}^{ +0.3}$ & $1.4_{ -1.0}^{ +1.0}$ & $4.1_{ -1.2}^{ +0.9}$\\\\\n4697 & 17.0(12.3)&$ 8.3_{ -0.7}^{ +0.4}$ & $ 1.6_{ -1.2}^{ +1.0}$ & $\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.03$\n& $0.3_{ -0.1}^{ +0.4}$ & $0.2_{ -0.1}^{ +0.1}$ & $2.3_{ -0.4}^{ +0.5}$\\\\\n\\hline\n\\end{tabular}\n\n$^1$ Stellar masses are computed from the best fit CXP (2BST)\nmodels. Estimated uncertainty $\\Delta\\log$M$_s\\sim 0.3$~dex (mostly\nfrom the assumption about the IMF).\n\\end{table*}\n\n\\subsection{Disentangling the stellar populations}\n\nThe comparison of the EWs is done by measuring the line strengths\ndirectly from the model SEDs after being resampled and smoothed to the\nresolution and velocity dispersion of the actual observations of each\ngalaxy, taken from \\citet{yam06}. Since the spectra of the models have\nalready been smoothed in terms of velocity dispersion to that of the\nreal galaxy, no correction for this need be applied. All galaxies are\nfitted using the age-sensitive indices, H$\\beta_{20}$, H$\\gamma_{20}$,\nH$\\delta_{20}$ and G4300$_{20}$ along with the metallicity\nindicator [MgFe]$_{20}$\\citep{gon93}. This is used instead of the\nnewer [MgFe]$^\\prime$ as defined by \\citet{tm03} since the original is\nalmost independent of $\\alpha\/$Fe and it is not clear whether the same\nfractions of the two Fe lines is still appropriate for the new BMC\nindices. We note that we have excluded the D4000 index \nsince it was found to be poorly modelled by BC03, a problem already \nidentified by both \\citet{wild07} and \\citet{sbla06c}. The exclusion (or inclusion) of \nthe D4000, does not significantly affect the final results obtained although \nit does significantly increase (decrease) the quality of the fit.\nThe removal of the D4000 also limits the constraints we can put on \nsecond order parameters (e.g. age and mass fraction of young stellar \npopulation in the 2-Burst model).\n\n\nAs indicated by \\citet{schiavon04} and later confirmed by\n\\citet{serra07}, a discrepancy found between the parameters derived\nthrough each of the three Balmer lines could be explained by the\nexistence of an underlying younger sub-population. \\citet{schiavon04}\nexplain that this may well be due to the dominance of the younger\npopulation at bluer wavelengths. However the mere fact that different\nindices would give different parameters, should suggest that the model\nbeing used has not captured all aspects of the true star formation\nhistory. Hence, in the analysis of EWs of our sample galaxies, we are\nnot only looking to determine the age and metallicity but also\nestimating whether it is possible to \\emph{rule out a single\npopulation} to describe an early-type galaxy. This could be done\nthrough discrepancies between the three Balmer lines targeted\nhere, a possibility highlighted by \\citet{schiavon07} on a single\nstacked spectra of low mass ellipticals, who found that the addition\nof a small young population alleviated such discrepancies.\n\nWith the same aim, we focus on possible discrepancies between \nindividual line strengths and explore a number of star formation\nhistories to determine the age distribution of our sample of elliptical\ngalaxies.\n\n\\subsection{Spectral Fitting}\nAs an alternative approach to targeted line strengths, we also\nconsider the full spectral energy distribution to constrain\nthe star formation history of our sample. In this paper\nwe want to test the consistency between the constraints imposed\nby the line strengths and those from a full spectral fit.\n\nWe use a maximum likelihood method to fit the spectra, which is\nanalysed over the spectral range between 3900\\AA\\ and 4500\\AA\\ (or\n4000--4500\\AA\\ for the six galaxies observed with WHT). The SEDs are\nnormalised over the same range. The data span a much wider spectral\nrange (out to 5500-5800\\AA ). However, we have chosen a smaller\nrange to avoid flux calibration errors, which could introduce\nimportant systematic changes in the predicted ages and metallicities. \nThe spectral range chosen straddles the D4000 break, since this is a\nvery sensitive indicator of the ages of the stellar populations.\n\n\n\n\\subsection{Results}\n\nFigures~\\ref{fig:maps1} and \\ref{fig:maps2} show the average age and\nmetallicity for all four models explored in this paper. The error bars\nare shown at the 69\\% confidence level (i.e. like a 1$\\sigma$ level\nalthough our analysis does not assume a Gaussian distribution). \nFor the SSPs the average age is trivially the age of the\npopulation. For the others, the average values are weighed by the\nstellar mass. Each error bar corresponds to the \\emph{individual fit}\nof an age-sensitive line (H$\\beta$, H$\\gamma$,\nH$\\delta$ or G4300) plus the [MgFe] as metallicity indicator\n(i.e. each point comes from the analysis involving 2 indices). The\ndiamond and its error bar correspond to the fit to the SED -- no\nadditional information from the line strengths is added to this\nlikelihood. In grey, the shaded areas correspond to the ages from\n\\citet{yam06} (using the H$\\gamma_\\sigma$ -- [MgFe] diagram on\nSSPs). One can see that SSPs (left panel of figure~\\ref{fig:maps1})\nshow the largest discrepancies between the ages and metallicities\nestimated using independently the different age-sensitive\nindices. This would imply that a joint likelihood putting all indices\ntogether would not be a consistent way of determining the ages of the\npopulations.\n\nNevertheless, as a comparison we show in table~\\ref{tab:Ages} the\nbest estimates of the average age and metallicity, combining all five\nindices (but not the fit of the SED). Table~\\ref{tab:BFit} shows the\nbest fit parameters for the 2BST and CXP models. The uncertaintes in\nboth tables are given at the 90\\% confidence level. It can be\nseen from table~\\ref{tab:Ages} that in a majority of cases the use of\na model with an age spread offers an improvement in the $\\chi^2$ value\nfor the five indices. Importantly the recovered values of metallicity\nremain consistent (within errors) across the single metallicity models\nconsidered, indicating its robustness against the inclusion of\nadditional populations. For some galaxies, such as NGC 4365, NGC 4387\nand NGC 4473, an SSP is as good as the other models presented here and\nin fact the parameters recovered by complex SFHs essentially reduce to\nan SSP. This is to be expected since the short formation timescale\nexpected for elliptical galaxies will in some cases be well\nrepresented by an SSP. However galaxies which have younger SSP ages\ngive very different (mass-weighted) ages when considering a 2-Burst\nmodel. One interesting result is that the CXP models, which\nincorporate a spread both in metallicity and age, find solutions\nthat are as good as -- and in the case of NGC4458, and NGC4467 and NGC\n4697 significantly better than -- the other models.\n\nIn addition, the discrepancies among individual measurements shown in\nfigure~\\ref{fig:maps1} reflects the fact that a single age and\nmetallicity scenario is a weaker, less consistent model of the\npopulations of a galaxy. Table~\\ref{tab:Ages} shows that the actual\nchange in the age when going beyond SSPs can be quite significant,\nespecially for younger ages. We emphasize here that the different ages\ndo not reflect a possible inclusion of a prior caused by the choice of\nparameters. All the composite models presented here (CSP, EXP and\n2-Burst) explore a range of parameters that includes the best-fit ages\nand metallicities obtained by the SSPs.\n\n\nIn some cases the $\\chi^2$ values are larger than might be\nexpected. In the case of NGC~4621, the largest contributor is the\nG-band. If we remove that indicator from the analysis, the reduced\n$\\chi^2$ decreases to $\\sim 1$. It is clear that the high S\/N of the\nspectra poses a challenge for current stellar population synthesis\nmodels. Certainly the errors associated with the models do not take\ninto consideration internal problems with the models themselves or the\npossible limitations of the stellar libraries from which they are\nassembled. \\cite{proct04} also consider that limitations in the models\nas the cause of the high $\\chi^2$ values. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f12.eps}\n\\caption{Ages and metallicities versus stellar mass (left)\nand velocity dispersion (right). The solid dots (grey diamonds)\ncorrespond to the estimates from the CXP (2-Burst) models.\nThe dashed lines in the rightmost\npanels are the age and metallicity scaling relation of \\citet{tm05}.}\n\\label{fig:AgeMet}\n\\end{center}\n\\end{figure}\n\nThe mismatch discussed above can be seen in more detail in figure~\\ref{fig:AgeHist},\nwhere the marginalised distribution of average age for all four models is shown\nfor a few galaxies from the sample, as labelled. The black solid, dashed, and dotted lines\ncorrespond to individual fits to H$\\beta$, H$\\gamma$, and H$\\delta$, respectively,\nalong with the G4300 (grey, dotted) and the SED (grey, solid). SSPs (leftmost panels) fail\nto give a consistent distribution, whereas any of the other models\ngive a more unified distribution. Notice that the galaxies shown in\nthe top panels (NGC 4239 and NGC 4489) have young populations, whereas\nthe bottom two galaxies (NGC 4464 and NGC4697) have overall older populations.\nThe top two galaxies are better fit by a 2-Burst model and the bottom two get\nbetter fits from an extended model like CXP. This would suggest that the youngest\npopulations in early-type galaxies are best fit by the assumption of \n``sprinkles'' of young stars, as suggested by \\citet{sct00} and modelled by \n\\citet{fs00} and \\citet{kav07}.\n\nThis explains why the models incorporating an age spread have\ngreater maximum likelihood values, where $\\mathcal{L}_{max} \\propto\n\\exp(-\\chi^2_{min}\/2)$. However, as was seen in table~\\ref{tab:Ages},\nthe fits of the SSPs are not substantially worse than the more complex\nmodels. This is not surprising given firstly, that the SFH of many\nelliptical galaxies is not different from a single episode of star\nformation, especially when it happens at high redshift. Secondly, as\nidentified by \\cite{serra07}, for bursts of mass fraction, $f_{\\rm Y}\n\\geq$ 10\\% the disagreement in the recovered ages from different\nBalmer lines dissappears. This indicates that there is no\ndifferential effect across the spectrum from which to determine an age\nspread. Nevertheless, it remains the case that the more complex models\nare physically well motivated, resulting in better average age\nestimates and fit the data more consistently. Under such criteria we\nselect the 2BST and CXP for further analysis.\n\nFigure~\\ref{fig:Ages} shows a comparison of our age estimates using\n2-Burst (\\emph{top}) and CXP models (\\emph{bottom}) with values taken\nfrom the literature, as given in the figure caption. It is important\nto notice that our models do not consider ages older than the age of\nthe Universe using a concordance $\\Lambda$CDM cosmology (i.e. 13.7\nGyr). Our ages are also closer to a mass weighted average due to\nthe underlying assumptions in the modelling in both cases. We find\nthat for the youngest galaxies, our analysis gives rather older ages\nthan previous estimates from the literature. Also notice that the\nolder galaxies appear younger because of the strong constraint imposed\nby the chosen cosmological parameters. Figure~\\ref{fig:AgeMet} shows\nthe average age and metallicity as a function of stellar mass (\\emph{\nleft}) and velocity dispersion (\\emph{right}), both for the CXP (black\ndots) and for the 2-Burst models (grey diamonds). The stellar masses\nare obtained by combining the absolute magnitude of the galaxies in\nthe $V$ band with $M\/L_V$ corresponding to the best fit CXP models\n(There is no significant difference if the best fit 2-Burst models are\nused instead). The dashed line is the fit to age and metallicity from\n\\citet{tm05}. Our age-mass relationship is compatible with recent\nestimates in the literature \\citep[see\ne.g. ][]{sct00,cald03,tm05,sbla06b}. We should emphasize that our\nreduced sample of Virgo \\emph{elliptical} galaxies excludes massive,\nyoung S0s in high density environments \\citep{tm05}, removing them as\na possible cause of the age scatter.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.2in]{BFPS_f13.eps}\n\\caption{The predictions of the CXP models for star formation timescale (\\emph{\ntop}) and formation redshift (\\emph{bottom}) are shown with respect to\nstellar mass (\\emph{left}) and velocity dispersion (\\emph{right}). \nThis speculative plot suggests that the age trend found in\nfigure~\\ref{fig:AgeMet} is mainly caused by a range of formation epochs (bottom).\nThe formation timescale (top) also correlates with mass or velocity dispersion,\nbut it does not have values larger than $\\tau_1\\sim 1.5$~Guyr.\nThe solid lines in the\nright panels are simple linear fits to the data (equations~\\ref{eq:zF}~\\&~\\ref{eq:Tau},\nsee text for details).}\n\\label{fig:zF}\n\\end{center}\n\\end{figure}\n\nFurthermore, notice in figure~\\ref{fig:AgeMet} the transition in the age\ndistribution around 150 km\/s in velocity dispersion or\n5$\\times$10$^{10}M_\\odot$ in stellar mass. Above this value the ages\nof the populations are overall old and with small scatter among\ngalaxies, whereas lower mass galaxies have a wide range of ages and\nmetallicities (not necessarily younger throughout). The trend is\nrobustly independent of the modelling, as shown both by the CXP fits\n(black dots) and the 2-Burst models (grey diamonds). This threshold is\nreminiscent of the one found by \\citet{kauf03} in the general\npopulation of SDSS galaxies at a stellar mass $3\\times 10^{10}M_\\odot$\nor the threshold in star formation efficiency of late-type galaxies at\nrotation velocities v$_c\\sim$140 km\/s based on their photometry\n\\citep{tfz} or on the presence of dust lanes \\citep{dal04}.\n\n\\begin{figure*}\n\\begin{minipage}{15.5cm}\n\\begin{center}\n\\includegraphics[width=5.2in]{BFPS_f14.eps}\n\\caption{Our proposed model for the star formation history \nof Virgo cluster elliptical galaxies. The parameter estimates suggest\na nearly constant star formation timescale with respect to velocity dispersion,\nwhereas formation \\emph{epoch} is strongly correlated. We find a stronger\ncorrelation with velocity dispersion (compared to stellar mass), so we label\nthe formation scenarios according to $\\sigma$. The bottom panel \nemphasizes the issue that the \\emph{observed} low-mass Es are best fit\nby later formation epochs, an issue which does not rule out the possibility\nof other low-mass Es in the past -- or elsewhere -- to have been formed \nearlier (grey Gaussians).}\n\\label{fig:SFH}\n\\end{center}\n\\end{minipage}\n\\end{figure*}\n\n\n\\section{Conclusions}\n\nWe have revisited the superb, high signal-to-noise spectra of 14\nelliptical galaxies in the Virgo cluster presented by\n\\citet{yam06,yam08}. Our main goal was two-fold: a) to give an optimal\nbut versatile definition of the equivalent width of spectral features\nthat would minimise the contribution from nearby lines, b) to explore \nthe possibility of discriminating between the standard treatment of\ngalaxy spectra either as simple stellar populations or composite models \nwith a distribution of ages and metallicities.\n\nThe former was achieved by defining a ``boosted median'' pseudo-continuum.\nThis method is very easy to implement on any spectral data and it\nimproves on the traditional side-band methods by reducing the final\nuncertainty of the EWs for the same spectra and reducing the\nage-metallicity degeneracy of age- and metal-sensitive lines by\nreducing the effect of unwanted, nearby spectral features.\nThe method only requires two numbers to determine the pseudo-continuum\n(i.e. confidence level of the boosted median and size of the kernel\nover which the analysis is performed). We propose 90\\% and 100\\AA\\\nfor these two parameters when dealing with stellar populations of\ngalaxies at moderate resolution (R$\\sim 1000-2000$).\n\nThe second goal -- going beyond SSPs -- is approached by comparing\nSSPs and three more sets of models which assume some distribution of\nages and metallicities. We find simple populations fail to give a\nconsistent marginalised distribution of ages when fitting\nindependently different age-sensitive lines. We propose either a\n2-Burst model or a $\\tau$ star formatiom history with a\nphenomenological prescription for chemical enrichment (CSP\nmodels). They give similar results, with a clear trend between average\nage and either stellar mass or velocity dispersion, as shown in\nfigure~\\ref{fig:AgeMet}.\n\nAs suggested above when discussing figure~\\ref{fig:AgeHist}, if\nyounger (older) galaxies are better fit by 2-Burst (CXP) models, one\nwould expect that young stars in elliptical galaxies appear in a\nrandom way, and not as a time-coherent, smooth distribution. This\nresult supports minor merging -- possibly involving metal-poor\nsub-systems -- as the main mechanism to generate recent\nstar formation in early-type galaxies \\citep{kav09}. This process\nwill be readily detected in low-mass galaxies, whereas a similar\namount of young stars in a massive galaxy will be harder to detect.\n\nDoubtlessly average ages and metallicities are the observables best\nconstrained by these models. Parameters like formation timescale and\nformation epoch can only be considered ``next-order'' observables,\nwhich will be fraught with larger uncertainties. This is largely\ndue to the massive degeneracies which are present in such parameters\nas mentioned in the introduction. Its is well known that the mass and\nage of a subpopulation are degenerate in that additional mass fraction\ncan compensate for an older age or vice versa \\citep{leo96}. Another\nconsiderable degeneracy also exists between the formation redshift and\nthe star formation timescale for both the EXP and CXP models\n\\citep[see e.g. ][]{gobat08}. While our indices cover a large\nwavelength range we find that these degeneracies are still\nsignificant. Figure~\\ref{fig:Degen} shows the degeneracy between\nparameters for a typical case using 2-Burst models (\\emph{left}) and\nCXP models (\\emph{right}). We take the probability weighted values\nacross the entire parameter space as a more robust measure of these\nparameters. Although the derived parameters are still subject to\nconsiderable uncertainty, as expressed by the error bars shown in\nfigure~\\ref{fig:zF}, we note that our grids sample the parameter space\nquite finely meaning our recovered likelihood distribution should be a\nreasonable approximation to the complete distribution.\n\nWe proceed in a more speculative way, taking the continuous CXP\nmodels at face value and putting some trust on the predictions of their\nformation epochs and timescales. Figure~\\ref{fig:zF} reveals an\nintriguing trend that suggests it is not just formation timescale but\nalso formation epoch which drives the mass-age relationship in early-type\ngalaxies. The solid lines in the top\/bottom right panels gives a simple\nlinear fit to the data, namely:\n\\begin{equation}\nz_F \\sim 1.8 + 3.9\\log\\sigma_{100}\n\\label{eq:zF}\n\\end{equation}\n\\begin{equation}\n\\tau ({\\rm Gyr}) \\sim 0.3 - 1.14\\log\\sigma_{100}\n\\label{eq:Tau}\n\\end{equation}\n\n\\noindent\nwith $\\sigma_{100}$ given in units of 100~km\/s. Figure~\\ref{fig:SFH}\nshows our proposed model for the star formation history of Virgo\ncluster elliptical galaxies. We model the formation histories as\nskewed Gaussians whose spread maps the star formation timescale, and\nwe include a correlation between formation epoch and velocity\ndispersion, as labelled. Notice that the \\emph{observed} low-mass\ngalaxies (black in the bottom panel) are best fit by late formation\nepochs, although this fact does not prevent smaller early-type\ngalaxies in the past to have formed at earlier epochs (represented\nby the grey Gaussians). Our sample is too small and too local to\nexplore this issue further.\n\nOne could argue that the lack of a correlation between formation\ntimescale ($\\tau_1$; \\emph{top}) and mass would contradict the observed\nrelationship between mass and abundance ratios \\citep[see\ne.g. ][]{tm05}. One possibility would involve minor merging of dwarf\ngalaxies. Those mergers will have a more prominent effect on the\nspectra of low-mass galaxies. Dwarf galaxies have a very extended\nperiod of star formation and eject a big fraction of their metals,\nresulting in metal-poor gas with solar or sub-solar abundance ratios\nthat will reduce the observed (luminosity-weighted) [$\\alpha$\/Fe] in\nlow-mass galaxies. Alternatively, notice that the formation epoch\n($z_F$; \\emph{bottom}) correlates quite strongly with velocity\ndispersion, implying that the structures that form the bulk of the\nstars in low-mass galaxies are more likely to be contaminated by the\nejecta from type~Ia supernova, thereby reducing the abundance ratios\nto reproduce the observed correlation between [$\\alpha$\/Fe] and\nmass. A more detailed analysis of the abundance ratios -- although\nbeyond the scope of this paper -- is under way to explore this very\ninteresting possibility.\n\n\n\\begin{figure*}\n\\begin{minipage}{18cm}\n\\includegraphics[width=3.2in]{BFPS_f15a.eps}\n\\includegraphics[width=3.2in]{BFPS_f15b.eps}\n\\caption{Probability distributions of NGC 4489 (left) and NGC 4467\n(right) for 2-Burst and CXP models, respectively. \\emph{Left:} The\ncontour plot shows the degeneracy between the age (t$_{\\rm Y}$) and\nmass fraction (f$_{\\rm Y}$) of the younger component. Contours are\nshown at the 75, 90 and 95\\% confidence level. The solid circle\nidentifies the best fit and quoted error bars. The inset shows the\nprobability distributions of the average age and metallicity, both of\nwhich are well constrained. \\emph{Right Panel} The degeneracy between\nthe formation redshift (z$_{\\rm F}$) and star formation timescale\n($\\tau_1$) is shown in the contour plot (75,90 and 95\\% confidence\nlevels).The probability distribution of the average age and\nmetallicity (inset) show that their values are well constrained.}\n\\label{fig:Degen}\n\\end{minipage}\n\\end{figure*}\n\n\\section*{Acknowledgments}\nWe would like to thank the referee, Ricardo Schiavon, for his very\ninsightful and extensive comments and suggestions. This work has made\nuse of the delos computer cluster in the physics department at King's\nCollege London.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\subsection{Sample Alignments}\nWe manually combed through libraries of HOL Light, PVS, Mizar and Coq to find alignments.\nSpecifically, we picked the mathematical areas of numbers, sets (as well as lists), abstract algebra, calculus, combinatorics, logic, topology, and graphs as a sample.\nThis produced around 900 declarations overall, from which we constructed the interface theories presented in Sect. \\ref{sec:interfaces}.\n\n\\begin{table}[ht]\\centering\\tiny\n\t\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n\t\tInterface & PVS (Standard)& HOL Light (Standard)& Mizar (Standard) & Coq (Standard)\\\\\\hline\n\t\t\\textsf{nat\\_lit} & \\textsf{naturalnumbers?naturalnumber} & \\textsf{nums?nums} & \\textsf{ORDINAL1?modenot.6}& \\textsf{Coq.Init.Datatypes?nat}\\\\\n\t\t\\textsf{succ} & \\textsf{naturalnumbers?succ} & \\textsf{nums?SUC} & \\textsf{ORDINAL1?func.1} & \\textsf{Coq.Init.Nat?succ}\\\\\n\t \t\\textsf{addition} & \\textsf{number\\_fields?+} & \\textsf{arith?ADD} & \\textsf{ORDINAL2?func.10} & \\textsf{Coq.Init.Nat?add}\\\\\n\t\t\\textsf{multiplication} & \\textsf{number\\_fields?*} & \\textsf{arith?MULT} & \\textsf{ORDINAL2?func.11}& \\textsf{Coq.Init.Nat?mul} \\\\\n\t\t\\textsf{lethan} & \\textsf{number\\_fields?$<=$} & \\textsf{arith?$<=$} & \\textsf{XXREAL\\_0?pred.1}& \\textsf{Coq.Init.Nat?leb} \\\\\\hline\n\t\\end{tabular}\n\t\\caption{Alignments to the interface theory \\textsf{NaturalNumbers} (libraries in brackets)}\\label{tbl:natalignments}\n\\end{table}\n\n\\begin{table}[ht]\\centering\\tiny\n\t\\begin{tabular}{|c|c|c|c|c|}\\hline\n\t\tInterface & PVS (NASA\\footnotemark ) & HOL Light (Standard)& Mizar (Standard) &Coq (coq-topology\\footnotemark)\\\\\\hline\n\t\t\\textsf{topology} & \\textsf{topology\\_prelim?topology} & \\textsf{topology?topology} & \\textsf{PRE\\_TOPC?modenot.1}& \\textsf{TopologicalSpaces?TopologicalSpace} \\\\\n\t\t\\textsf{open} & \\textsf{topology?open?} & \\textsf{topology?open\\_in} & \\textsf{PRE\\_TOPC?attr.3} & \\textsf{TopologicalSpaces?open} \\\\\n\t\t\\textsf{closed} & \\textsf{topology?closed?} & \\textsf{topology?closed\\_in} & \\textsf{PRE\\_TOPC?attr.4} & \\textsf{TopologicalSpaces?closed}\\\\\n\t\t\\textsf{interior} & \\textsf{topology?interior} & \\textsf{topology?interior} & \\textsf{TOPS\\_1?func.1}& \\textsf{InteriorsClosures?interior} \\\\\n\t\t\\textsf{closure} & \\textsf{topology?Cl} & \\textsf{topology?closure} & \\textsf{PRE\\_TOPC?func.2 }& \\textsf{InteriorsClosures?closure} \\\\\\hline\n\t\\end{tabular}\n\t\\caption{Alignments to the interface theory \\textsf{Topology} (libraries in brackets)}\\label{tbl:topalignments}\n\\end{table}\n\nFor example, Table \\ref{tbl:natalignments} and~\\ref{tbl:topalignments} show some of these alignments for the interface theories for natural numbers from Figure \\ref{fig:intgraph} and topology. The URIs for PVS consist of the name of the containing theory in PVS (the alignments from the first table are to PVS's prelude, the alignments from the second are to the NASA library) and the name of the symbol within the interface, separated by a question mark\\footnotemark. The ? appearing at the end of some of the URIs are actually a part of the name of the symbol in PVS. The URIs for HOL Light (prelude) and Mizar (mll) consist of the filename and the symbol name within the file. \n\n \\addtocounter{footnote}{-3}\n \\stepcounter{footnote}\\footnotetext{See http:\/\/github.com\/nasa\/pvslib}\\label{fn:nasa}\n \\stepcounter{footnote}\\footnotetext{See http:\/\/github.com\/coq-contribs\/topology}\\label{fn:coq-topology}\n\\stepcounter{footnote}\\footnotetext{The general structure of MMT URIs is \\textsf{$<$Namespace$>$?$<$Theory name$>$?$<$Symbol name$>$}, see e.g. \\cite{RabKoh:WSMSML13}}\nAs mentioned earlier, alignments in topology pose some additional difficulties. Firstly, HOL Light defines a topology on some \\emph{subset} of the universal set of the type, whereas PVS defines it on the universal set of the type directly. Thus, the alignment from HOL Light to the interface theory is unidirectional. Secondly, Mizar does not define the notion of a topology, but instead the notion of a topological space. Therefore, we align them to two different symbols in the interface (\\expr{topology} and \\expr{topological\\_space}) and define \\expr{topological\\_space} based on \\expr{topology}, so that \\textsc{Mmt}\\xspace can still translate expressions based of these two definitions. \n\nThe set of all alignments we found can be inspected at \\url{https:\/\/gl.mathhub.info\/alignments\/Public\/tree\/master\/manual}. \n\\subsection{Alignment Directions}\nFor translation purposes, we distinguish among three kinds of alignments (which are different from the categories in \\cite{MueGauKal:cacfms17}) based on the possible directions of translations between the concepts in the interface theory and the prover library.\n\\paragraph{Bidirectional Alignments} This includes perfect alignments. For example, the translation between the definitions of set union in the interface theory and PVS library is bidirectional:\n\\subparagraph{Interface}\\expr{union\\ :\\ \\{A\\}\\ set\\ A \\to set\\ A \\to set\\ A\\ \\#\\ 2\\cup \\ 3}\n\\subparagraph{PVS}\\expr{union(a,\\ b):\\ set\\ =\\ {x\\ |\\ member(x,\\ a)\\ OR\\ member(x,\\ b)}}\n\\paragraph{Unidirectional Alignments} This includes alignments up to totality of functions, alignments up to certain arguments and alignments up to associativity. For example, in PVS the operator \\expr{+} is used universally for all number fields ($\\mathbb N$, $\\mathbb R$, $\\mathbb C$), but in the interface theory \\expr{plus} is defined for each type. Thus we can only translate from the interface theory to PVS.\n\\paragraph{Other Alignments} Due to the limitation of \\textsc{Mmt}\\xspace's current implementations, there are some alignments which cannot be used at the moment but are potentially directional. For example, the \\expr{>} operator in Mizar is not explicitly declared, but instead defined as the so-called \\expr{antonym} of the \\expr{<=} operator. It is redirected to the \\expr{<=} operator whenever used: \\[\\expr{antonym\\ a\\ >\\ b\\ for\\ a\\ <=\\ b;}\\] However, since \\textsc{Mmt}\\xspace cannot handle alignments up to negation, this cannot be used for translation yet.\n\n\\begin{table}[ht]\\centering\\footnotesize\n\t\\begin{tabular}{|c|c|c|c|c|}\n\t \\hline \n\t Topic & HOL Light & PVS & Mizar & Coq\\\\\n\t\t\\hline \\textsf{Algebra}& \\textsf{0\/0} & \\textsf{18\/1} & \\textsf{17\/0} & \\textsf{14\/0}\\\\\n\t\t\\hline \\textsf{Calculus}& \\textsf{15\/0} & \\textsf{14\/0} & \\textsf{16\/0} & \\textsf{5\/15} \\\\\n\t\t\\hline \\textsf{Categories} & \\textsf{0\/0} & \\textsf{0\/0} & \\textsf{9\/1} & \\textsf{5\/0}\\\\\n\t\t\\hline \\textsf{Combinatorics}& \\textsf{24\/0} & \\textsf{15\/0} & \\textsf{1\/0} & \\textsf{1\/0} \\\\\n\t\t\\hline \\textsf{Complex Numbers} & \\textsf{9\/2} & \\textsf{4\/6} & \\textsf{7\/2} & \\textsf{11\/2}\\\\\n\t\t\\hline \\textsf{Graphs} & \\textsf{5\/5} & \\textsf{17\/0} & \\textsf{20\/0} & \\textsf{7\/2}\\\\\n\t\t\\hline \\textsf{Integers}& \\textsf{10\/0 }& \\textsf{0\/0 }& \\textsf{5\/2} & \\textsf{47\/3} \\\\\n\t\t\\hline \\textsf{Lists}& \\textsf{16\/0} & \\textsf{9\/0} & \\textsf{8\/0} & \\textsf{36\/2} \\\\\n\t\t\\hline \\textsf{Logic}& \\textsf{7\/0} & \\textsf{7\/5} & \\textsf{7\/0}& \\textsf{24\/1} \\\\\n\t\t\\hline \\textsf{Natural Numbers} & \\textsf{19\/0} & \\textsf{8\/10} & \\textsf{9\/0}& \\textsf{34\/1} \\\\\n\t\t\\hline \\textsf{Polynomials} & \\textsf{4\/0} & \\textsf{1\/0} & \\textsf{7\/0}& \\textsf{0\/0} \\\\\n\t\t\\hline \\textsf{Rational Numbers} & \\textsf{0\/14} & \\textsf{2\/11} & \\textsf{0\/10}& \\textsf{14\/3} \\\\\n\t\t\\hline \\textsf{Real Numbers} & \\textsf{13\/2} & \\textsf{3\/10} & \\textsf{7\/4} & \\textsf{12\/2}\\\\\n\t\t\\hline \\textsf{Relations} & \\textsf{4\/0} & \\textsf{16\/5} & \\textsf{18\/3} & \\textsf{1\/12}\\\\\n\t\t\\hline \\textsf{Sets} & \\textsf{23\/0} & \\textsf{28\/0} & \\textsf{18\/0} & \\textsf{19\/0}\\\\\n\t\t\\hline \\textsf{Topology} & \\textsf{15\/0} & \\textsf{10\/0} & \\textsf{9\/0}& \\textsf{17\/1} \\\\\n\t\t\\hline \\textsf{Vectors} & \\textsf{13\/0} & \\textsf{7\/0} & \\textsf{15\/0}& \\textsf{0\/0} \\\\\n\t\t\\hline \\textbf{Sum} & \\textbf{177\/23} & \\textbf{159\/48} & \\textbf{173\/22}& \\textbf{240\/42} \\\\ \n\t\t\\hline \n\t\\end{tabular}\n\t\\caption{Number of bidirectional\/unidirectional alignments per library}\\label{tbl:alignmenttotal}\n\\end{table}\n\n\n\\subsection{Causes for Imperfect Alignments}\nThe most common causes for imperfect alignments are subtyping and partiality.\n\\paragraph{Subtyping} In the PVS library, the arithmetic operations on all the number fields are defined on a common supertype \\expr{numfields}. Therefore, a translation from PVS to the other two languages may not be viable. \n\\paragraph{Partiality} The result of division by zero in the libraries of HOL Light and Mizar is defined as zero; in PVS, however, the divisor must be nonzero. Therefore, certain theorems in HOL Light and Mizar involving division no longer hold in PVS, and the translation is unidirectional.\n\n\n\\paragraph{}\nIn order to translate an expression from one library to another, the concepts in the expression must at least exist in both libraries. This creates the need to inspect the intersection of the concepts in these libraries.\nTable~\\ref{tbl:intersectionConcepts} gives an overview of the library intersection for various interface theories.\n\\begin{table}[ht]\\centering\\footnotesize\t\n\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\\hline Topic & 1 System & 2 Systems & 3 Systems & 4 Systems\\\\\n\t\t\\hline \\textsf{Algebra}& \\textsf{17} & \\textsf{9} & \\textsf{5} & \\textsf{0}\\\\\n\t\t\\hline \\textsf{Calculus}& \\textsf{35} & \\textsf{7} & \\textsf{8} & \\textsf{0} \\\\\n\t\t\\hline \\textsf{Categories} & \\textsf{4} & \\textsf{5} & \\textsf{0} & \\textsf{0}\\\\\n\t\t\\hline \\textsf{Combinatorics}& \\textsf{25} & \\textsf{6} & \\textsf{0} & \\textsf{1} \\\\\n\t\t\\hline \\textsf{Complex Numbers} & \\textsf{10} & \\textsf{5} & \\textsf{3} & \\textsf{3}\\\\\n\t\t\\hline \\textsf{Graphs} & \\textsf{72} & \\textsf{6} & \\textsf{3} & \\textsf{0}\\\\\n\t\t\\hline \\textsf{Integers}& \\textsf{52 }& \\textsf{2 }& \\textsf{7} & \\textsf{0} \\\\\n\t\t\\hline \\textsf{Lists}& \\textsf{28} & \\textsf{8} & \\textsf{9} & \\textsf{0} \\\\\n\t\t\\hline \\textsf{Logic}& \\textsf{18} & \\textsf{0} & \\textsf{2}& \\textsf{5} \\\\\n\t\t\\hline \\textsf{Natural Numbers} & \\textsf{53} & \\textsf{2} & \\textsf{10}& \\textsf{2} \\\\\n\t\t\\hline \\textsf{Polynomials} & \\textsf{12} & \\textsf{0} & \\textsf{0}& \\textsf{0} \\\\\n\t\t\\hline \\textsf{Rational Numbers} & \\textsf{11} & \\textsf{4} & \\textsf{2}& \\textsf{7} \\\\\n\t\t\\hline \\textsf{Real Numbers} & \\textsf{9} & \\textsf{3} & \\textsf{5} & \\textsf{5}\\\\\n\t\t\\hline \\textsf{Relations} & \\textsf{21} & \\textsf{15} & \\textsf{4} & \\textsf{0}\\\\\n\t\t\\hline \\textsf{Sets} & \\textsf{56} & \\textsf{10} & \\textsf{9} & \\textsf{10}\\\\\n\t\t\\hline \\textsf{Topology} & \\textsf{62} & \\textsf{2} & \\textsf{8}& \\textsf{0} \\\\\n\t\t\\hline \\textsf{Vectors} & \\textsf{25} & \\textsf{5} & \\textsf{0}& \\textsf{0} \\\\\n\t\t\\hline \\textbf{Sum} & \\textbf{510} & \\textbf{91} & \\textbf{79}& \\textbf{33} \\\\ \n\t\t\\hline \n\t\\end{tabular}\n\t\\caption{Number of concepts found in exactly one, two, three or four systems}\n\t\\label{tbl:intersectionConcepts}\n\\end{table}\n\n\n\n\\section{More Examples of Interface Theories}\\label{sec:interface_examples}\n \\begin{figure}[ht]\\centering\n \\fbox{\\includegraphics[width=0.9\\textwidth]{analysis}}\n \\caption{An interface theory for analysis}\\label{fig:anatheory}\n\\end{figure}\n\n\\begin{figure}[ht]\\centering\n \\fbox{\\includegraphics[width=0.7\\textwidth]{setstheory}}\n \\caption{An interface theory for typed sets}\\label{fig:setstheory}\n\\end{figure}\n\n\n\n\\subsection{Example: Natural Numbers}\n\nIn Mizar~\\cite{mizar}, which is based on Tarski-Grothendieck set theory, the set of natural numbers \\expr{NAT} is defined as \\textsf{omega}, the set of finite ordinals. Arithmetic operations are defined directly on those.\n\nIn contrast, in PVS~\\cite{OwRu92} natural numbers are defined as a specific subtype of the integers, which in turn are a subtype of the rationals etc. up to an abstract type \\textsf{number} which serves as a maximal supertype to all number types. The arithmetic operations are inherited from a subtype \\textsf{number\\_field} of \\textsf{number}.\n\nThese are two fundamentally different approaches to describe and implement an abstract mathematical concept, but for all practical purposes the concept they describe is the same; namely the natural numbers. The interface theory for both variants would thus only contain the symbols that are relevant to the abstract concept itself, independent of their specific implementation -- hence, things like the type of naturals, the arithmetic operations and the Peano axioms. The interface theory thus provides everything we need to work with natural numbers, and at the same time everything we know about them independently of the logical foundation or their specific implementation within any given formal system.\n\n\\paragraph{}However, there is an additional layer of abstraction here, namely that in stating that the natural numbers in Mizar are the finite ordinals we have already ignored the system dialect (in the sense of p.\\ref{lbl:dialect}). This step of abstraction (from the concrete definition using only Mizar-specific symbols) yields another interface theory for finite ordinals, which in turn can be aligned not just with Mizar natural numbers, but also e.g. with MetaMath~\\cite{MetaMath:on}, which is built on ZFC set theory.\n\nFigure \\ref{fig:intgraph} illustrates this situation. Blue arrows point from more detailed theories to their interfaces. The arrows from PVS or Mizar to interfaces merely strip away the system dialects; the arrows within Interfaces abstract away more fundamental differences in definition or implementation.\n\n\\begin{figure\n\\begin{center}\n\\resizebox{0.6\\textwidth}{!}{\n\\begin{tikzpicture}\n\n\\node (PVS) at (1,8.5) {PVS};\n\n\\node (PVNat) at (1,7) {\\textsf{Nat}};\n\\node (PVdots) at (1,5) {$\\vdots$};\n\\node (PVnf) at (1,3) {\\textsf{number\\_field}};\n\\draw[\\arrowtipmono-\\arrowtip] (PVnf) -- (PVdots);\n\\draw[\\arrowtipmono-\\arrowtip] (PVdots) -- (PVNat);\n \\node[draw,thick,fit=(PVNat) (PVdots) (PVnf),\n rounded corners=.55cm, inner sep=5pt] (PVScloud) {};\n\n\\node (Int) at (6,8.5) {Interfaces};\n\n\\node (ST) at (7.6,4.2) {Ordinals};\n\n\\node (Reals) at (4.7,4.8) {\\textsf{Numbers}};\n\\node (Nat) at (4.7,6) {$\\mathbb N$};\n\\draw[\\arrowtipmono-\\arrowtip] (Reals) -- (Nat);\n\n\\node (FOrd) at (7.2,5.5) {Finite Ordinals};\n\n\\node (PA) at (6,7) {Peano Axioms};\n\\draw[->,blue] (FOrd) -- (PA);\n\\draw[->,blue] (Nat) -- (PA);\n\\draw[\\arrowtipmono-\\arrowtip] (ST) -- (FOrd);\n\n \\node[draw,thick,fit=(ST) (Reals) (Nat) (FOrd) (PA),\n rounded corners=.55cm, inner sep=10pt] (PVScloud) {};\n\n\\node (Miz) at (11,8.5) {Mizar};\n\n\\node (MNat) at (11,7) {\\textsf{Nat}};\n\\node (MOrd) at (11,3) {\\textsf{Ordinals}};\n\\draw[\\arrowtipmono-\\arrowtip] (MOrd) -- (MNat);\n\n \\node[draw,thick,fit=(MNat) (MOrd),\n rounded corners=.55cm, inner sep=5pt] (PVScloud) {};\n\n\\draw[->,blue] (MOrd) -- (ST);\n\\draw[->,blue] (PVNat) -- (Nat);\n\\draw[->,blue] (PVnf) -- (Reals);\n\\draw[->,blue] (MNat) -- (FOrd);\n\n\n\\end{tikzpicture}\n}\n\\end{center}\n\\caption{A graph showing different theories for natural numbers}\\label{fig:intgraph}\n\\end{figure}\n\n\\paragraph{} Consider again Figure \\ref{fig:natinterface}, a possible interface theory for natural numbers. Note, that symbols such as \\textsf{leq} \\emph{could} be defined, but don't actually need to be. Since they are only interfaces, all we need is for the symbols to exist. \n \n In fact, the more abstract the interface, the less we \\emph{want} to define the symbols -- given that there's usually more than one way to define symbols, definitions are just one more thing we might want to abstract away from completely.\n \n The symbols in this interface theory can then be aligned either with symbols in other formal systems directly, or with additional interfaces in between, such as a theory for Peano arithmetic, or the intersection of all inductive subsets of the real numbers, or finite ordinals or any other possible formalization of the natural numbers.\n\n\\subsection{Additional Interface Theories}\n\nThe foundation independent nature of \\textsc{Mmt}\\xspace allows us to implement interface theories with almost arbitrary levels of detail and for vastly different foundational settings.\n\nWe have started a repository of interface theories specifically for translation purposes~\\cite{MitMInter:on} and also aligned to already existing interfaces (as in the case of arithmetics, see below) in a second and third \\textsf{MathHub}\\xspace repository~\\cite{MitMsmglom:on} and~\\cite{MitMFoundation:on} extending them when necessary. \nCrucially, this interface repository contains interface theories for basic type-related symbols like the function type constructors (see Figure \\ref{fig:typeinterface}\\footnote{This interface theory, like most formalizations of foundations, uses types as terms (via the symbols \\expr{tp} and \\expr{tm}), whereas the interface theories above (like \\expr{NaturalNumbers}) use the universe of types provided by the logical framework directly. During translation, special \\emph{higher-order abstract syntax rules} take care of eliminating or inserting the corresponding symbols appropriately to make aligning between the two formalization levels possible.}), that are aligned with the respective symbols in HOL Light and PVS. These symbols are so basic as to be primitive in systems based on type theory, and consequently they occur in the vast majority of expressions. To have these symbols aligned is strictly necessary to get any further use of alignments off the ground.\n\n\\begin{figure}[ht]\\centering\n \\fbox{\\includegraphics[width=0.6\\textwidth]{typeinterface}}\n \\caption{Interface theories for type-theoretical foundations\\label{fig:typeinterface}}\n\\end{figure}\n\nHere, a \\emph{structure} is used to include the theory for simple function types in the theory for dependent function types, while providing definitions for the symbols in terms of the latter. This automatically yields a translation from the simple to the dependent variant. \n\nTable~\\ref{tbl:alignmenttotal} shows the total number of alignments we found for PVS, HOL Light and Mizar. The following are some additional examples of mathematical areas covered by the current interface theories:\n\n \\begin{itemize}\n \\item[\\textbf{Calculus}] contains the following 3 subinterfaces:\n \\begin{itemize}\n \t\\item[\\textbf{Limits}] currently contains 17 symbols related to sequences and limits, including \\expr{metric\\_space} and \\expr{complete} (metric spaces and their completeness).\n \t\\item[\\textbf{Differentiation}] currently contains 4 symbols, namely differentiability in a point and on a set and the derivative in a point and as a function\n \t\\item[\\textbf{Integration}] currently contains 6 symbols, namely integrability and the integral over a set for Riemann, Lebesgue and Gauge-integration.\n \\end{itemize}\n \\item[\\textbf{Arithmetics}] is an already existing interface theory from ~\\cite{MitMsmglom:on}. It contains the interfaces for below number arithmetics (each split into two interfaces for the basic number type definitions and the arithmetics on them). \n \\begin{itemize}\n \\item[\\textbf{Complex Numbers}] currently contains 11 symbols for complex numbers aligned to their counterparts in HOL Light, PVS and Mizar. Besides the usual arithmetic operations similar to \\expr{NaturalNumbers}, it contains \\expr{i} (the imaginary unit), \\expr{abs} (the modulus of a complex number) and \\expr{Re}, \\expr{Im} (the real and imaginary parts of a complex number).\n \\item[\\textbf{Integers}] currently contains 9 symbols for the usual arithmetic operations on integers and for comparison between two integers.\n \\item[\\textbf{Natural Numbers}] currently contains 21 symbols and is already described above. \n \\item[\\textbf{Real Numbers}] currently contains 15 symbols, again very similar in nature to the other number spaces.\n \\end{itemize}\n \\item[\\textbf{Lists}] currently contains the 13 most important symbols for lists, including \\expr{head}, \\expr{tail}, \\expr{concat}, \\expr{length} and \\expr{filter} (filter a list using another list) as well as some auxiliary definitions.\n There are no lists in Mizar, instead finite sequences are used. These however deserve their own interface.\n \\item[\\textbf{Logic}] is an already existing interface in the~\\cite{MitMFoundation:on} repository. It contains 9 symbols for boolean algebra that are all perfectly aligned to HOL Light, PVS and Mizar the like and sometimes also to Coq.\n \\item[\\textbf{Sets}] is again an already existing interface theory from ~\\cite{MitMsmglom:on} split into many subtheories. Currently, 28 of the contained symbols have been aligned. sets contains symbols for typed sets as in a type theoretical setting, including axioms and theorems. Here we have the most alignments so far. It also contains the following two interfaces:\n \\begin{itemize}\n \t\\item[\\textbf{Relations}] currently contains 23 symbols for alignments to relations and their properties, including orders. \n \t\\item[\\textbf{Functions}] currently contains 7 symbols for alignments to functions and their relations, that are not already contained in relations. \n \\end{itemize}\n \\item[\\textbf{Topology}] currently contains 25 symbols for both general topological spaces as well as the standard topology on $\\mathbb R^n$ specifically. Since this yields additional difficulties, it will be examined in more detail in the next section.\n \\end{itemize}\n \\paragraph{} As additional examples, the interface theories for limits and sets can be found in Appendix \\ref{sec:interface_examples}, Figure \\ref{fig:anatheory} and Figure \\ref{fig:setstheory}.\n\n\n\n\n\\subsection{Alignments}\n\n\\paragraph{} We speak of \\emph{alignment} if the same (or a closely related) concept occurs in different libraries, possibly with slightly different names, notations, or formal definitions. Two aligned symbols thus can be thought to represent ``the same\" abstract mathematical concept while differing only in implementation details. The notion of alignments is intentionally broad as to cover a wide variety of ways in which two symbols can be seen as ``morally the same\".\n\nFor example, consider untyped and typed equality, e.g. in a set-theoretical and a dependently typed language respectively:\n\\begin{align*}\n\\textsf{eq}_1 &: \\textsf{Set}\\to\\textsf{Set}\\to\\textsf{bool} \\\\\n\\textsf{eq}_2 &:(T:\\textsf{Type})\\to T\\to T\\to \\textsf{bool}.\n\\end{align*}\n\nObviously, these two symbols are not easily interchangeable since they don't even have the same type; nevertheless, if we assume either symbol belonging to a distinct formal system, they clearly represent the same mathematical concept (namely \\emph{equality}). Furthermore, if we want to translate any expression from one system to the other, we will clearly need to replace occurrences of one kind of equality by the respective other one.\n\nIt thus makes sense to think of both equalities as being \\emph{aligned}.\n\n\nFor example, the alignment between $\\textsf{eq}_1$ and $\\textsf{eq}_2$ can be given as\n\t\\[\\textsf{system1:eq}_1\\textsf{ system2:eq}_2\\textsf{ arguments=``(1,2)(2,3)\" direction=``both\"}\\]\nFor details on alignments we refer to~\\cite{MueGauKal:cacfms17}, where alignments are classified, discussed and their implementations in \\textsc{Mmt}\\xspace are described in detail.\n\n\\subsection{The \\textsc{Mmt}\\xspace Framework}\n\n\\textsc{Mmt}\\xspace \\cite{RabKoh:WSMSML13,rabe:howto:14} is a wide-coverage representation language for formal mathematical knowledge.\nIt can be seen as a triple of the fragment of OMDoc \\cite{Kohlhase:OMDoc1.2} dealing with logical and related knowledge, a rigorous semantics for that fragment, and a mature implementation.\n\\omdoc\/\\mmt is designed to\n\\begin{compactitem}\n \\item avoid a commitment to any particular foundational type system or logic,\n \\item allow for highly modular representations of foundational systems or domain knowledge,\n \\item support interoperability across foundations, tools, and libraries.\n\\end{compactitem}\nThat makes it an ideal choice for describing interface theories.\n\nIn the sequel, we explain by example the key features of \\textsc{Mmt}\\xspace that are relevant for our purposes.\nFigure \\ref{fig:mmtlogic} shows an example theory in \\textsc{Mmt}\\xspace surface syntax that can serve as interface for basic logical constants.\n\\begin{figure}[ht]\\centering\n \\fbox{\\includegraphics[width=0.45\\textwidth]{logic}}\n \\caption{An interface theory for logic in \\textsc{Mmt}\\xspace}\\label{fig:mmtlogic}\n\\end{figure}\n\nHere, a new theory \\expr{Logic} is declared with URI \\textsf{http:\/\/mathhub.info\/MitM\/Foundation?Logic}, which is composed of the namespace declared in the first line and the name of the theory. Afterwards, three constants are declared and given a type; namely the type of booleans \\expr{bool} of type \\expr{type}. The symbol \\expr{type} is provided via a logical framework. The constants \\expr{true} and \\expr{false} are declared to be of type \\expr{bool}.\n\nIn general, \\textsc{Mmt}\\xspace constants have the form \\expr{c [:TYPE] [=DEF] [\\#NOT]}, though we will not need definitions in this paper. The individual components (type, definition, notation, respectively) are all optional and can be provided in any order.\n\nThe symbol \\expr{ded} will serve as a function from propositions to the type of their proofs to make use of the Curry-Howard correspondence. It is given the appropriate type \\expr{bool \\to type} and a notation \\expr{\\vdash\\textsf{ 1 }}. The latter allows for writing \\expr{\\vdash\\textsf A} for the type of proofs of a proposition $A$. Furthermore, the notation is provided with a precedence, to allow for omitting brackets.\n\nThe next symbols provide a typed equality and the basic logical connectives. The curly braces denote the dependent function type, providing for example \\expr{eq} with the type $\\prod_{\\textsf{A:type}}\\textsf A\\to\\textsf A\\to\\textsf{bool}$. The notation \\expr{2 \\doteq 3} omits the first argument \\expr{A:type}, leaving it implicit and to be inferred by the system.\n\n\\paragraph{}\nWe can use this theory as one (out of many possible) meta-theory for various interface theories.\n\\begin{figure}[ht]\\centering\n \\fbox{\\includegraphics[width=0.5\\textwidth]{natinterface}}\n \\caption{An (excerpt of an) interface theory for natural numbers with \\textsf{Logic} as meta-theory}\\label{fig:natinterface}\n\\end{figure}\nFigure \\ref{fig:natinterface} shows an example of a simple theory of natural numbers, using the theory \\expr{Logic} and the symbols declared therein.\n\n\\paragraph{} Using this modular approach as well as the foundation-independent nature of \\omdoc\/\\mmt, the core primitives of various formal systems (such as interactive theorem provers)\ncan be (and in some cases have been, see e.g. \\cite{OAF:on} or \\cite{KohRab:qrtpflmk15} for the big picture) represented in \\omdoc\/\\mmt alongside their libraries by using a formalization of the former as \nmeta-theory for the theories in the latter, making either equally accessible to the system. This is what makes \\textsc{Mmt}\\xspace a ``meta-system'' suitable for our purposes, instead of the $n+1$st competing standard.\n\n\n\n\n\\section{Introduction and Related Work}\\label{sec:intro}\n \\input{intro}\n \n \\section{Preliminaries}\\label{sec:mmt}\n \\input{mmt}\n \n \\section{Interface Theories}\\label{sec:interfaces}\n \\input{interfaces}\n \n \\section{Alignments}\\label{sec:alignments}\n \\input{alignments}\n \n \n \\section{An Implementation of Alignment-based Translations}\\label{sec:implementations}\n \\input{translations}\n \\paragraph{}\\input{implementation}\n\n\\section{Conclusion}\\label{sec:conclusion}\n\\input{conclusion}\n\\bibliographystyle{eptcsalpha}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Calculation of the connectivity and average contour length}\\label{appendix_connectivity}\n\nTo estimate the connectivity of a given assembled structure, we consider a reduced network defined by (1) merging each pair of crosslinked \"linker\" nodes into a single vertex and then (2) merging each filament section separating two vertices into a single edge. This is sketched in Fig. \\ref{figS1}. \n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Fig_S1.pdf}\n\\caption{ \\label{figS1} To calculate a network's average connectivity $z$, we consider a reduced network structure that treats each crosslink as a node and each inter-crosslink (non-dangling) filament section as an edge. On the right, a few example nodes in the reduced network are labeled according to their connectivity. \n}\n\\end{figure}\n\n\n\\section{Parameters}\\label{appendix_parameters}\n\nIn the interest of drawing comparison to experimental results in Ref. \\citenum{gardel_elastic_2004}, we have chosen parameter values roughly appropriate for F-actin in laboratory conditions. Specific values are listed in Table \\ref{table1}.\n\n\\begin{table}[htb!]\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{quantity} & \\textbf{symbol} & \\textbf{value} \\\\ \n\\hline\nnumber of filaments & $N_f$ & $500$ \\\\\nfilament length & $\\ell_f$ & $9\\;\\si{\\micro\\meter}$ \\\\\nfilament length per volume & $\\rho$ & $2.6\\;\\si{\\micro\\meter}^{-2}$ \\\\\nnumber of nodes per filament & $n$ & $10$ \\\\\n\nfilament persistence length & $\\ell_p$ & $17\\;\\si{\\micro\\meter}$ \\\\\ncrosslink rest length & $\\ell_{0,c\\ell}$ & $0.2\\;\\si{\\micro\\meter}$ \\\\\nfilament stretching rigidity & $\\mu$ & $588\\;\\si{\\pico\\newton}$ \\\\\nthermal energy scale & $k_B T$ & $4.11\\times 10^{-3}\\;\\si{\\pico\\newton}\\cdot\\si{\\micro\\meter}$ \\\\\nsolvent viscosity & $\\eta_s$ & $0.001\\;\\si{\\pico\\newton}\\cdot\\si{\\micro\\meter}^{-2}\\cdot \\si{s}$\\\\\ntimestep & $\\Delta t$ & $9.42\\times10^{-7}\\;\\mathrm{s}$ \\\\\ntotal time, assembly & $\\tau_\\mathrm{a}$ & $60\\;\\mathrm{s}$ \\\\ \ntotal time, rheology & $\\tau_\\mathrm{tot}$ & $30\\;\\mathrm{s}$ \\\\ \n\\hline\n\\end{tabular}\n\\caption{ \\label{table1} Independent parameters, in real units.}\n\\end{table}\n\n\\begin{table}[htb!]\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{quantity} & \\textbf{symbol} & \\textbf{value} \\\\ \n\\hline\nfilament bond rest length & $\\ell_0=\\ell_f\/(n-1)$ & $1\\;\\si{\\micro\\meter}$ \\\\\nsimulation box edge length & $L=(N_f\\ell_f\/\\rho)^{1\/3}$ & $12\\;\\si{\\micro\\meter}$ \\\\\nfilament bending rigidity & $\\kappa = k_B T \\ell_p$ & $6.99\\times10^{-2}\\;\\si{\\pico\\newton}\\cdot\\si{\\micro\\meter}^2$ \\\\\n\\hline\n\\end{tabular}\n\\caption{ \\label{table2} Parameter-dependent quantities.}\n\\end{table}\n Assuming a molecular mass of $42 \\;\\si{\\kilo\\dalton}$ for actin\\cite{ballweber_polymerisation_2002}, the mass per length is $\\sim 16\\times10^3\\; \\si{\\kilo\\dalton}\/\\si{\\micro\\meter}$, which translates to $0.63\\;\\si{\\micro \\mol}\/\\si{\\micro\\meter}$. Therefore, our chosen length density of $\\rho=2.6\\;\\si{\\micro\\meter}^{-2}$ corresponds to an actin concentration of $c_A = 1.63\\;\\si{\\micro M}$.\n\nQuantities used in simulations are nondimensionalized by characteristic length, force, and drag coefficient values: $\\ell^*=1\\;\\si{\\micro\\meter}$, $f^*=1\\;\\si{\\pico\\newton}$, and $\\zeta^*=6\\pi\\eta_s(\\ell_0\/2)=9.42\\times10^{-3}\\;\\si{\\pico\\newton}\\cdot\\si{\\micro\\meter}\\cdot \\mathrm{s} $ (for example, the simulation timestep is $\\Delta t_\\mathrm{sim}=\\Delta t\/(\\zeta^* \\ell^*\/f^*)=10^{-4}$. Note that one can define a dimensionless bending rigidity $\\tilde{\\kappa}=\\kappa\/(\\mu\\ell_0^2)$, as in past work.\\cite{sharma_strain-controlled_2016} With our parameters, $\\tilde{\\kappa}=1.18\\times10^{-4}$.\n\n\n\n\\section{Fits of K to the expected scaling form}\\label{appendix_fits}\nWe assume that the differential shear modulus $K=\\partial\\sigma\/\\partial\\gamma$ can be written as\n\\begin{equation} \\label{Eq1}\nK = a\\mathcal{K}(\\tilde{\\kappa},\\gamma)\n\\end{equation}in which $a$ is some prefactor with units of stress, $\\tilde{\\kappa}$ is a dimensionless bending rigidity, and $\\mathcal{K}$ is a dimensionless function of strain that scales with the distance to a critical strain $\\gamma_c$ as\\cite{sharma_strain-controlled_2016}\n\\begin{equation}\n\\mathcal{K} \\propto|\\gamma-\\gamma_c|^f\\mathcal{G}_\\pm\\left(\\frac{\\tilde{\\kappa}}{|\\gamma-\\gamma_c|^\\phi}\\right).\n\\end{equation}This scaling is reproduced by the constitutive equation\n\\begin{equation}\n\\frac{\\tilde{\\kappa}}{|\\gamma-\\gamma_c|^\\phi} \\sim \\frac{\\mathcal{K}}{|\\gamma-\\gamma_c|^f}\\left(\\pm1 + \\frac{\\mathcal{K}^{1\/f}}{|\\gamma-\\gamma_c|}\\right)^{\\phi-f}\n\\end{equation}\nin which the plus and minus correspond to the regions below and above the critical strain, respectively.\nWe compute $\\mathcal{K}(\\gamma)$ by numerical inversion of $\\tilde{\\kappa}\\left(\\mathcal{K}\\right)$:\n\\begin{equation}\\label{eq4} \\tilde{\\kappa}=b\\mathcal{K}\\left(\\pm|\\gamma-\\gamma_c|+\\mathcal{K}^{1\/f}\\right)^{\\phi-f},\n\\end{equation}\nsetting $b=1$. Fit parameters for the data in Fig. 7a in the main text are shown in Fig. \\ref{figS2}.\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Fig_S2.pdf}\n\\caption{ \\label{figS2} Best-fit parameters for systems with varying crosslinker coverage fraction $p$, corresponding to the solid curves in Fig. 7a in the main text.}\n\\end{figure}\n\n\n\\bibliographystyle{apsrev}\n\n\\section{Introduction\n}\n\nThe formation of living things involves the energy-intensive assembly of complex structures from scarce resources. Yet, the demanding physical processes of life require robust materials that remain intact and functional under significant and often repetitive applied stresses and strains. Quasi-one-dimensional or filamentous structures address these challenges efficiently by supporting significant tensile stresses with minimal material cost.\\cite{burla_mechanical_2019} Biological polymers and fibers are generally also semiflexible,\\cite{janmey_mechanical_1991} meaning that they resist modes of deformation that induce bending, such as applied compression. In examples spanning a wide range of length scales, including information-storing DNA and RNA, actin and intermediate filaments in the cell cytoskeleton, and extracellular collagen and elastin fibers in tissues, these mechanical features consistently play crucial roles in biological function. \n\nThe cytoskeleton and extracellular matrix are quintessential examples of disordered networks, a common class of higher-order structure in living materials. Building upon the qualities of their underlying semiflexible filaments, these networks act as responsive elastic scaffolds that resist potentially damaging deformation while leaving ample space for the transport and storage of functional components, such as interstitial fluids and cells. Unlike conventional elastic solids, their mechanical properties are scale-dependent\\cite{tyznik_length_2019,proestaki_modulus_2019} and extremely sensitive to changes in applied stress or strain,\\cite{onck_alternative_2005,picu_poissons_2018,ban_strong_2019} to which they respond with dramatic stiffening, alignment, and changes in local filament density, enabling essential biological phenomena such as long-range force transmission by cells\\cite{notbohm_microbuckling_2015,wang_long-range_2015,han_cell_2018,alisafaei_long-range_2021,grill_directed_2021} and the contraction of muscle fibers.\\cite{huxley_mechanism_1969,gunst_first_2003} Recent work has suggested that the nonlinear viscoelastic properties of these and related fibrous networks are governed by an underlying mechanical phase transition\\cite{sharma_strain-controlled_2016,jansen_role_2018} associated with the onset of stretching-dominated rigidity under tension-generating (i.e. shear or extensile) applied strains. Within this framework, elastic properties are predicted to exhibit a power law dependence on applied strain in the vicinity of a critical strain, and growing nonaffine (inhomogeneous) strain fluctuations are expected to drive a significant slowing of stress relaxation as the transition is approached.\\cite{shivers_nonaffinity_2022} The location of this transition (the magnitude of the critical strain) depends sensitively on key features of the underlying network architecture.\\cite{wyart_elasticity_2008, broedersz_filament-length-controlled_2012}\n\nExtensive simulation-based studies have explored the nonlinear rheological properties of disordered networks of crosslinked stiff or semiflexible polymers,\\cite{onck_alternative_2005,picu_mechanics_2011,carrillo_nonlinear_2013,freedman_versatile_2017} in some cases with realistic three-dimensional geometries produced either by physical assembly processes\\cite{huisman_three-dimensional_2007,kim_computational_2009,dobrynin_universality_2011,muller_rheology_2014,muller_resolution_2015,zagar_two_2015,debenedictis_structure_2020,grill_directed_2021} or artificial generation procedures.\\cite{huisman_monte_2008,lindstrom_biopolymer_2010,huisman_internal_2011,huisman_frequency-dependent_2010,lindstrom_finite-strain_2013,amuasi_nonlinear_2015} However, efforts to specifically connect network structure with strain-controlled critical behavior have typically focused on simplified random spring networks.\\cite{wyart_elasticity_2008,vermeulen_geometry_2017,jansen_role_2018,rens_micromechanical_2018,merkel_minimal-length_2019,shivers_normal_2019,shivers_nonlinear_2020,burla_connectivity_2020,burla_stress_2019,dussi_athermal_2020,tauber_stretchy_2022} In spring networks, the critical strain coincident with the onset of stretching-dominated mechanics can be tuned by changing the average connectivity $z$, defined as the average number of bonds joined at each network junction.\\cite{wyart_elasticity_2008} For a network of crosslinked filaments, $z$ is controlled by the typical number of crosslinks formed per filament, approaching an upper bound of $z\\to 4$ in the high crosslink-density limit.\\cite{broedersz_filament-length-controlled_2012} In biopolymer gels, small changes in the concentration of available crosslinkers can drive dramatic changes in rheological properties,\\cite{broedersz_modeling_2014} including changes in the linear elastic modulus \\cite{piechocka_multi-scale_2016} and shifts in the critical strain corresponding to the onset of stretching-dominated mechanics.\\cite{gardel_elastic_2004} These changes are naively consistent with a tendency of the connectivity to increase with crosslinker concentration. However, the quantitative relationships between crosslinker concentration and connectivity, and the structural characteristics of assembled networks more generally, remain mostly unknown. Improving our knowledge of how the concentration-dependent microscopic structural details of self-assembled networks translate into strain-dependent macroscopic rheological properties is essential for understanding the forces at play in important biological processes, such as wound healing and cancer metastasis, and for the design of biomimetic synthetic materials.\\cite{romera_strain_2017}\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_1.pdf}\n\\caption{ \\label{fig_1} Model system and rheological approach. (a) Coarse-grained semiflexible filaments are decorated with randomly assigned sticky sites (light blue spheres in the image on the right) with coverage fraction $p$. When two sticky sites meet, they are connected by a permanent crosslink (royal blue dumbbell). Solutions of these filaments diffusively self-assemble into percolating disordered networks with macroscopic elasticity. (b) After network assembly, the structure is fixed and the strain-dependent rheological properties are determined from the fluctuating shear stress $\\sigma(\\gamma, t)$ measured under constant macroscopic simple shear strain $\\gamma$.}\n\\end{figure}\n\nIn this study, we consider a system composed of coarse-grained semiflexible filaments that diffusively self-assemble into a system-spanning network through the formation of permanent inter-filament crosslinks. We begin with randomly positioned free filaments with a specified coverage fraction $p$ of \"sticky\" or linker-decorated sites. The linker coverage fraction serves as a proxy for the ratio of crosslinker and filament concentrations in a real system. Allowing diffusive motion to proceed, we add permanent crosslinks (short elastic bonds) between sticky sites whose pairwise distance decreases beneath a designated crosslink formation distance. After the rate of diffusion-driven formation of new crosslinks becomes sufficiently slow, we halt the assembly process and analyze the structure of the fixed network, measuring the dependence of both the connectivity and the average inter-crosslink contour length on the linker coverage fraction.\n\nWe then transition to the rheology stage, in which we observe the network's steady state behavior under constant simple shear strain $\\gamma$. We obtain time-series measurements of the thermally fluctuating shear stress $\\sigma(\\gamma, t)$ in the mechanically equilibrated state, as shown schematically in Fig. \\ref{fig_1}b. Repeating these measurements over a range of strains for each set of input parameters, we compute relevant elastic quantities such as the differential shear modulus $K_\\mathrm{eq}$ and the critical strain $\\gamma_c$, at which the macroscopic system transitions between bending-dominated and stretching-dominated mechanical regimes. Appropriate physical parameters are chosen to enable comparison with prior experimental measurements of the elasticity of irreversibly crosslinked networks of F-actin,\\cite{gardel_elastic_2004} an essential component of the cytoskeleton of eukaryotic cells. Using the same time-series measurements, we then characterize the dynamics of stress relaxation via time correlations in the stress fluctuations. Building upon recent work,\\cite{shivers_nonaffinity_2022} we demonstrate that the excess differential viscosity, a measure of energy dissipation reflected in a system's finite-temperature stress fluctuations, is directly proportional to the same system's corresponding quasistatic, athermal differential nonaffinity, a measure of the strain heterogeneity induced by a small perturbation in the quasistatic, athermal limit. Since, in disordered networks, the quasistatic nonaffinity is highly strain-dependent and reaches a maximum at the critical strain, analogous behavior is expected in the excess differential viscosity and the slowest viscoelastic relaxation time. Our simulations confirm this expectation, providing crucial insight into potentially measurable effects of nonaffine fluctuations, which have generally proven challenging to experimentally quantify.\n\n\n\n\n\n\n\\section{Model definition and network assembly}\n\nWe imagine a system that begins as a solution of free semiflexible filaments covered to some extent with bound linker proteins that are capable of dimerizing to form permanent elastic crosslinks. In experiments, the linker coverage could be controlled by varying the relative concentration of crosslinking protein and filament monomer, as in Ref. \\citenum{gardel_elastic_2004}. If the filament concentration and linker coverage in such a solution are sufficiently high, the formation of crosslinks between diffusively migrating filaments eventually produces a macroscopic network.\n\nTo capture this behavior, we turn to a simplified computational model. We specify the number of filaments $N_f$, filament length $\\ell_f$, and filament length density $\\rho$, which together determine the size of the periodic simulation box, $L=(N_f \\ell_f\/\\rho)^{1\/3}$. To ensure that the (initially straight) filaments do not span the entire system, we require $L>\\ell_f$, or equivalently $\\ell_f < \\sqrt{N_f\/\\rho}$. We further specify the number of evenly-spaced nodes per filament, $n$, which determines the filament bond rest length $\\ell_0 = \\ell_f\/(n-1)$. The total number of nodes in the system is then $N = N_f n$. We then designate a fraction $p$ of the nodes, chosen randomly, as sticky, or capable of forming a crosslink bond with another node of the same type. Crosslinks are permanent bonds with a shorter rest length $\\ell_{0,c\\ell}$ than the filament bonds. All bonds, including crosslinks, are treated as harmonic springs with stretching stiffness $\\mu$, and harmonic bending interactions with stiffness $\\kappa = k_B T \\ell_p$ act between adjacent filament bonds. Here, $\\ell_p$ denotes the filament persistence length. In terms of the $3N$-dimensional vector of node positions $\\mathbf{x}$, we write the elastic potential energy of this system as\n\\begin{equation}\n U(\\mathbf{x})=\\frac{\\mu}{2}\\sum_{ij}\\frac{\\left(\\ell_{ij}-\\ell_{ij,0}\\right)^2}{\\ell_{ij,0}}+\\frac{\\kappa}{2}\\sum_{ijk}\\frac{\\left(\\theta_{ijk}-\\theta_{ijk,0}\\right)^2}{\\ell_{ijk,0}}\n\\end{equation}\nin which $\\ell_{ij}=\\left|\\mathbf{x}_j-\\mathbf{x}_i\\right|$, $\\theta_{ijk}$ is the angle between bonds $ij$ and $jk$, $\\ell_{ijk}=(\\ell_{ij}+\\ell_{jk})\/2$, and the subscript $0$ denotes rest values. The first sum is taken over all bonds $ij$, including crosslink bonds, and the second over all pairs of connected bonds $ij$ and $jk$ along the backbone of each filament. Because we are considering crosslinked networks with very low filament volume fractions, we deem it acceptable to ignore steric interactions between filaments. Neglecting inertia, as is appropriate for the timescales studied here, the system obeys the overdamped Langevin equation,\\cite{allen_computer_2017,frenkel_understanding_2002} such that the forces acting on all nodes satisfy\n\\begin{equation} \\label{eqn:forces}\n \\mathbf{F}_N+\\mathbf{F}_D+\\mathbf{F}_B=\\mathbf{0},\n\\end{equation}\nin which the terms on the left represent the network forces, drag forces, and Brownian forces, respectively. The force due to the network is\n\\begin{equation}\n \\mathbf{F}_N=-\\frac{\\partial U(\\mathbf{x})}{\\partial \\mathbf{x}}.\n\\end{equation}\nNodes are subjected to a Stokes drag force\n\\begin{equation}\n \\mathbf{F}_D=-\\zeta\\frac{\\partial \\mathbf{x}}{\\partial t}\n\\end{equation}\nwith drag coefficient $\\zeta=6\\pi\\eta_s a$, in which $\\eta_s$ is the solvent viscosity and $a$ is an effective node radius of half the filament bond length, $a=\\ell_0\/2$. Finally, the Brownian force is\n\\begin{equation}\n \\mathbf{F}_B=\\sqrt{2\\zeta k_B T}\\dot{\\mathbf{w}},\n\\end{equation}\nin which each component of $\\dot{\\mathbf{w}}(t)$ is a Gaussian random variable with zero mean and unit variance.\\cite{allen_computer_2017} Equation \\ref{eqn:forces} can be rewritten as\n\\begin{equation}\\label{eqn:eom}\n \\frac{\\partial \\mathbf{x}}{\\partial t}=-\\frac{1}{\\zeta}\\frac{\\partial U(\\mathbf{x})}{\\partial \\mathbf{x}} + \\sqrt{\\frac{2k_B T}{\\zeta}}\\dot{\\mathbf{w}}(t).\n\\end{equation}\nDuring each timestep in the assembly stage, we check whether the distance separating any pair of sticky nodes has decreased below a specified crosslink formation distance $r_c$, in which case we connect the two with a crosslink bond. Each sticky node can form a maximum of one crosslink, and connections between crosslinks and filaments are treated as freely hinging. In other words, we do not include a bending potential between adjacent crosslink and filament bonds. We halt the assembly stage after a total time $\\tau_\\mathrm{a}$ has elapsed, ensuring that $\\tau_\\mathrm{a}$ is suitably long to neglect the small fraction of crosslinks forming at longer times. Here, we use $\\tau_\\mathrm{a} = 6 \\times 10^7 \\Delta t$, corresponding to approximately one minute in real units. For the parameters specified in Table S1 (see Supplemental Material), we deem this $\\tau_\\mathrm{a}$ sufficiently long for the rate of crosslink formation to become negligible, meaning that the network structure has stabilized. Note that crosslink formation does not take place after the assembly stage; during the rheology stage, described in the next section, the network topology remains fixed.\n\nAfter assembly, we analyze the structural features of each network. To determine the connectivity $z$, or average number of connections at each network junction, we consider a reduced version of the simulated network. Each pair of crosslinked nodes in the original network corresponds to a single node in the reduced network, and each filament section between two crosslinked nodes in the original network corresponds to an edge (see sketch in Fig. S1 and further details in Supplemental Material, Section S1). Dangling ends, or filament sections in the original network connected to only one crosslink, are therefore neglected. This is a reasonable choice, as dangling ends do not contribute to the network's zero-frequency elastic response. We then calculate the connectivity from the number of edges $n_\\mathrm{edges}$ and nodes $n_\\mathrm{nodes}$ present in the reduced network structure as $z=2n_\\mathrm{edges}\/n_\\mathrm{nodes}$. \n\nIn practice, we construct systems with filament length $\\ell_f = 9 \\;\\si{\\micro\\meter}$ and filament persistence length $\\ell_p = 17\\; \\si{\\micro\\meter}$, chosen to approximate F-actin, with filament length per volume $\\rho = 2.6\\; \\si{\\micro\\meter}^{-2}$ (for F-actin, this corresponds to a concentration of $c_A = \n1.6\\;\\si{\\micro M}$). Additional parameters are specified in Table S1 (see Supplemental Material). Unless stated otherwise, reported measurements throughout this work are averaged over three randomly generated network samples, and error bars represent $\\pm 1$ standard deviation. All simulations are performed using the open-source molecular dynamics simulation tool LAMMPS.\\cite{thompson_lammps_2022}\n\nWe first consider the effects of adjusting the crosslinker coverage fraction $p$ on the assembled network structure. Varying $p$ from $0.4$ to $0.9$, we find that the connectivity of the fully assembled network structures ranges from $z\\in[2.8,3.6]$, increasing monotonically with $p$ (see Fig. \\ref{fig_2}a). These values are similar to those measured for collagen and fibrin networks in prior experimental work.\\cite{lindstrom_biopolymer_2010,beroz_physical_2017,jansen_role_2018,xia_anomalous_2021} Next, we compute the average inter-crosslink contour length $\\ell_c$, which we define as the average length of filament sections in the original network corresponding to edges in the reduced network. As shown in Fig. \\ref{fig_2}b, we find that the average inter-crosslink contour length $\\ell_c$ decreases monotonically with $p$. \n\n\n\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_2.pdf}\n\\caption{ \\label{fig_2} Linker coverage-dependence of key structural characteristics of assembled networks. With increasing linker coverage fraction $p$, we observe (a) increasing average connectivity $z$ and (b) decreasing average inter-crosslink contour length $\\ell_c$.}\n\\end{figure}\n\n\\section{Nonlinear rheology}\n\n\n\nAfter the assembly stage is halted, we proceed to the rheology stage, in which the strain-dependent steady-state viscoelastic properties of the system are measured via the time-dependent shear stress as the system fluctuates about the mechanically equilibrated state under a fixed shear strain. We impose a constant macroscopic simple shear strain $\\gamma$ using Lees-Edwards periodic boundary conditions \\cite{lees_computer_1972} and, using the conjugate gradient method, initially obtain the energy-minimizing configuration of the network, corresponding to the mechanically equilibrated state at $T = 0$. Then, we evolve the system according to Eq. \\ref{eqn:eom} at temperature $T>0$, specified in Table S1 in Supplemental Material, for a total run time $\\tau_\\mathrm{tot} = 3 \\times 10^7 \\Delta t$, discarding the first half of the trajectory to avoid initialization effects. For a given configuration of the system, we compute the instantaneous virial stress tensor,\n\\begin{equation}\n \\sigma_{\\alpha \\beta}=\\frac{1}{2V}\\sum_{i>j}{f_{ij\\alpha} r_{ij\\beta}}\n\\end{equation}in which $\\mathbf{r}_{ij}=\\mathbf{x}_j-\\mathbf{x}_i$ and $\\mathbf{f}_{ij}$ is the force acting on node $i$ due to its interaction with node $j$. Because we will focus on macroscopic simple shear oriented along the $x$-axis with gradient direction $z$, we now define $\\sigma\\equiv\\sigma_{xz}$ to simplify notation. Once we have obtained a time series of the shear stress $\\sigma(\\gamma, t)$ at strain $\\gamma$, we calculate the time-averaged shear stress $\\langle \\sigma(\\gamma, t)\\rangle_t $. After repeating this procedure over many strains in the interval $\\gamma\\in[0,1]$, we compute the strain-dependent differential shear modulus,\n\\begin{equation}\n K_\\mathrm{eq}(\\gamma)=\\frac{\\partial \\langle \\sigma(\\gamma, t)\\rangle_t}{\\partial \\gamma},\n\\end{equation} which measures the apparent stiffness of the sample under macroscopic strain $\\gamma$ in response to an infinitesimal additional shear strain step. For sufficiently small strains, this yields the corresponding linear shear modulus, $G_0=\\lim_{\\gamma\\to0}K_\\mathrm{eq}(\\gamma)$. \n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_3.pdf}\n\\caption{ \\label{fig_3} (a) Time-averaged shear stress $\\langle \\sigma(\\gamma, t)\\rangle_t$ as a function of applied shear strain $\\gamma$, for a single network sample with linker coverage fraction $p=0.9$. (b) Differential shear modulus $K_\\mathrm{eq}=\\partial \\langle \\sigma(\\gamma, t)\\rangle_t\/\\partial \\gamma$. The linear shear modulus $G_0$ and critical strain $\\gamma_c$ are indicated by dashed lines, and the solid line is a fit to the equation of state from Ref. \\citenum{sharma_strain-controlled_2016}, described in Supplemental Material, Section S3. }\n\\end{figure}\n\nIn Fig. \\ref{fig_3}a and b, we plot the mean stress $\\langle\\sigma(\\gamma,t)\\rangle_t$ and differential shear modulus $K_\\mathrm{eq}$ as a function of strain for a single network sample with the parameters specified in Table S1 (see Supplemental Material) and linker coverage fraction $p=0.9$. We determine the critical strain $\\gamma_c$, which indicates the transition between the bending-dominated and stretching-dominated mechanical regimes, as the inflection point of the $\\log{K_\\mathrm{eq}}$ vs. $\\log{\\gamma}$ curve. To avoid issues associated with differentiating noisy data, we can alternatively find $G_0$ and $\\gamma_c$ by fitting the entire $K_\\mathrm{eq}$ vs. $\\gamma$ curve to an Ising-like equation of state discussed in prior work,\\cite{sharma_strain-controlled_2016} as we describe in further detail in Supplemental Material, Section S3. Such a fit is shown in Fig. \\ref{fig_3}b. For the data presented here, both methods yield effectively equivalent values of $G_0$ and $\\gamma_c$.\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_4.pdf}\n\\caption{ \\label{fig_4} Effects of linker coverage on features of the linear and nonlinear stiffness in shear. a) Differential shear modulus $K_\\mathrm{eq}$ for networks with varying linker coverage fraction $p$ as a function of strain and (b) as a function of shear stress. Increasing the linker coverage fraction $p$ drives (c) an increase in the linear modulus $G_0=\\lim_{\\gamma\\to 0} K_\\mathrm{0}$, with $G_0\\propto p^3$, and (d) a decrease in the critical strain $\\gamma_c$ as $\\gamma_c\\propto p^{-1}$. }\n\\end{figure}\n\n\nIn Fig \\ref{fig_4}, we report the strain-dependence of the differential shear modulus for networks with varying linker coverage fraction $p$. As $p$ increases, the critical strain evidently decreases (stiffening occurs at lower applied strains) and the linear shear modulus increases. These observations agree qualitatively with both the observed crosslinker concentration-dependence of the linear modulus $G_0$ and rupture strain $\\gamma_\\mathrm{max}$ (which we expect to be proportional to $\\gamma_c$) of F-actin gels reported in Ref. \\citenum{gardel_elastic_2004}, and with the connectivity dependence of the linear modulus and critical strain observed in spring network simulations.\\cite{wyart_elasticity_2008} The same differential shear modulus data are plotted in Fig. \\ref{fig_4}b as a function of stress. The linear modulus $G_0$ and critical strain $\\gamma_c$ values extracted from these curves are plotted as functions of the linker coverage fraction $p$ in Fig. \\ref{fig_4}c and d, respectively. We find that each exhibits a power law dependence on $p$, with $G_0\\propto p^3$ and $\\gamma_c\\propto p^{-1}$. In Ref. \\citenum{gardel_elastic_2004}, in which the analogous quantity $R$ (the ratio between the concentrations of actin and the crosslinking protein scruin) is varied, the authors report $G_0\\propto R^2$ and $\\gamma_c\\propto R^{-0.6}$. These observations may be consistent with ours, provided that the linker coverage fraction in our system maps to the experimental crosslinker concentration ratio in Ref. \\citenum{gardel_elastic_2004} as $p\\propto R^{x}$ with $x\\sim 0.6$. Separately, the authors of Ref. \\citenum{tharmann_viscoelasticity_2007} reported that, for networks of actin filaments crosslinked by heavy meromyosin, $G_0\\propto R^{1.2}$ and $\\gamma_c\\propto R^{-0.4}$, consistent with our observed dependence of both quantities on $p$ if $x\\sim 0.4$. However, meaningfully relating $p$ and $R$ will require further investigation. Here, for example, no two sticky sites on the same filament can reside closer than a distance $\\ell_0$ ($1\\; \\si{\\micro\\meter}$ with our parameters) from one another. This is obviously not the case in real F-actin networks. \n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_5.pdf}\n\\caption{ \\label{fig_5} The linear shear modulus $G_0$ (a) increases dramatically with connectivity, shown here for systems with varying linker coverage fraction $p$, and (b) decreases with increasing average inter-crosslink contour length $\\ell_c$. For small $\\ell_c$ (high $p$), we find $G_0 \\propto \\ell_c^{-3}$, consistent with the naive scaling predicted for affine, isotropic semiflexible filament networks.\\cite{mackintosh_elasticity_1995} Deviation from this scaling appears at larger $\\ell_c$. (c) The critical strain $\\gamma_c$, in contrast, decreases approximately linearly with $z$. A linear fit suggests $\\gamma_c\\to0$ as $z\\to z^*\\sim 4$, roughly consistent with the upper bound for $z$ in the high crosslink density limit.\\cite{broedersz_filament-length-controlled_2012} (d) The critical strain increases approximately linearly with increasing average inter-crosslink contour length, $\\gamma_c\\propto\\ell_c$, again in apparent agreement with the scaling expected for the naive affine model. }\n\\end{figure}\n\n\nIn Fig. \\ref{fig_5}, we plot the same extracted linear shear modulus and critical strain measurements as functions of the $p$-dependent structural quantities discussed in the previous section, the average network connectivity $z$ and average inter-crosslink contour length $\\ell_c$. Notably, we find that the linear shear modulus $G_0$ decreases with increasing inter-crosslink contour length approximately as $G_0\\propto \\ell_c^{-3}$. This exponent is consistent with the analytically predicted dependence for an affinely deforming isotropic gel of semiflexible filaments, $G_0 \\propto \\rho k_B T \\ell_p^2\/\\ell_c^3$.\\cite{mackintosh_elasticity_1995} This agreement may be coincidental, as, for our range of parameters, the typical inter-crosslink contour lengths are presumably too small for bending modes to be properly resolved. This warrants further investigation in simulations with reduced coarse graining, i.e. smaller $\\ell_0\/\\ell_f$. In Fig. \\ref{fig_5}c, we show that $\\gamma_c$ approaches 0 as $z$ approaches an intercept value $z^*\\sim 4$, in apparent agreement with the upper limit of $z$ for crosslinked networks with high crosslinking density.\\cite{broedersz_filament-length-controlled_2012} However, we expect some dependence of this limit on the total filament length $\\ell_f$ and the bending rigidity $\\kappa$, which both necessarily influence the structure of the assembled network. We also find that the critical strain increases linearly with the average inter-crosslink contour length $\\ell_c$ (see Fig. \\ref{fig_5}d), in apparent agreement with the expected scaling of the critical strain with contour length for an affinely deforming isotropic gel of semiflexible filaments, $\\gamma_c \\propto \\ell_c\/\\ell_p$; as with the dependence of $G_0$ on $\\ell_c$, this may be a coincidence.\n\n\n\nFurther information is contained in the fluctuations of the instantaneous stress about its average value,\\cite{wittmer_shear_2013}\n\\begin{equation}\n \\delta\\sigma(\\gamma,t)=\\sigma(\\gamma,t)-\\langle\\sigma(\\gamma,t)\\rangle_t,\n\\end{equation}\nfrom which we compute the stress fluctuation autocorrelation function,\n\\begin{equation}\n C(\\gamma,\\tau)=\\frac{V}{k_B T}\\langle\\delta\\sigma(\\gamma,t)\\delta\\sigma(\\gamma,t+\\tau)\\rangle_t,\n\\end{equation}\nwhich reveals useful details about the time-dependence of energy dissipation.\\cite{wittmer_fluctuation-dissipation_2015,wittmer_shear-stress_2015}\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_6.pdf}\n\\caption{ \\label{fig_6} The relaxation of stress fluctuations slows dramatically near the critical strain as rearrangements become increasingly nonaffine. (a) The shear stress fluctuation autocorrelation function $C(\\gamma, \\tau)$ measured at representative strains below (purple), near (red), and above (orange) the critical strain $\\gamma_c$ (see the corresponding $K_\\mathrm{eq}$ data in the inset of panel (b)) reveals this slowing, which is captured quantitatively by (b) a peak in the apparent slowest relaxation time $\\tau_R$ (Eq. \\ref{eqn:relaxationtime}). These data correspond to the same sample as in Fig. \\ref{fig_3} (see inset), and the labeled strains are indicated by the same colors in all panels. (c) At long lag times, the time-dependent apparent excess viscosity $\\delta\\eta(\\gamma, \\tau)$ computed with Eq. \\ref{eqn:visc} grows much larger near $\\gamma_c$ than elsewhere. This behavior is especially clear when (d) the apparent excess zero-shear viscosity $\\delta\\eta(\\tau=\\tau_\\mathrm{max})$ is plotted as a function of $\\gamma$, revealing a peak at the $p$- (and thus $z$-) dependent critical strain. These observations are supported by independent measurements of the strain-dependent quasistatic, athermal differential nonaffinity $\\delta\\Gamma_\\infty$, which is quantitatively related to $\\delta\\eta$ by Eq. \\ref{eqn:nonaff_viscosity} (solid blue line). }\n\\end{figure}\n\n\nIn Fig. \\ref{fig_6}a, we show representative $C(\\gamma,\\tau)$ data for a single sample (the same sample as in Fig. \\ref{fig_3}) under applied strains below, near, and above the critical strain $\\gamma_c$. We find that the stress fluctuations decay slowly when the macroscopic applied strain is near $\\gamma_c$, in contrast to a much faster decay for strains below or above the critical regime. We can quantify this strain-dependent change in relaxation dynamics by integrating the shear stress autocorrelation function $C(\\gamma,\\tau)$ over a sufficiently long range of lag times $\\tau$. In practice, the range of lag times for which we can reliably estimate $C(\\gamma,\\tau)$ is limited by the simulation run time. Assuming $C(\\gamma,\\tau)$ is known for lag times below a maximum $\\tau_\\mathrm{max}$, we can estimate the system's slowest relaxation time as\n\\begin{equation}\\label{eqn:relaxationtime}\n \\tau_\\mathrm{R}(\\gamma, \\tau_\\mathrm{max}) = \\frac{\\int_0^{\\tau_\\mathrm{max}}{\\tau C(\\gamma,\\tau)d\\tau}}{\\int_0^{\\tau_\\mathrm{max}}{C(\\gamma,\\tau)d\\tau}}\n\\end{equation}\nIn the limit $\\tau_\\mathrm{max}\\to\\infty$, this converges to the true slowest relaxation time $\\tau_\\mathrm{R,0}(\\gamma)$. In Fig. \\ref{fig_6}b, we plot the apparent slowest relaxation time $\\tau_\\mathrm{R}$ for the same system, with marker colors indicating the strains plotted in Fig. \\ref{fig_6}a. Note that the corresponding stiffening curve from Fig. \\ref{fig_3} is also shown in the inset. We observe that $\\tau_\\mathrm{R}$ grows substantially as the critical strain $\\gamma_c$ is approached from either side, reaching a maximum at $\\gamma_c$. In fact, we see consistent behavior -- that is, growth of the slowest relaxation time by an order of magnitude or more with a peak at $p$-dependent critical strain $\\gamma_c$ -- in networks over the entire range of $p$ considered. This is shown in Fig. \\ref{fig_7}d, in which the applied strain is normalized by the $p$-dependent critical strain $\\gamma_c$, revealing maxima in $\\tau_\\mathrm{R}$ at $\\gamma\/\\gamma_c=1$ for all $p$.\n\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_7.pdf}\n\\caption{ \\label{fig_7} Shared features of stiffening and slowing down at the structure-dependent critical strain. (a) For networks with varying crosslinker coverage fraction $p$, the stiffening regime of the normalized differential shear modulus $K\/G_0$ collapses onto a single curve when plotted vs. $\\gamma\/\\gamma_c$. Lines correspond to fits to the equation of state described in Ref. \\citenum{sharma_strain-controlled_2016} (see Supplemental Material, Section S3), and the associated fit parameters are plotted as a function of $p$ in Fig. S2. (b) The mean-squared stress fluctuations $\\langle \\delta\\sigma(\\gamma, t)^2\\rangle_t$ are largest at the $p$- (and thus $z$-) dependent critical strain. Solid lines correspond to the difference between the measured strain-dependent affine and equilibrium differential shear moduli, $K_\\mathrm{aff}$ and $K_\\mathrm{eq}$, according to Eq. \\ref{eqn:stressfluc}. Both the (a) apparent excess zero-shear viscosity $\\delta\\eta$ and (b) apparent slowest relaxation time $\\tau_\\mathrm{R}$ likewise exhibit large peaks at the $p$-dependent critical strain $\\gamma_c$.}\n\\end{figure}\n\n\nThe stress fluctuation autocorrelation function is likewise related to the system's lag-dependent excess differential viscosity \\cite{wittmer_fluctuation-dissipation_2015} $\\delta\\eta(\\gamma,\\tau)\\equiv\\eta(\\gamma,\\tau)-\\eta_s$ measured at strain $\\gamma$,\n\\begin{equation} \\label{eqn:visc}\n \\delta\\eta(\\gamma,\\tau) =\\int_0^{\\tau} C(\\gamma,\\tau') d\\tau',\n\\end{equation}\nwhich we plot in Fig. \\ref{fig_6}c for the same labeled strain values. It is clear that $\\delta\\eta(\\gamma,\\tau)$ grows far more dramatically near the critical strain than elsewhere.\nIn recent work,\\cite{shivers_nonaffinity_2022} it was suggested that the low-frequency or ``zero-shear'' excess differential viscosity $\\delta\\eta_0(\\gamma) = \\lim_{\\tau\\to \\infty} \\delta\\eta(\\gamma, \\tau)$ of prestrained disordered networks is controlled by nonaffinity. Specifically, $\\delta\\eta_0$ was shown to be related to the quasistatic, athermal differential nonaffinity $\\delta\\Gamma_\\infty(\\gamma)$, defined as\n\\begin{equation}\\label{eqn:nonaffinity}\n \\delta\\Gamma_\\infty(\\gamma)=\\lim_{\\delta\\gamma\\to 0}\\frac{1}{N\\ell_0^2\\delta\\gamma^2}\\sum_i\\left|\\delta\\mathbf{x}_i^\\mathrm{NA}\\right|^2\n\\end{equation}\nin which the sum is taken over all network nodes and the vector $\\delta\\mathbf{x}_i^\\mathrm{NA}$ represents the nonaffine component of the displacement vector of node $i$ under a macroscopic, quasistatically applied strain step $\\delta\\gamma$. The viscosity-nonaffinity relationship in Ref. \\citenum{shivers_nonaffinity_2022} can be stated concisely as\n\\begin{equation}\\label{eqn:nonaff_viscosity}\n \\delta\\eta_0(\\gamma)=\\frac{N}{V}\\zeta\\ell_0^2\\delta\\Gamma_\\infty(\\gamma).\n\\end{equation}\nThe left side of Eq. \\ref{eqn:nonaff_viscosity} reflects the dynamics of the stress fluctuations at finite temperature, while the right side reflects the heterogeneous nature of the strictly quasistatic deformation field, which can be obtained by comparing energy-minimized, $T=0$ system configurations under varying applied $\\gamma$.\n\nSimulations have suggested that, for disordered filament networks, the differential nonaffinity generically reaches a maximum at the structure-dependent critical strain,\\cite{rens_nonlinear_2014,sharma_strain-driven_2016} at which the system macroscopically transitions between bending-dominated (or floppy, for $\\kappa=0$) and stretching-dominated regimes. Thus, according to Eq. \\ref{eqn:nonaff_viscosity}, we should generically see a proportional peak in the excess differential viscosity at $\\gamma_c$. Testing this prediction requires measuring the quasistatic, athermal differential nonaffinity. To do so, for a given network at strain $\\gamma$, we first obtain the energy-minimizing configuration at $T=0$ using the conjugate gradient method. To this configuration, we then apply a small additional strain step $d\\gamma = 0.01$, after which we repeat the energy minimization procedure. Comparing the positions in the energy-minimizing configurations at $\\gamma$ and $\\gamma + d\\gamma$, we compute the quasistatic differential nonaffinity $\\delta\\Gamma_\\infty$ using Eq. \\ref{eqn:nonaffinity}. In Fig. \\ref{fig_6}d, we plot $\\delta\\Gamma_\\infty$ along with the apparent excess differential zero-shear viscosity for the same system, demonstrating excellent agreement with Eq. \\ref{eqn:nonaff_viscosity}. To demonstrate that this behavior is preserved as the structure is varied, we show in Fig. \\ref{fig_7}c that a clear peak in $\\delta\\eta$ occurs at $\\gamma\/\\gamma_c = 1$ over the entire range of $p$ considered.\n\nWe can also use the stress fluctuations to determine the differential relaxation modulus,\\cite{wittmer_shear-strain_2015}\n\\begin{equation}\n K(\\gamma,\\tau)=K_\\mathrm{eq}(\\gamma)+C(\\gamma,\\tau),\n\\end{equation}\nwhich quantifies the time-dependent apparent stiffness of the system at prestrain $\\gamma$ in response to an instantaneous additional strain step.\\cite{rubinstein_polymer_2006,doi_theory_1988}\nFor sufficiently short times, the differential relaxation modulus approaches an upper limit of $\\lim_{\\tau\\to 0}K(\\gamma,\\tau) = K_\\mathrm{aff}(\\gamma)$, corresponding to the apparent stiffness of the energy-minimized, athermal equivalent of the system at strain $\\gamma$ under an instantaneous, homogeneous infinitesimal shear strain step. Consequently, the equilibrium stiffness of the system under applied strain $\\gamma$ is simply $K_\\mathrm{aff}(\\gamma)$ reduced by $C(\\gamma,0)$,\\cite{squire_isothermal_1969,lutsko_generalized_1989,wittmer_shear_2013} or equivalently\n\\begin{equation} \\label{eqn:stressfluc}\n K_\\mathrm{eq}(\\gamma)=K_\\mathrm{aff}(\\gamma)-\\frac{V}{k_B T}\\langle\\left(\\delta\\sigma(\\gamma,t)\\right)^2\\rangle_t.\n\\end{equation}\nWe find that this relationship provides a useful estimate of the mean squared stress fluctuations, as shown in Fig. \\ref{fig_7}b, and we observe that the mean squared stress fluctuations robustly reach a maximum at $\\gamma\/\\gamma_c = 1$.\n\n\n\n\n\\section{Conclusions}\n\nWe have investigated the assembly and mechanical testing of disordered networks of crosslinked semiflexible polymers via Brownian dynamics simulations. Such networks serve as essential mechanical constituents of a wide variety of biological materials spanning many length scales. We explored the structural and rheological consequences of varying the crosslinker coverage fraction $p$, an analog of the experimental ratio between the concentrations of crosslinker and filament proteins. Using physical parameters intended to mimic the cytoskeletal polymer F-actin, we measured the effects of increasing linker coverage fraction $p$ on the average connectivity $z$ of self-assembled networks, observing a corresponding increase in connectivity between $z\\in[2.8,3.6]$.\n\nWe then investigated the relationship between these changes in connectivity and associated changes in the strain-dependent rheological properties of assembled networks, obtaining extended time series measurements of the fluctuating shear stress $\\sigma(\\gamma, t)$ for systems held under fixed shear strain $\\gamma$. Analyzing many such trajectories gathered over a range of applied strains $\\gamma$, we computed the strain-dependent differential shear modulus $K_\\mathrm{eq}$, from which we extracted both the linear shear modulus $G_0$ and the critical strain $\\gamma_c$. We described the dependence of these quantities on the crosslinker coverage fraction $p$, demonstrating qualitative agreement with the experimentally observed dependence of the same quantities on crosslinker concentration ratio $R$ reported in Ref. \\citenum{gardel_elastic_2004}. Specifically, we found that increasing $p$ produces inherently stiffer networks (having an increased linear shear modulus $G_0$) that simultaneously exhibit an earlier tendency to strain-stiffen (having a lower critical strain $\\gamma_c$). Drawing comparisons between the scaling of both $G_0$ and $\\gamma_c$ with $p$ and the analogous experimentally-observed scaling of the same quantities with $R$, we suggested a simple power law relationship between $p$ and $R$. We next described the apparent dependence of these elastic features on the $p$-dependent structural quantities $z$ and $\\ell_c$, notably showing that the critical strain decreases linearly with $z$. A linear fit suggests that $\\gamma_c$ approaches $0$ as the connectivity $z$ approaches a limiting value $z^*\\to z_c$ near $4$, the theoretical upper bound for $z$ at high crosslinking density.\n \nWe then extended our observations beyond strictly elastic properties by analyzing the stress fluctuation autocorrelation function $C(\\gamma,\\tau)$, which revealed the development of extremely slow dynamics in systems subjected to applied shear strains near the critical strain $\\gamma_c$. From $C(\\gamma,\\tau)$, we obtained estimates of the slowest relaxation time $\\tau_{\\mathrm{R},0}$ and the excess differential zero-shear viscosity $\\delta\\eta_0$, both of which we showed are consistently maximized at the $p$-dependent critical strain, irrespective of the details of the underlying network. Building upon results from Ref. \\citenum{shivers_nonaffinity_2022}, we demonstrated that the excess differential viscosity in these finite temperature systems is controlled quantitatively by the athermal, quasistatic differential nonaffinity $\\delta\\Gamma_\\infty$, a measure of the inherent tendency of the strained network to deform heterogeneously. Importantly, our results suggest that one should expect measurable dynamical signatures of transition-associated nonaffine fluctuations to appear in semiflexible polymer networks with biologically relevant elastic properties, e.g. those of the F-actin cytoskeleton, under physiologically relevant applied strains. In other words, one should generically expect to observe slowing stress relaxation in biopolymer networks under prestrain levels near the critical strain $\\gamma_c$ marking the macroscopic transition between bending-dominated and stretching-dominated elasticity. Since $\\gamma_c$ is controlled by the average connectivity $z$, which is controlled in turn by the availability of crosslinking sites (here, $p$), our results suggest a route to experimentally control the strain-dependence of major features of both network elasticity and the dynamics of stress relaxation.\n\n\n\n\\begin{acknowledgments}\n\nThis study was supported in part by the National Science\nFoundation Division of Materials Research (Grant No. DMR-1826623) and the National Science Foundation Center for Theoretical Biological Physics (Grant No. PHY-2019745). This work began as a summer research experience for undergraduates (REU) project supported by the Frontiers in Science (FIS) program of the Center for Theoretical Biological Physics.\n\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev}\n\n\\section{Introduction\n}\n\nThe formation of living things involves the energy-intensive assembly of complex structures from scarce resources. Yet, the demanding physical processes of life require robust materials that remain intact and functional under significant and often repetitive applied stresses and strains. Quasi-one-dimensional or filamentous structures address these challenges efficiently by supporting significant tensile stresses with minimal material cost.\\cite{burla_mechanical_2019} Biological polymers and fibers are generally also semiflexible,\\cite{janmey_mechanical_1991} meaning that they resist modes of deformation that induce bending, such as applied compression. In examples spanning a wide range of length scales, including information-storing DNA and RNA, actin and intermediate filaments in the cell cytoskeleton, and extracellular collagen and elastin fibers in tissues, these mechanical features consistently play crucial roles in biological function. \n\nThe cytoskeleton and extracellular matrix are quintessential examples of disordered networks, a common class of higher-order structure in living materials. Building upon the qualities of their underlying semiflexible filaments, these networks act as responsive elastic scaffolds that resist potentially damaging deformation while leaving ample space for the transport and storage of functional components, such as interstitial fluids and cells. Unlike conventional elastic solids, their mechanical properties are scale-dependent\\cite{tyznik_length_2019,proestaki_modulus_2019} and extremely sensitive to changes in applied stress or strain,\\cite{onck_alternative_2005,picu_poissons_2018,ban_strong_2019} to which they respond with dramatic stiffening, alignment, and changes in local filament density, enabling essential biological phenomena such as long-range force transmission by cells\\cite{notbohm_microbuckling_2015,wang_long-range_2015,han_cell_2018,alisafaei_long-range_2021,grill_directed_2021} and the contraction of muscle fibers.\\cite{huxley_mechanism_1969,gunst_first_2003} Recent work has suggested that the nonlinear viscoelastic properties of these and related fibrous networks are governed by an underlying mechanical phase transition\\cite{sharma_strain-controlled_2016,jansen_role_2018} associated with the onset of stretching-dominated rigidity under tension-generating (i.e. shear or extensile) applied strains. Within this framework, elastic properties are predicted to exhibit a power law dependence on applied strain in the vicinity of a critical strain, and growing nonaffine (inhomogeneous) strain fluctuations are expected to drive a significant slowing of stress relaxation as the transition is approached.\\cite{shivers_nonaffinity_2022} The location of this transition (the magnitude of the critical strain) depends sensitively on key features of the underlying network architecture.\\cite{wyart_elasticity_2008, broedersz_filament-length-controlled_2012}\n\nExtensive simulation-based studies have explored the nonlinear rheological properties of disordered networks of crosslinked stiff or semiflexible polymers,\\cite{onck_alternative_2005,picu_mechanics_2011,carrillo_nonlinear_2013,freedman_versatile_2017} in some cases with realistic three-dimensional geometries produced either by physical assembly processes\\cite{huisman_three-dimensional_2007,kim_computational_2009,dobrynin_universality_2011,muller_rheology_2014,muller_resolution_2015,zagar_two_2015,debenedictis_structure_2020,grill_directed_2021} or artificial generation procedures.\\cite{huisman_monte_2008,lindstrom_biopolymer_2010,huisman_internal_2011,huisman_frequency-dependent_2010,lindstrom_finite-strain_2013,amuasi_nonlinear_2015} However, efforts to specifically connect network structure with strain-controlled critical behavior have typically focused on simplified random spring networks.\\cite{wyart_elasticity_2008,vermeulen_geometry_2017,jansen_role_2018,rens_micromechanical_2018,merkel_minimal-length_2019,shivers_normal_2019,shivers_nonlinear_2020,burla_connectivity_2020,burla_stress_2019,dussi_athermal_2020,tauber_stretchy_2022} In spring networks, the critical strain coincident with the onset of stretching-dominated mechanics can be tuned by changing the average connectivity $z$, defined as the average number of bonds joined at each network junction.\\cite{wyart_elasticity_2008} For a network of crosslinked filaments, $z$ is controlled by the typical number of crosslinks formed per filament, approaching an upper bound of $z\\to 4$ in the high crosslink-density limit.\\cite{broedersz_filament-length-controlled_2012} In biopolymer gels, small changes in the concentration of available crosslinkers can drive dramatic changes in rheological properties,\\cite{broedersz_modeling_2014} including changes in the linear elastic modulus \\cite{piechocka_multi-scale_2016} and shifts in the critical strain corresponding to the onset of stretching-dominated mechanics.\\cite{gardel_elastic_2004} These changes are naively consistent with a tendency of the connectivity to increase with crosslinker concentration. However, the quantitative relationships between crosslinker concentration and connectivity, and the structural characteristics of assembled networks more generally, remain mostly unknown. Improving our knowledge of how the concentration-dependent microscopic structural details of self-assembled networks translate into strain-dependent macroscopic rheological properties is essential for understanding the forces at play in important biological processes, such as wound healing and cancer metastasis, and for the design of biomimetic synthetic materials.\\cite{romera_strain_2017}\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_1.pdf}\n\\caption{ \\label{fig_1} Model system and rheological approach. (a) Coarse-grained semiflexible filaments are decorated with randomly assigned sticky sites (light blue spheres in the image on the right) with coverage fraction $p$. When two sticky sites meet, they are connected by a permanent crosslink (royal blue dumbbell). Solutions of these filaments diffusively self-assemble into percolating disordered networks with macroscopic elasticity. (b) After network assembly, the structure is fixed and the strain-dependent rheological properties are determined from the fluctuating shear stress $\\sigma(\\gamma, t)$ measured under constant macroscopic simple shear strain $\\gamma$.}\n\\end{figure}\n\nIn this study, we consider a system composed of coarse-grained semiflexible filaments that diffusively self-assemble into a system-spanning network through the formation of permanent inter-filament crosslinks. We begin with randomly positioned free filaments with a specified coverage fraction $p$ of \"sticky\" or linker-decorated sites. The linker coverage fraction serves as a proxy for the ratio of crosslinker and filament concentrations in a real system. Allowing diffusive motion to proceed, we add permanent crosslinks (short elastic bonds) between sticky sites whose pairwise distance decreases beneath a designated crosslink formation distance. After the rate of diffusion-driven formation of new crosslinks becomes sufficiently slow, we halt the assembly process and analyze the structure of the fixed network, measuring the dependence of both the connectivity and the average inter-crosslink contour length on the linker coverage fraction.\n\nWe then transition to the rheology stage, in which we observe the network's steady state behavior under constant simple shear strain $\\gamma$. We obtain time-series measurements of the thermally fluctuating shear stress $\\sigma(\\gamma, t)$ in the mechanically equilibrated state, as shown schematically in Fig. \\ref{fig_1}b. Repeating these measurements over a range of strains for each set of input parameters, we compute relevant elastic quantities such as the differential shear modulus $K_\\mathrm{eq}$ and the critical strain $\\gamma_c$, at which the macroscopic system transitions between bending-dominated and stretching-dominated mechanical regimes. Appropriate physical parameters are chosen to enable comparison with prior experimental measurements of the elasticity of irreversibly crosslinked networks of F-actin,\\cite{gardel_elastic_2004} an essential component of the cytoskeleton of eukaryotic cells. Using the same time-series measurements, we then characterize the dynamics of stress relaxation via time correlations in the stress fluctuations. Building upon recent work,\\cite{shivers_nonaffinity_2022} we demonstrate that the excess differential viscosity, a measure of energy dissipation reflected in a system's finite-temperature stress fluctuations, is directly proportional to the same system's corresponding quasistatic, athermal differential nonaffinity, a measure of the strain heterogeneity induced by a small perturbation in the quasistatic, athermal limit. Since, in disordered networks, the quasistatic nonaffinity is highly strain-dependent and reaches a maximum at the critical strain, analogous behavior is expected in the excess differential viscosity and the slowest viscoelastic relaxation time. Our simulations confirm this expectation, providing crucial insight into potentially measurable effects of nonaffine fluctuations, which have generally proven challenging to experimentally quantify.\n\n\n\n\n\n\n\\section{Model definition and network assembly}\n\nWe imagine a system that begins as a solution of free semiflexible filaments covered to some extent with bound linker proteins that are capable of dimerizing to form permanent elastic crosslinks. In experiments, the linker coverage could be controlled by varying the relative concentration of crosslinking protein and filament monomer, as in Ref. \\citenum{gardel_elastic_2004}. If the filament concentration and linker coverage in such a solution are sufficiently high, the formation of crosslinks between diffusively migrating filaments eventually produces a macroscopic network.\n\nTo capture this behavior, we turn to a simplified computational model. We specify the number of filaments $N_f$, filament length $\\ell_f$, and filament length density $\\rho$, which together determine the size of the periodic simulation box, $L=(N_f \\ell_f\/\\rho)^{1\/3}$. To ensure that the (initially straight) filaments do not span the entire system, we require $L>\\ell_f$, or equivalently $\\ell_f < \\sqrt{N_f\/\\rho}$. We further specify the number of evenly-spaced nodes per filament, $n$, which determines the filament bond rest length $\\ell_0 = \\ell_f\/(n-1)$. The total number of nodes in the system is then $N = N_f n$. We then designate a fraction $p$ of the nodes, chosen randomly, as sticky, or capable of forming a crosslink bond with another node of the same type. Crosslinks are permanent bonds with a shorter rest length $\\ell_{0,c\\ell}$ than the filament bonds. All bonds, including crosslinks, are treated as harmonic springs with stretching stiffness $\\mu$, and harmonic bending interactions with stiffness $\\kappa = k_B T \\ell_p$ act between adjacent filament bonds. Here, $\\ell_p$ denotes the filament persistence length. In terms of the $3N$-dimensional vector of node positions $\\mathbf{x}$, we write the elastic potential energy of this system as\n\\begin{equation}\n U(\\mathbf{x})=\\frac{\\mu}{2}\\sum_{ij}\\frac{\\left(\\ell_{ij}-\\ell_{ij,0}\\right)^2}{\\ell_{ij,0}}+\\frac{\\kappa}{2}\\sum_{ijk}\\frac{\\left(\\theta_{ijk}-\\theta_{ijk,0}\\right)^2}{\\ell_{ijk,0}}\n\\end{equation}\nin which $\\ell_{ij}=\\left|\\mathbf{x}_j-\\mathbf{x}_i\\right|$, $\\theta_{ijk}$ is the angle between bonds $ij$ and $jk$, $\\ell_{ijk}=(\\ell_{ij}+\\ell_{jk})\/2$, and the subscript $0$ denotes rest values. The first sum is taken over all bonds $ij$, including crosslink bonds, and the second over all pairs of connected bonds $ij$ and $jk$ along the backbone of each filament. Because we are considering crosslinked networks with very low filament volume fractions, we deem it acceptable to ignore steric interactions between filaments. Neglecting inertia, as is appropriate for the timescales studied here, the system obeys the overdamped Langevin equation,\\cite{allen_computer_2017,frenkel_understanding_2002} such that the forces acting on all nodes satisfy\n\\begin{equation} \\label{eqn:forces}\n \\mathbf{F}_N+\\mathbf{F}_D+\\mathbf{F}_B=\\mathbf{0},\n\\end{equation}\nin which the terms on the left represent the network forces, drag forces, and Brownian forces, respectively. The force due to the network is\n\\begin{equation}\n \\mathbf{F}_N=-\\frac{\\partial U(\\mathbf{x})}{\\partial \\mathbf{x}}.\n\\end{equation}\nNodes are subjected to a Stokes drag force\n\\begin{equation}\n \\mathbf{F}_D=-\\zeta\\frac{\\partial \\mathbf{x}}{\\partial t}\n\\end{equation}\nwith drag coefficient $\\zeta=6\\pi\\eta_s a$, in which $\\eta_s$ is the solvent viscosity and $a$ is an effective node radius of half the filament bond length, $a=\\ell_0\/2$. Finally, the Brownian force is\n\\begin{equation}\n \\mathbf{F}_B=\\sqrt{2\\zeta k_B T}\\dot{\\mathbf{w}},\n\\end{equation}\nin which each component of $\\dot{\\mathbf{w}}(t)$ is a Gaussian random variable with zero mean and unit variance.\\cite{allen_computer_2017} Equation \\ref{eqn:forces} can be rewritten as\n\\begin{equation}\\label{eqn:eom}\n \\frac{\\partial \\mathbf{x}}{\\partial t}=-\\frac{1}{\\zeta}\\frac{\\partial U(\\mathbf{x})}{\\partial \\mathbf{x}} + \\sqrt{\\frac{2k_B T}{\\zeta}}\\dot{\\mathbf{w}}(t).\n\\end{equation}\nDuring each timestep in the assembly stage, we check whether the distance separating any pair of sticky nodes has decreased below a specified crosslink formation distance $r_c$, in which case we connect the two with a crosslink bond. Each sticky node can form a maximum of one crosslink, and connections between crosslinks and filaments are treated as freely hinging. In other words, we do not include a bending potential between adjacent crosslink and filament bonds. We halt the assembly stage after a total time $\\tau_\\mathrm{a}$ has elapsed, ensuring that $\\tau_\\mathrm{a}$ is suitably long to neglect the small fraction of crosslinks forming at longer times. Here, we use $\\tau_\\mathrm{a} = 6 \\times 10^7 \\Delta t$, corresponding to approximately one minute in real units. For the parameters specified in Table S1 (see Supplemental Material), we deem this $\\tau_\\mathrm{a}$ sufficiently long for the rate of crosslink formation to become negligible, meaning that the network structure has stabilized. Note that crosslink formation does not take place after the assembly stage; during the rheology stage, described in the next section, the network topology remains fixed.\n\nAfter assembly, we analyze the structural features of each network. To determine the connectivity $z$, or average number of connections at each network junction, we consider a reduced version of the simulated network. Each pair of crosslinked nodes in the original network corresponds to a single node in the reduced network, and each filament section between two crosslinked nodes in the original network corresponds to an edge (see sketch in Fig. S1 and further details in Supplemental Material, Section S1). Dangling ends, or filament sections in the original network connected to only one crosslink, are therefore neglected. This is a reasonable choice, as dangling ends do not contribute to the network's zero-frequency elastic response. We then calculate the connectivity from the number of edges $n_\\mathrm{edges}$ and nodes $n_\\mathrm{nodes}$ present in the reduced network structure as $z=2n_\\mathrm{edges}\/n_\\mathrm{nodes}$. \n\nIn practice, we construct systems with filament length $\\ell_f = 9 \\;\\si{\\micro\\meter}$ and filament persistence length $\\ell_p = 17\\; \\si{\\micro\\meter}$, chosen to approximate F-actin, with filament length per volume $\\rho = 2.6\\; \\si{\\micro\\meter}^{-2}$ (for F-actin, this corresponds to a concentration of $c_A = \n1.6\\;\\si{\\micro M}$). Additional parameters are specified in Table S1 (see Supplemental Material). Unless stated otherwise, reported measurements throughout this work are averaged over three randomly generated network samples, and error bars represent $\\pm 1$ standard deviation. All simulations are performed using the open-source molecular dynamics simulation tool LAMMPS.\\cite{thompson_lammps_2022}\n\nWe first consider the effects of adjusting the crosslinker coverage fraction $p$ on the assembled network structure. Varying $p$ from $0.4$ to $0.9$, we find that the connectivity of the fully assembled network structures ranges from $z\\in[2.8,3.6]$, increasing monotonically with $p$ (see Fig. \\ref{fig_2}a). These values are similar to those measured for collagen and fibrin networks in prior experimental work.\\cite{lindstrom_biopolymer_2010,beroz_physical_2017,jansen_role_2018,xia_anomalous_2021} Next, we compute the average inter-crosslink contour length $\\ell_c$, which we define as the average length of filament sections in the original network corresponding to edges in the reduced network. As shown in Fig. \\ref{fig_2}b, we find that the average inter-crosslink contour length $\\ell_c$ decreases monotonically with $p$. \n\n\n\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_2.pdf}\n\\caption{ \\label{fig_2} Linker coverage-dependence of key structural characteristics of assembled networks. With increasing linker coverage fraction $p$, we observe (a) increasing average connectivity $z$ and (b) decreasing average inter-crosslink contour length $\\ell_c$.}\n\\end{figure}\n\n\\section{Nonlinear rheology}\n\n\n\nAfter the assembly stage is halted, we proceed to the rheology stage, in which the strain-dependent steady-state viscoelastic properties of the system are measured via the time-dependent shear stress as the system fluctuates about the mechanically equilibrated state under a fixed shear strain. We impose a constant macroscopic simple shear strain $\\gamma$ using Lees-Edwards periodic boundary conditions \\cite{lees_computer_1972} and, using the conjugate gradient method, initially obtain the energy-minimizing configuration of the network, corresponding to the mechanically equilibrated state at $T = 0$. Then, we evolve the system according to Eq. \\ref{eqn:eom} at temperature $T>0$, specified in Table S1 in Supplemental Material, for a total run time $\\tau_\\mathrm{tot} = 3 \\times 10^7 \\Delta t$, discarding the first half of the trajectory to avoid initialization effects. For a given configuration of the system, we compute the instantaneous virial stress tensor,\n\\begin{equation}\n \\sigma_{\\alpha \\beta}=\\frac{1}{2V}\\sum_{i>j}{f_{ij\\alpha} r_{ij\\beta}}\n\\end{equation}in which $\\mathbf{r}_{ij}=\\mathbf{x}_j-\\mathbf{x}_i$ and $\\mathbf{f}_{ij}$ is the force acting on node $i$ due to its interaction with node $j$. Because we will focus on macroscopic simple shear oriented along the $x$-axis with gradient direction $z$, we now define $\\sigma\\equiv\\sigma_{xz}$ to simplify notation. Once we have obtained a time series of the shear stress $\\sigma(\\gamma, t)$ at strain $\\gamma$, we calculate the time-averaged shear stress $\\langle \\sigma(\\gamma, t)\\rangle_t $. After repeating this procedure over many strains in the interval $\\gamma\\in[0,1]$, we compute the strain-dependent differential shear modulus,\n\\begin{equation}\n K_\\mathrm{eq}(\\gamma)=\\frac{\\partial \\langle \\sigma(\\gamma, t)\\rangle_t}{\\partial \\gamma},\n\\end{equation} which measures the apparent stiffness of the sample under macroscopic strain $\\gamma$ in response to an infinitesimal additional shear strain step. For sufficiently small strains, this yields the corresponding linear shear modulus, $G_0=\\lim_{\\gamma\\to0}K_\\mathrm{eq}(\\gamma)$. \n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_3.pdf}\n\\caption{ \\label{fig_3} (a) Time-averaged shear stress $\\langle \\sigma(\\gamma, t)\\rangle_t$ as a function of applied shear strain $\\gamma$, for a single network sample with linker coverage fraction $p=0.9$. (b) Differential shear modulus $K_\\mathrm{eq}=\\partial \\langle \\sigma(\\gamma, t)\\rangle_t\/\\partial \\gamma$. The linear shear modulus $G_0$ and critical strain $\\gamma_c$ are indicated by dashed lines, and the solid line is a fit to the equation of state from Ref. \\citenum{sharma_strain-controlled_2016}, described in Supplemental Material, Section S3. }\n\\end{figure}\n\nIn Fig. \\ref{fig_3}a and b, we plot the mean stress $\\langle\\sigma(\\gamma,t)\\rangle_t$ and differential shear modulus $K_\\mathrm{eq}$ as a function of strain for a single network sample with the parameters specified in Table S1 (see Supplemental Material) and linker coverage fraction $p=0.9$. We determine the critical strain $\\gamma_c$, which indicates the transition between the bending-dominated and stretching-dominated mechanical regimes, as the inflection point of the $\\log{K_\\mathrm{eq}}$ vs. $\\log{\\gamma}$ curve. To avoid issues associated with differentiating noisy data, we can alternatively find $G_0$ and $\\gamma_c$ by fitting the entire $K_\\mathrm{eq}$ vs. $\\gamma$ curve to an Ising-like equation of state discussed in prior work,\\cite{sharma_strain-controlled_2016} as we describe in further detail in Supplemental Material, Section S3. Such a fit is shown in Fig. \\ref{fig_3}b. For the data presented here, both methods yield effectively equivalent values of $G_0$ and $\\gamma_c$.\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_4.pdf}\n\\caption{ \\label{fig_4} Effects of linker coverage on features of the linear and nonlinear stiffness in shear. a) Differential shear modulus $K_\\mathrm{eq}$ for networks with varying linker coverage fraction $p$ as a function of strain and (b) as a function of shear stress. Increasing the linker coverage fraction $p$ drives (c) an increase in the linear modulus $G_0=\\lim_{\\gamma\\to 0} K_\\mathrm{0}$, with $G_0\\propto p^3$, and (d) a decrease in the critical strain $\\gamma_c$ as $\\gamma_c\\propto p^{-1}$. }\n\\end{figure}\n\n\nIn Fig \\ref{fig_4}, we report the strain-dependence of the differential shear modulus for networks with varying linker coverage fraction $p$. As $p$ increases, the critical strain evidently decreases (stiffening occurs at lower applied strains) and the linear shear modulus increases. These observations agree qualitatively with both the observed crosslinker concentration-dependence of the linear modulus $G_0$ and rupture strain $\\gamma_\\mathrm{max}$ (which we expect to be proportional to $\\gamma_c$) of F-actin gels reported in Ref. \\citenum{gardel_elastic_2004}, and with the connectivity dependence of the linear modulus and critical strain observed in spring network simulations.\\cite{wyart_elasticity_2008} The same differential shear modulus data are plotted in Fig. \\ref{fig_4}b as a function of stress. The linear modulus $G_0$ and critical strain $\\gamma_c$ values extracted from these curves are plotted as functions of the linker coverage fraction $p$ in Fig. \\ref{fig_4}c and d, respectively. We find that each exhibits a power law dependence on $p$, with $G_0\\propto p^3$ and $\\gamma_c\\propto p^{-1}$. In Ref. \\citenum{gardel_elastic_2004}, in which the analogous quantity $R$ (the ratio between the concentrations of actin and the crosslinking protein scruin) is varied, the authors report $G_0\\propto R^2$ and $\\gamma_c\\propto R^{-0.6}$. These observations may be consistent with ours, provided that the linker coverage fraction in our system maps to the experimental crosslinker concentration ratio in Ref. \\citenum{gardel_elastic_2004} as $p\\propto R^{x}$ with $x\\sim 0.6$. Separately, the authors of Ref. \\citenum{tharmann_viscoelasticity_2007} reported that, for networks of actin filaments crosslinked by heavy meromyosin, $G_0\\propto R^{1.2}$ and $\\gamma_c\\propto R^{-0.4}$, consistent with our observed dependence of both quantities on $p$ if $x\\sim 0.4$. However, meaningfully relating $p$ and $R$ will require further investigation. Here, for example, no two sticky sites on the same filament can reside closer than a distance $\\ell_0$ ($1\\; \\si{\\micro\\meter}$ with our parameters) from one another. This is obviously not the case in real F-actin networks. \n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_5.pdf}\n\\caption{ \\label{fig_5} The linear shear modulus $G_0$ (a) increases dramatically with connectivity, shown here for systems with varying linker coverage fraction $p$, and (b) decreases with increasing average inter-crosslink contour length $\\ell_c$. For small $\\ell_c$ (high $p$), we find $G_0 \\propto \\ell_c^{-3}$, consistent with the naive scaling predicted for affine, isotropic semiflexible filament networks.\\cite{mackintosh_elasticity_1995} Deviation from this scaling appears at larger $\\ell_c$. (c) The critical strain $\\gamma_c$, in contrast, decreases approximately linearly with $z$. A linear fit suggests $\\gamma_c\\to0$ as $z\\to z^*\\sim 4$, roughly consistent with the upper bound for $z$ in the high crosslink density limit.\\cite{broedersz_filament-length-controlled_2012} (d) The critical strain increases approximately linearly with increasing average inter-crosslink contour length, $\\gamma_c\\propto\\ell_c$, again in apparent agreement with the scaling expected for the naive affine model. }\n\\end{figure}\n\n\nIn Fig. \\ref{fig_5}, we plot the same extracted linear shear modulus and critical strain measurements as functions of the $p$-dependent structural quantities discussed in the previous section, the average network connectivity $z$ and average inter-crosslink contour length $\\ell_c$. Notably, we find that the linear shear modulus $G_0$ decreases with increasing inter-crosslink contour length approximately as $G_0\\propto \\ell_c^{-3}$. This exponent is consistent with the analytically predicted dependence for an affinely deforming isotropic gel of semiflexible filaments, $G_0 \\propto \\rho k_B T \\ell_p^2\/\\ell_c^3$.\\cite{mackintosh_elasticity_1995} This agreement may be coincidental, as, for our range of parameters, the typical inter-crosslink contour lengths are presumably too small for bending modes to be properly resolved. This warrants further investigation in simulations with reduced coarse graining, i.e. smaller $\\ell_0\/\\ell_f$. In Fig. \\ref{fig_5}c, we show that $\\gamma_c$ approaches 0 as $z$ approaches an intercept value $z^*\\sim 4$, in apparent agreement with the upper limit of $z$ for crosslinked networks with high crosslinking density.\\cite{broedersz_filament-length-controlled_2012} However, we expect some dependence of this limit on the total filament length $\\ell_f$ and the bending rigidity $\\kappa$, which both necessarily influence the structure of the assembled network. We also find that the critical strain increases linearly with the average inter-crosslink contour length $\\ell_c$ (see Fig. \\ref{fig_5}d), in apparent agreement with the expected scaling of the critical strain with contour length for an affinely deforming isotropic gel of semiflexible filaments, $\\gamma_c \\propto \\ell_c\/\\ell_p$; as with the dependence of $G_0$ on $\\ell_c$, this may be a coincidence.\n\n\n\nFurther information is contained in the fluctuations of the instantaneous stress about its average value,\\cite{wittmer_shear_2013}\n\\begin{equation}\n \\delta\\sigma(\\gamma,t)=\\sigma(\\gamma,t)-\\langle\\sigma(\\gamma,t)\\rangle_t,\n\\end{equation}\nfrom which we compute the stress fluctuation autocorrelation function,\n\\begin{equation}\n C(\\gamma,\\tau)=\\frac{V}{k_B T}\\langle\\delta\\sigma(\\gamma,t)\\delta\\sigma(\\gamma,t+\\tau)\\rangle_t,\n\\end{equation}\nwhich reveals useful details about the time-dependence of energy dissipation.\\cite{wittmer_fluctuation-dissipation_2015,wittmer_shear-stress_2015}\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_6.pdf}\n\\caption{ \\label{fig_6} The relaxation of stress fluctuations slows dramatically near the critical strain as rearrangements become increasingly nonaffine. (a) The shear stress fluctuation autocorrelation function $C(\\gamma, \\tau)$ measured at representative strains below (purple), near (red), and above (orange) the critical strain $\\gamma_c$ (see the corresponding $K_\\mathrm{eq}$ data in the inset of panel (b)) reveals this slowing, which is captured quantitatively by (b) a peak in the apparent slowest relaxation time $\\tau_R$ (Eq. \\ref{eqn:relaxationtime}). These data correspond to the same sample as in Fig. \\ref{fig_3} (see inset), and the labeled strains are indicated by the same colors in all panels. (c) At long lag times, the time-dependent apparent excess viscosity $\\delta\\eta(\\gamma, \\tau)$ computed with Eq. \\ref{eqn:visc} grows much larger near $\\gamma_c$ than elsewhere. This behavior is especially clear when (d) the apparent excess zero-shear viscosity $\\delta\\eta(\\tau=\\tau_\\mathrm{max})$ is plotted as a function of $\\gamma$, revealing a peak at the $p$- (and thus $z$-) dependent critical strain. These observations are supported by independent measurements of the strain-dependent quasistatic, athermal differential nonaffinity $\\delta\\Gamma_\\infty$, which is quantitatively related to $\\delta\\eta$ by Eq. \\ref{eqn:nonaff_viscosity} (solid blue line). }\n\\end{figure}\n\n\nIn Fig. \\ref{fig_6}a, we show representative $C(\\gamma,\\tau)$ data for a single sample (the same sample as in Fig. \\ref{fig_3}) under applied strains below, near, and above the critical strain $\\gamma_c$. We find that the stress fluctuations decay slowly when the macroscopic applied strain is near $\\gamma_c$, in contrast to a much faster decay for strains below or above the critical regime. We can quantify this strain-dependent change in relaxation dynamics by integrating the shear stress autocorrelation function $C(\\gamma,\\tau)$ over a sufficiently long range of lag times $\\tau$. In practice, the range of lag times for which we can reliably estimate $C(\\gamma,\\tau)$ is limited by the simulation run time. Assuming $C(\\gamma,\\tau)$ is known for lag times below a maximum $\\tau_\\mathrm{max}$, we can estimate the system's slowest relaxation time as\n\\begin{equation}\\label{eqn:relaxationtime}\n \\tau_\\mathrm{R}(\\gamma, \\tau_\\mathrm{max}) = \\frac{\\int_0^{\\tau_\\mathrm{max}}{\\tau C(\\gamma,\\tau)d\\tau}}{\\int_0^{\\tau_\\mathrm{max}}{C(\\gamma,\\tau)d\\tau}}\n\\end{equation}\nIn the limit $\\tau_\\mathrm{max}\\to\\infty$, this converges to the true slowest relaxation time $\\tau_\\mathrm{R,0}(\\gamma)$. In Fig. \\ref{fig_6}b, we plot the apparent slowest relaxation time $\\tau_\\mathrm{R}$ for the same system, with marker colors indicating the strains plotted in Fig. \\ref{fig_6}a. Note that the corresponding stiffening curve from Fig. \\ref{fig_3} is also shown in the inset. We observe that $\\tau_\\mathrm{R}$ grows substantially as the critical strain $\\gamma_c$ is approached from either side, reaching a maximum at $\\gamma_c$. In fact, we see consistent behavior -- that is, growth of the slowest relaxation time by an order of magnitude or more with a peak at $p$-dependent critical strain $\\gamma_c$ -- in networks over the entire range of $p$ considered. This is shown in Fig. \\ref{fig_7}d, in which the applied strain is normalized by the $p$-dependent critical strain $\\gamma_c$, revealing maxima in $\\tau_\\mathrm{R}$ at $\\gamma\/\\gamma_c=1$ for all $p$.\n\n\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig_7.pdf}\n\\caption{ \\label{fig_7} Shared features of stiffening and slowing down at the structure-dependent critical strain. (a) For networks with varying crosslinker coverage fraction $p$, the stiffening regime of the normalized differential shear modulus $K\/G_0$ collapses onto a single curve when plotted vs. $\\gamma\/\\gamma_c$. Lines correspond to fits to the equation of state described in Ref. \\citenum{sharma_strain-controlled_2016} (see Supplemental Material, Section S3), and the associated fit parameters are plotted as a function of $p$ in Fig. S2. (b) The mean-squared stress fluctuations $\\langle \\delta\\sigma(\\gamma, t)^2\\rangle_t$ are largest at the $p$- (and thus $z$-) dependent critical strain. Solid lines correspond to the difference between the measured strain-dependent affine and equilibrium differential shear moduli, $K_\\mathrm{aff}$ and $K_\\mathrm{eq}$, according to Eq. \\ref{eqn:stressfluc}. Both the (a) apparent excess zero-shear viscosity $\\delta\\eta$ and (b) apparent slowest relaxation time $\\tau_\\mathrm{R}$ likewise exhibit large peaks at the $p$-dependent critical strain $\\gamma_c$.}\n\\end{figure}\n\n\nThe stress fluctuation autocorrelation function is likewise related to the system's lag-dependent excess differential viscosity \\cite{wittmer_fluctuation-dissipation_2015} $\\delta\\eta(\\gamma,\\tau)\\equiv\\eta(\\gamma,\\tau)-\\eta_s$ measured at strain $\\gamma$,\n\\begin{equation} \\label{eqn:visc}\n \\delta\\eta(\\gamma,\\tau) =\\int_0^{\\tau} C(\\gamma,\\tau') d\\tau',\n\\end{equation}\nwhich we plot in Fig. \\ref{fig_6}c for the same labeled strain values. It is clear that $\\delta\\eta(\\gamma,\\tau)$ grows far more dramatically near the critical strain than elsewhere.\nIn recent work,\\cite{shivers_nonaffinity_2022} it was suggested that the low-frequency or ``zero-shear'' excess differential viscosity $\\delta\\eta_0(\\gamma) = \\lim_{\\tau\\to \\infty} \\delta\\eta(\\gamma, \\tau)$ of prestrained disordered networks is controlled by nonaffinity. Specifically, $\\delta\\eta_0$ was shown to be related to the quasistatic, athermal differential nonaffinity $\\delta\\Gamma_\\infty(\\gamma)$, defined as\n\\begin{equation}\\label{eqn:nonaffinity}\n \\delta\\Gamma_\\infty(\\gamma)=\\lim_{\\delta\\gamma\\to 0}\\frac{1}{N\\ell_0^2\\delta\\gamma^2}\\sum_i\\left|\\delta\\mathbf{x}_i^\\mathrm{NA}\\right|^2\n\\end{equation}\nin which the sum is taken over all network nodes and the vector $\\delta\\mathbf{x}_i^\\mathrm{NA}$ represents the nonaffine component of the displacement vector of node $i$ under a macroscopic, quasistatically applied strain step $\\delta\\gamma$. The viscosity-nonaffinity relationship in Ref. \\citenum{shivers_nonaffinity_2022} can be stated concisely as\n\\begin{equation}\\label{eqn:nonaff_viscosity}\n \\delta\\eta_0(\\gamma)=\\frac{N}{V}\\zeta\\ell_0^2\\delta\\Gamma_\\infty(\\gamma).\n\\end{equation}\nThe left side of Eq. \\ref{eqn:nonaff_viscosity} reflects the dynamics of the stress fluctuations at finite temperature, while the right side reflects the heterogeneous nature of the strictly quasistatic deformation field, which can be obtained by comparing energy-minimized, $T=0$ system configurations under varying applied $\\gamma$.\n\nSimulations have suggested that, for disordered filament networks, the differential nonaffinity generically reaches a maximum at the structure-dependent critical strain,\\cite{rens_nonlinear_2014,sharma_strain-driven_2016} at which the system macroscopically transitions between bending-dominated (or floppy, for $\\kappa=0$) and stretching-dominated regimes. Thus, according to Eq. \\ref{eqn:nonaff_viscosity}, we should generically see a proportional peak in the excess differential viscosity at $\\gamma_c$. Testing this prediction requires measuring the quasistatic, athermal differential nonaffinity. To do so, for a given network at strain $\\gamma$, we first obtain the energy-minimizing configuration at $T=0$ using the conjugate gradient method. To this configuration, we then apply a small additional strain step $d\\gamma = 0.01$, after which we repeat the energy minimization procedure. Comparing the positions in the energy-minimizing configurations at $\\gamma$ and $\\gamma + d\\gamma$, we compute the quasistatic differential nonaffinity $\\delta\\Gamma_\\infty$ using Eq. \\ref{eqn:nonaffinity}. In Fig. \\ref{fig_6}d, we plot $\\delta\\Gamma_\\infty$ along with the apparent excess differential zero-shear viscosity for the same system, demonstrating excellent agreement with Eq. \\ref{eqn:nonaff_viscosity}. To demonstrate that this behavior is preserved as the structure is varied, we show in Fig. \\ref{fig_7}c that a clear peak in $\\delta\\eta$ occurs at $\\gamma\/\\gamma_c = 1$ over the entire range of $p$ considered.\n\nWe can also use the stress fluctuations to determine the differential relaxation modulus,\\cite{wittmer_shear-strain_2015}\n\\begin{equation}\n K(\\gamma,\\tau)=K_\\mathrm{eq}(\\gamma)+C(\\gamma,\\tau),\n\\end{equation}\nwhich quantifies the time-dependent apparent stiffness of the system at prestrain $\\gamma$ in response to an instantaneous additional strain step.\\cite{rubinstein_polymer_2006,doi_theory_1988}\nFor sufficiently short times, the differential relaxation modulus approaches an upper limit of $\\lim_{\\tau\\to 0}K(\\gamma,\\tau) = K_\\mathrm{aff}(\\gamma)$, corresponding to the apparent stiffness of the energy-minimized, athermal equivalent of the system at strain $\\gamma$ under an instantaneous, homogeneous infinitesimal shear strain step. Consequently, the equilibrium stiffness of the system under applied strain $\\gamma$ is simply $K_\\mathrm{aff}(\\gamma)$ reduced by $C(\\gamma,0)$,\\cite{squire_isothermal_1969,lutsko_generalized_1989,wittmer_shear_2013} or equivalently\n\\begin{equation} \\label{eqn:stressfluc}\n K_\\mathrm{eq}(\\gamma)=K_\\mathrm{aff}(\\gamma)-\\frac{V}{k_B T}\\langle\\left(\\delta\\sigma(\\gamma,t)\\right)^2\\rangle_t.\n\\end{equation}\nWe find that this relationship provides a useful estimate of the mean squared stress fluctuations, as shown in Fig. \\ref{fig_7}b, and we observe that the mean squared stress fluctuations robustly reach a maximum at $\\gamma\/\\gamma_c = 1$.\n\n\n\n\n\\section{Conclusions}\n\nWe have investigated the assembly and mechanical testing of disordered networks of crosslinked semiflexible polymers via Brownian dynamics simulations. Such networks serve as essential mechanical constituents of a wide variety of biological materials spanning many length scales. We explored the structural and rheological consequences of varying the crosslinker coverage fraction $p$, an analog of the experimental ratio between the concentrations of crosslinker and filament proteins. Using physical parameters intended to mimic the cytoskeletal polymer F-actin, we measured the effects of increasing linker coverage fraction $p$ on the average connectivity $z$ of self-assembled networks, observing a corresponding increase in connectivity between $z\\in[2.8,3.6]$.\n\nWe then investigated the relationship between these changes in connectivity and associated changes in the strain-dependent rheological properties of assembled networks, obtaining extended time series measurements of the fluctuating shear stress $\\sigma(\\gamma, t)$ for systems held under fixed shear strain $\\gamma$. Analyzing many such trajectories gathered over a range of applied strains $\\gamma$, we computed the strain-dependent differential shear modulus $K_\\mathrm{eq}$, from which we extracted both the linear shear modulus $G_0$ and the critical strain $\\gamma_c$. We described the dependence of these quantities on the crosslinker coverage fraction $p$, demonstrating qualitative agreement with the experimentally observed dependence of the same quantities on crosslinker concentration ratio $R$ reported in Ref. \\citenum{gardel_elastic_2004}. Specifically, we found that increasing $p$ produces inherently stiffer networks (having an increased linear shear modulus $G_0$) that simultaneously exhibit an earlier tendency to strain-stiffen (having a lower critical strain $\\gamma_c$). Drawing comparisons between the scaling of both $G_0$ and $\\gamma_c$ with $p$ and the analogous experimentally-observed scaling of the same quantities with $R$, we suggested a simple power law relationship between $p$ and $R$. We next described the apparent dependence of these elastic features on the $p$-dependent structural quantities $z$ and $\\ell_c$, notably showing that the critical strain decreases linearly with $z$. A linear fit suggests that $\\gamma_c$ approaches $0$ as the connectivity $z$ approaches a limiting value $z^*\\to z_c$ near $4$, the theoretical upper bound for $z$ at high crosslinking density.\n \nWe then extended our observations beyond strictly elastic properties by analyzing the stress fluctuation autocorrelation function $C(\\gamma,\\tau)$, which revealed the development of extremely slow dynamics in systems subjected to applied shear strains near the critical strain $\\gamma_c$. From $C(\\gamma,\\tau)$, we obtained estimates of the slowest relaxation time $\\tau_{\\mathrm{R},0}$ and the excess differential zero-shear viscosity $\\delta\\eta_0$, both of which we showed are consistently maximized at the $p$-dependent critical strain, irrespective of the details of the underlying network. Building upon results from Ref. \\citenum{shivers_nonaffinity_2022}, we demonstrated that the excess differential viscosity in these finite temperature systems is controlled quantitatively by the athermal, quasistatic differential nonaffinity $\\delta\\Gamma_\\infty$, a measure of the inherent tendency of the strained network to deform heterogeneously. Importantly, our results suggest that one should expect measurable dynamical signatures of transition-associated nonaffine fluctuations to appear in semiflexible polymer networks with biologically relevant elastic properties, e.g. those of the F-actin cytoskeleton, under physiologically relevant applied strains. In other words, one should generically expect to observe slowing stress relaxation in biopolymer networks under prestrain levels near the critical strain $\\gamma_c$ marking the macroscopic transition between bending-dominated and stretching-dominated elasticity. Since $\\gamma_c$ is controlled by the average connectivity $z$, which is controlled in turn by the availability of crosslinking sites (here, $p$), our results suggest a route to experimentally control the strain-dependence of major features of both network elasticity and the dynamics of stress relaxation.\n\n\n\n\\begin{acknowledgments}\n\nThis study was supported in part by the National Science\nFoundation Division of Materials Research (Grant No. DMR-1826623) and the National Science Foundation Center for Theoretical Biological Physics (Grant No. PHY-2019745). This work began as a summer research experience for undergraduates (REU) project supported by the Frontiers in Science (FIS) program of the Center for Theoretical Biological Physics.\n\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev}\n\n\\section{Calculation of the connectivity and average contour length}\\label{appendix_connectivity}\n\nTo estimate the connectivity of a given assembled structure, we consider a reduced network defined by (1) merging each pair of crosslinked \"linker\" nodes into a single vertex and then (2) merging each filament section separating two vertices into a single edge. This is sketched in Fig. \\ref{figS1}. \n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Fig_S1.pdf}\n\\caption{ \\label{figS1} To calculate a network's average connectivity $z$, we consider a reduced network structure that treats each crosslink as a node and each inter-crosslink (non-dangling) filament section as an edge. On the right, a few example nodes in the reduced network are labeled according to their connectivity. \n}\n\\end{figure}\n\n\n\\section{Parameters}\\label{appendix_parameters}\n\nIn the interest of drawing comparison to experimental results in Ref. \\citenum{gardel_elastic_2004}, we have chosen parameter values roughly appropriate for F-actin in laboratory conditions. Specific values are listed in Table \\ref{table1}.\n\n\\begin{table}[htb!]\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{quantity} & \\textbf{symbol} & \\textbf{value} \\\\ \n\\hline\nnumber of filaments & $N_f$ & $500$ \\\\\nfilament length & $\\ell_f$ & $9\\;\\si{\\micro\\meter}$ \\\\\nfilament length per volume & $\\rho$ & $2.6\\;\\si{\\micro\\meter}^{-2}$ \\\\\nnumber of nodes per filament & $n$ & $10$ \\\\\n\nfilament persistence length & $\\ell_p$ & $17\\;\\si{\\micro\\meter}$ \\\\\ncrosslink rest length & $\\ell_{0,c\\ell}$ & $0.2\\;\\si{\\micro\\meter}$ \\\\\nfilament stretching rigidity & $\\mu$ & $588\\;\\si{\\pico\\newton}$ \\\\\nthermal energy scale & $k_B T$ & $4.11\\times 10^{-3}\\;\\si{\\pico\\newton}\\cdot\\si{\\micro\\meter}$ \\\\\nsolvent viscosity & $\\eta_s$ & $0.001\\;\\si{\\pico\\newton}\\cdot\\si{\\micro\\meter}^{-2}\\cdot \\si{s}$\\\\\ntimestep & $\\Delta t$ & $9.42\\times10^{-7}\\;\\mathrm{s}$ \\\\\ntotal time, assembly & $\\tau_\\mathrm{a}$ & $60\\;\\mathrm{s}$ \\\\ \ntotal time, rheology & $\\tau_\\mathrm{tot}$ & $30\\;\\mathrm{s}$ \\\\ \n\\hline\n\\end{tabular}\n\\caption{ \\label{table1} Independent parameters, in real units.}\n\\end{table}\n\n\\begin{table}[htb!]\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{quantity} & \\textbf{symbol} & \\textbf{value} \\\\ \n\\hline\nfilament bond rest length & $\\ell_0=\\ell_f\/(n-1)$ & $1\\;\\si{\\micro\\meter}$ \\\\\nsimulation box edge length & $L=(N_f\\ell_f\/\\rho)^{1\/3}$ & $12\\;\\si{\\micro\\meter}$ \\\\\nfilament bending rigidity & $\\kappa = k_B T \\ell_p$ & $6.99\\times10^{-2}\\;\\si{\\pico\\newton}\\cdot\\si{\\micro\\meter}^2$ \\\\\n\\hline\n\\end{tabular}\n\\caption{ \\label{table2} Parameter-dependent quantities.}\n\\end{table}\n Assuming a molecular mass of $42 \\;\\si{\\kilo\\dalton}$ for actin\\cite{ballweber_polymerisation_2002}, the mass per length is $\\sim 16\\times10^3\\; \\si{\\kilo\\dalton}\/\\si{\\micro\\meter}$, which translates to $0.63\\;\\si{\\micro \\mol}\/\\si{\\micro\\meter}$. Therefore, our chosen length density of $\\rho=2.6\\;\\si{\\micro\\meter}^{-2}$ corresponds to an actin concentration of $c_A = 1.63\\;\\si{\\micro M}$.\n\nQuantities used in simulations are nondimensionalized by characteristic length, force, and drag coefficient values: $\\ell^*=1\\;\\si{\\micro\\meter}$, $f^*=1\\;\\si{\\pico\\newton}$, and $\\zeta^*=6\\pi\\eta_s(\\ell_0\/2)=9.42\\times10^{-3}\\;\\si{\\pico\\newton}\\cdot\\si{\\micro\\meter}\\cdot \\mathrm{s} $ (for example, the simulation timestep is $\\Delta t_\\mathrm{sim}=\\Delta t\/(\\zeta^* \\ell^*\/f^*)=10^{-4}$. Note that one can define a dimensionless bending rigidity $\\tilde{\\kappa}=\\kappa\/(\\mu\\ell_0^2)$, as in past work.\\cite{sharma_strain-controlled_2016} With our parameters, $\\tilde{\\kappa}=1.18\\times10^{-4}$.\n\n\n\n\\section{Fits of K to the expected scaling form}\\label{appendix_fits}\nWe assume that the differential shear modulus $K=\\partial\\sigma\/\\partial\\gamma$ can be written as\n\\begin{equation} \\label{Eq1}\nK = a\\mathcal{K}(\\tilde{\\kappa},\\gamma)\n\\end{equation}in which $a$ is some prefactor with units of stress, $\\tilde{\\kappa}$ is a dimensionless bending rigidity, and $\\mathcal{K}$ is a dimensionless function of strain that scales with the distance to a critical strain $\\gamma_c$ as\\cite{sharma_strain-controlled_2016}\n\\begin{equation}\n\\mathcal{K} \\propto|\\gamma-\\gamma_c|^f\\mathcal{G}_\\pm\\left(\\frac{\\tilde{\\kappa}}{|\\gamma-\\gamma_c|^\\phi}\\right).\n\\end{equation}This scaling is reproduced by the constitutive equation\n\\begin{equation}\n\\frac{\\tilde{\\kappa}}{|\\gamma-\\gamma_c|^\\phi} \\sim \\frac{\\mathcal{K}}{|\\gamma-\\gamma_c|^f}\\left(\\pm1 + \\frac{\\mathcal{K}^{1\/f}}{|\\gamma-\\gamma_c|}\\right)^{\\phi-f}\n\\end{equation}\nin which the plus and minus correspond to the regions below and above the critical strain, respectively.\nWe compute $\\mathcal{K}(\\gamma)$ by numerical inversion of $\\tilde{\\kappa}\\left(\\mathcal{K}\\right)$:\n\\begin{equation}\\label{eq4} \\tilde{\\kappa}=b\\mathcal{K}\\left(\\pm|\\gamma-\\gamma_c|+\\mathcal{K}^{1\/f}\\right)^{\\phi-f},\n\\end{equation}\nsetting $b=1$. Fit parameters for the data in Fig. 7a in the main text are shown in Fig. \\ref{figS2}.\n\n\\begin{figure}[htb!]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Fig_S2.pdf}\n\\caption{ \\label{figS2} Best-fit parameters for systems with varying crosslinker coverage fraction $p$, corresponding to the solid curves in Fig. 7a in the main text.}\n\\end{figure}\n\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\nThe following two-point boundary value problem is considered for the\nsingularly perturbed linear system of second order differential\nequations\n\\begin{equation}\\label{BVP}\n-E\\vec{u}''(x)+A(x)\\vec{u}(x)=\\vec{f}(x),\\;\\;\\;\nx\\in(0,1),\\;\\;\\;\\vec{u}(0)\\;\\tx{and}\\;\\vec{u}(1)\\;\\tx{given.}\n\\end{equation}\nHere $\\;\\vec{u}\\;$ is a column $\\;n-\\tx{vector},\\;E\\;$ and\n$\\;A(x)\\;$ are $\\;n\\times n\\;$ matrices, $\\;E =\n\\tx{diag}(\\vec{\\eps}),\\;\\vec\\eps = (\\eps_1,\\;\\cdots,\\;\\eps_n)\\;$\nwith $\\;0\\;<\\;\\eps_i\\;\\le\\;1\\;$ for all $\\;i=1,\\ldots,n$. The\n$\\eps_i$ are assumed to be distinct and,\nfor convenience, to have the ordering \\[\\eps_1\\;<\\;\\cdots\\;<\\;\\eps_n.\\] Cases with some of the parameters coincident are not considered here.\\\\\n\\noindent The problem can also be written in the operator form\n\\[\\vec L\\vec u\\; = \\;\\vec{f},\\;\\;\\;\\vec{u}(0)\\;\\tx{and}\\;\\vec{u}(1)\\;\\tx{given}\\]\nwhere the operator $\\;\\vec{L}\\;$ is defined by\n\\[\\vec{L}\\;=\\;-E D^2 + A(x)\\;\\;\\;\\tx{and}\\;\\;\\;D^2 = \\dfrac{d^2}{dx^2}.\\]\nFor all $x \\in [0,1]$ it is assumed that the components $a_{ij}(x)$\nof $A(x)$\nsatisfy the inequalities \\\\\n\\begin{eqnarray}\\label{a1} a_{ii}(x) > \\displaystyle{\\sum_{^{j\\neq\ni}_{j=1}}^{n}}|a_{ij}(x)| \\; \\; \\rm{for}\\;\\; 1 \\le i \\le n, \\;\\;\n\\rm{and} \\;\\;a_{ij}(x) \\le 0 \\;\\; \\rm{for} \\; \\; i \\neq\nj\\end{eqnarray} and, for some $\\alpha$,\n\\begin{eqnarray}\\label{a2} 0 <\\alpha <\n\\displaystyle{\\min_{^{x \\in [0,1]}_{1 \\leq i \\leq n}}}(\\sum_{j=1}^n\na_{ij}(x)).\n\\end{eqnarray} Wherever necessary the required smoothness of the problem data is assumed.\nIt is also assumed, without loss of generality, that\n\\begin{eqnarray}\\label{a3}\n\\max_{1 \\leq i \\leq n} \\sqrt{\\eps_i} \\leq \\frac{\\sqrt{\\alpha}}{6}.\n\\end{eqnarray}\nThe norms $\\parallel \\vec{V} \\parallel =\\max_{1 \\leq k \\leq n}|V_k|$\nfor any n-vector $\\vec{V}$, $\\parallel y\n\\parallel =\\sup_{0\\leq x\\leq 1}|y(x)|$ for any\nscalar-valued function $y$ and $\\parallel \\vec{y}\n\\parallel=\\max_{1 \\leq k \\leq n}\\parallel y_{k}\n\\parallel$ for any vector-valued function $\\vec{y}$ are introduced.\nThroughout the paper $C$ denotes a generic positive constant, which\nis independent of $x$ and of all singular perturbation and\ndiscretization parameters. Furthermore, inequalities between vectors\nare understood in the componentwise sense.\\\\\n\n\\noindent For a general introduction to parameter-uniform numerical\nmethods for singular perturbation problems, see \\cite{MORS},\n\\cite{RST} and \\cite{FHMORS}. Parameter-uniform numerical methods\nfor various special cases of (\\ref{BVP}) are examined in, for\nexample, \\cite{MMORS}, \\cite{MS} and \\cite{HV}. For (\\ref{BVP})\nitself parameter-uniform numerical methods of first and second order\nare considered in \\cite{LM}. However, the present paper differs from\n\\cite{LM} in two important ways. First of all, the meshes, and hence\nthe numerical methods, used are different from those in \\cite{LM};\nthe transition points between meshes of differing resolution are\ndefined in a similar but different manner. The piecewise-uniform\nShishkin meshes $M_{\\vec{b}}$ in the present paper have the elegant\nproperty that they reduce to uniform meshes whenever\n$\\vec{b}=\\vec{0}$. Secondly, the proofs given here do not require\nthe use of Green's function techniques, as is the case in \\cite{LM}.\nThe significance of this is that it is more likely that such\ntechniques can be extended in future to problems in higher\ndimensions and to nonlinear problems, than is the case for proofs\ndepending on Green's functions. It is also satisfying to demonstrate\nthat the methods of proof pioneered by G. I. Shishkin can be\nextended successfully to systems of this kind.\n\nThe plan of the paper is as follows. In the next section both standard and novel bounds on the smooth and singular components of the exact solution are obtained.\nThe sharp estimates for the singular component in Lemma \\ref{lsingular}\n are proved by mathematical induction, while interesting orderings of the points $x_{i,j}$ are\nestablished in Lemma \\ref{layers}. In Section 4\n piecewise-uniform Shishkin meshes are introduced, the\ndiscrete problem is defined and the discrete maximum principle and\ndiscrete stability properties are established. In Section 6\n an expression for the local\ntruncation error and a standard estimate are stated.\nIn Section 7 parameter-uniform estimates for the local truncation error of the smooth and singular components\nare obtained in a sequence of theorems. The section\nculminates with the statement and proof of the essentially second order parameter-uniform error\nestimate.\\\\\n\n\\section{Standard analytical results}\nThe operator $\\vec L$ satisfies the following maximum principle\n\\begin{lemma}\\label{max} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2}). Let\n$\\;\\vec{\\psi}\\;$ be any function in the domain of $\\;\\vec L\\;$ such\nthat $\\;\\vec{\\psi}(0)\\ge\\vec{0}\\;$ and\n$\\;\\vec{\\psi}(1)\\ge\\vec{0}.\\;$ Then $\\;\\vec\nL\\vec{\\psi}(x)\\ge\\vec{0}\\;$ for all $\\;x\\;\\in\\;(0,1)\\;$ implies that\n$\\;\\vec{\\psi}(x)\\ge\\vec{0}\\;$ for all $\\;x\\;\\in\\;[0,1]$.\n\\end{lemma}\n\\begin{proof}Let $i^*, x^*$ be such that $\\psi_{i^*}(x^{*})=\\min_{i,x}\\psi_i(x)$\nand assume that the lemma is false. Then $\\psi_{i^*}(x^{*})<0$ .\nFrom the hypotheses we have $x^* \\not\\in\\;\\{0,1\\}$ and $\\psi^{\\prime\n\\prime}_{i^*}(x^*)\\geq 0$. Thus\n\\begin{equation*}(\\vec{L}\\vec \\psi(x^*))_{i^*}=\n-\\eps_{i^*}\\psi^{\\prime\\prime}_{i^*}(x^*)+\\sum_{j=1}^n\na_{i^*,j}(x^{*})\\psi_j(x^*)<0,\n\\end{equation*} which contradicts the assumption and proves the\nresult for $\\vec{L}$.\\eop \\end{proof}\n\nLet $\\tilde{A}(x)$ be any principal sub-matrix of $A(x)$ and\n$\\vec{\\tilde{L}}$ the corresponding operator. To see that any\n$\\vec{\\tilde{L}}$ satisfies the same maximum principle as $\\vec{L}$,\nit suffices to observe that the elements of $\\tilde{A}(x)$ satisfy\n\\emph{a fortiori} the same inequalities as those of $A(x)$.\n\\begin{lemma}\\label{stab} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2}). If $\\vec{\\psi}$ is any function in the domain of $\\;\\vec L,\\;$\n then for each $i, \\; 1 \\leq i \\leq n $, \\[|\\vec{\\psi}_i(x)| \\le\\;\n\\max\\ds\\left\\{\\parallel\\vec \\psi(0)\\parallel,\\parallel\\vec\n\\psi(1)\\parallel, \\dfrac{1}{\\alpha}\\parallel \\vec L\\vec\n\\psi\\parallel\\right\\},\\s x\\in [0,1].\\]\n\\end{lemma}\n\\begin{proof}Define the two functions \\[\\vec \\theta^\\pm(x)\\;=\\;\n\\max\\ds\\left\\{\\parallel\\vec \\psi(0)\\parallel,\\;\\parallel\\vec\n\\psi(1)\\parallel,\\;\\dfrac{1}{\\alpha}\\parallel\\vec{L}\\vec\n\\psi\\parallel\\right\\}\\vec e\\;\\pm\\;\\vec \\psi(x)\\] where $\\;\\vec\ne\\;=\\;(1,\\;\\ldots,\\;1)^T\\;$ is the unit column vector. Using the\nproperties of $\\;A\\;$ it is not hard to verify that $\\;\\vec\n\\theta^\\pm(0)\\;\\ge\\;\\vec 0,\\;\\;\\vec \\theta^\\pm(1)\\;\\ge\\;\\vec 0\\;$\nand $\\;\\vec L\\vec \\theta^\\pm(x)\\;\\ge\\;\\vec 0.\\;$ It follows from\nLemma \\ref{max} that $\\;\\vec \\theta^\\pm(x)\\;\\ge\\;\\vec 0\\;$ for all\n$\\;x\\;\\in\\;[0,\\;1].\\;$\\eop\n\\end{proof}\nA standard estimate of the exact solution and its derivatives is\ncontained in the following lemma.\n\\begin{lemma}\\label{lexact} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2})\nand let $\\vec u$ be the exact solution of (\\ref{BVP}). Then, for\neach $i=1\\; \\dots \\; n$, all $x\\in [0,1]$ \\; and \\; $k=0,1,2$,\n\\[|u_i^{(k)}(x)| \\leq\nC\\eps_i^{-\\frac{k}{2}}(||\\vec{u}(0)||+||\\vec{u}(1)||+||\\vec{f}||)\\]\n\\[|u_i^{(3)}(x)| \\leq\nC\\eps_i^{-\\frac{3}{2}}(||\\vec{u}(0)||+||\\vec{u}(1)||+||\\vec{f}||+\\sqrt{\\eps_i}||\\vec{f}^\\prime||)\\]\nand\n\\[|u_i^{(4)}(x)|\\leq\nC\\eps_i^{-2}(||\\vec{u}(0)||+||\\vec{u}(1)||+||\\vec{f}||+\\eps_i||\\vec{f}^{\\prime\\prime}||).\n\\]\n\\end{lemma}\n\\begin{proof}\nThe bound on $\\vec{u}$ is an immediate consequence of Lemma\n\\ref{stab} and the differential equation.\\\\ To bound\n$u_i^\\prime(x)$, for all $i$ and any $x$, consider an interval\n$N_x=[a,a+\\sqrt{\\eps_i}]$ such that $x \\in N_x$. Then, by the mean\nvalue theorem, for some $y\\in N_x$,\n\\[u_i^\\prime(y)=\\frac{u_i(a+\\sqrt{\\eps_i})-u_i(a)}{\\sqrt{\\eps_i}}\\]\nand it follows that \\[|u_i^\\prime(y)|\\leq\n2\\eps_i^{-\\frac{1}{2}}||u_i||.\n\\]\nNow\n\\[\\vec{u}^\\prime(x)=\\vec{u}^\\prime(y)+\\int_y^x \\vec{u}^{\\prime\\prime}(s)ds=\n\\vec{u}^\\prime(y)+E^{-1}\\int_y^x(-\\vec{f}(s)+A(s)\\vec{u}(s))ds\\] and\nso\n\\[|u_i^\\prime(x)|\\leq |u_i^\\prime(y)|+C\\eps_i^{-1}(||f_i||+||\\vec u||)\\int_y^x ds\n\\leq C\\eps_i^{-\\frac{1}{2}}(||f_i||+||\\vec u||)\\] from which the\nrequired bound follows.\\\\\nRewriting and differentiating the differential equation gives $\\vec u^{\\prime\\prime}= E^{-1}(A\\vec u-\\vec f), $\\; \\; $\\vec u^{(3)}= E^{-1}(A\\vec\nu^\\prime+A^\\prime\\vec u-\\vec f^\\prime),$ $\\vec u^{(4)}= E^{-1}(A\\vec u^{\\prime\\prime}+2A^\\prime\\vec u^{\\prime}+A^{\\prime\\prime}\\vec{u}-\\vec f^{\\prime\\prime}),$ and\nthe bounds on $u_i^{\\prime\\prime}$, $u_i^{(3)}$, $u_i^{(4)}$ follow.\\eop\\end{proof} The reduced solution $\\vec{u}_0$ of (\\ref{BVP}) is the solution of the reduced\nequation $A\\vec {u}_0=\\vec f$. The Shishkin decomposition of the exact solution $\\;\\vec{u}\\;$ of (\\ref{BVP}) is $\\;\\vec{u}=\\vec{v}+\\vec{w}\\;$ where the smooth\ncomponent $\\;\\vec v\\;$ is the solution of $\\;\\vec L\\vec v = \\vec f\\;$ with $\\;\\vec v(0) = \\vec u_0(0)\\;$ and $\\;\\vec v(1) = \\vec u_0(1)\\;$ and the singular\ncomponent $\\;\\vec w\\;$ is the solution of $\\vec L\\vec w\\;=\\;\\vec 0$ with $\\vec w(0)=\\vec u(0)-\\vec v(0)$ and $\\vec w(1)=\\vec u(1)-\\vec v(1).$ For convenience the\nleft and right boundary layers of $\\vec w$ are separated using the further decomposition $\\vec w =\\vec{w}^l+\\vec{w}^r$ where $\\vec{L}\\vec{w}^l=\\vec{0},\\;\n\\vec{w}^l(0)=\\vec u(0)-\\vec v(0),\\; \\vec{w}^l(1)=\\vec{0}$ and $\\vec{L}\\vec{w}^r= \\vec{0},\\;\n\\vec{w}^r(0)=\\vec{0},\\; \\vec{w}^r(1)=\\vec u(1)-\\vec v(1).$\\\\\nBounds on the smooth component and its derivatives are contained in\n\\begin{lemma}\\label{lsmooth} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2}).\nThen the smooth component $\\vec v$ and its derivatives satisfy, for all $x\\in [0,1]$, \\; $i=1,\\; \\dots \\; n$ \\;and \\; $k=0, \\;\\dots \\;4$,\n\\[|v_i^{(k)}(x)| \\leq C(1+\\eps_i^{1-\\frac{k}{2}}).\\]\n\\end{lemma}\n\\begin{proof}\nThe bound on $\\vec{v}$ is an immediate consequence of the defining\nequations for $\\vec{v}$ and Lemma\n\\ref{stab}.\\\\\nThe bounds on $\\vec{v}^{\\prime}$ and $\\vec{v}^{\\prime\\prime}$ are\nfound as follows. Differentiating twice the equation for $\\vec{v}$,\nit is not hard to see that $\\vec{v}^{\\prime\\prime}$ satisfies\n\\begin{equation}\\label{4a}\n\\vec{L}\\vec{v}^{\\prime\\prime}=\\vec{g},\\;\\; \\text{where} \\;\\;\n\\vec{g}=\n\\vec{f}^{\\prime\\prime}-A^{\\prime\\prime}\\vec{v}-2A^{\\prime}\\vec{v}^{\\prime.}\n\\end{equation} Also the defining equations for $\\vec{v}$ yield at $x=0,\\;\\; x=1$\n\\begin{equation}\\label{4b}\n\\vec{v}^{\\prime\\prime}(0)=\\vec{0}, \\;\\;\n\\vec{v}^{\\prime\\prime}(1)=\\vec{0}.\n\\end{equation} Applying Lemma \\ref{stab} to $\\vec{v}^{\\prime\\prime}$ then gives\n\\begin{equation}\\label{smooth1} ||\\vec{v}^{\\prime\\prime}|| \\leq C(1+||\\vec{v}^{\\prime}||).\n\\end{equation}\nChoosing $i^*,\\; x^*,$ such that $1 \\leq i^* \\leq n,\\; x^* \\in\n(0,1)$ and\n\\begin{equation}\\label{smooth2} v_{i^*}^{\\prime}(x^*)=||\\vec{v}^{\\prime}|| \\end{equation}\nand using a Taylor expansion it follows that, for any $y \\in\n[0,1-x^*]$ and some $\\eta$, $x^*\\;<\\;\\eta\\;<\\;x^*+y$,\n\\begin{equation}\\label{smooth3} v_{i^*}(x^*+y) = v_{i^*}(x^*)+y\\;v_{i^*}'(x^*)+\\dfrac{y^2}{2}\\;v_{i^*}^{\\prime\\prime}(\\eta).\\end{equation}\nRearranging (\\ref{smooth3}) yields\n\\begin{equation}\\label{smooth4}v_{i^*}^{\\prime}(x^*)=\\frac{v_{i^*}(x^*+y)-v_{i^*}(x^*)}{y}-\\frac{y}{2}v_{i^*}^{\\prime\\prime}(\\eta)\n\\end{equation}\nand so, from (\\ref{smooth2}) and (\\ref{smooth4}),\n\\begin{equation}\\label{smooth5}\n||\\vec{v}^{\\prime}|| \\leq \\frac{2}{y}||\\vec{v}||+\n\\frac{y}{2}||\\vec{v}^{\\prime\\prime}||.\n\\end{equation}\nUsing (\\ref{smooth5}), (\\ref{smooth1}) and the bound on $\\vec{v}$\nyields\n\\begin{equation}\\label{smooth6}\n(1-\\frac{Cy}{2})||\\vec{v}^{\\prime\\prime}|| \\leq C(1+\\frac{2}{y}).\n\\end{equation}\nChoosing $y=\\min(\\frac{1}{C},1-x^*)$, (\\ref{smooth6}) then gives $||\\vec{v}^{\\prime\\prime}|| \\leq C$ and (\\ref{smooth5}) gives $||\\vec{v}^{\\prime}|| \\leq C$ as\nrequired. The bounds on $\\vec{v}^{(3)}, \\vec{v}^{(4)}$ are obtained\nby a similar argument.\\eop\\end{proof}\n\\section{Improved estimates}\nThe layer functions $B^{l}_{i}, \\; B^{r}_{i}, \\; B_{i}, \\; i=1,\\;\n\\dots , \\; n,\\;$, associated with the solution $\\;\\vec u$, are\ndefined on $[0,1]$ by\n\\[B^{l}_{i}(x) = e^{-x\\sqrt{\\alpha\/\\eps_i}},\\;B^{r}_{i}(x) =\nB^{l}_{i}(1-x),\\;B_{i}(x) = B^{l}_{i}(x)+B^{r}_{i}(x).\\] The following elementary properties of these layer functions, for all $1 \\leq i < j \\leq n$ and $0 \\leq x <\ny \\leq\n1$, should be noted:\\\\\n(a)\\;$B^{l}_i(x)\\; <\\; B^{l}_j(x),\\;\\;B^{l}_i(x)\\;\n>\\; B^{l}_i(y), \\;\\;0\\;<\\;B^{l}_i(x)\\;\\leq\\;1$.\\\\\n(b)\\;$B^{r}_i(x)\\; <\\; B^{r}_j(x),\\;\\;B^{r}_i(x)\\; <\\;\nB^{r}_i(y), \\;\\;0\\;<\\;B^{r}_i(x)\\;\\leq\\;1$.\\\\\n(c)\\;$B_{i}(x)$ is monotone decreasing (increasing) for\nincreasing $x \\in [0,\\frac{1}{2}] ([\\frac{1}{2},1])$.\\\\\n(d)\\;$B_{i}(x) \\leq 2B_{i}^{l}(x)$ for $x \\in [0,\\frac{1}{2}]$.\n\n\n\\begin{definition}\nFor $B_i^l$, $B_j^l$, each $i,j, \\;\\;1 \\leq i \\neq j \\leq n$ and\neach $s, s>0$, the point $x^{(s)}_{i,j}$ is defined by\n\\begin{equation}\\label{x1}\\frac{B^l_i(x^{(s)}_{i,j})}{\\varepsilon^s _i}=\n\\frac{B^l_j(x^{(s)}_{i,j})}{\\varepsilon^s _j}. \\end{equation}\n\\end{definition}\nIt is remarked that\n\\begin{equation}\\label{x2}\\frac{B^r_i(1-x^{(s)}_{i,j})}{\\varepsilon^s _i}=\n\\frac{B^r_j(1-x^{(s)}_{i,j})}{\\varepsilon^s _j}. \\end{equation} In\nthe next lemma the existence and uniqueness of the points\n$x^{(s)}_{i,j}$ are shown. Various properties are also established.\n\\begin{lemma}\\label{layers} For all $i,j$, such that $1 \\leq i < j \\leq\nn$ and $0 \\frac{B^l_{j}(x)}{\\eps^s _j},\\;\\; x\n\\in [0,x^{(s)}_{i,j}),\\;\\; \\frac{B^l_{i}(x)}{\\eps^s _i} <\n\\frac{B^l_{j}(x)}{\\eps^s _j}, \\; x \\in (x^{(s)}_{i,j}, 1].\\end{equation}\\\\\nMoreover\n\\begin{equation}\\label{x4}x^{(s)}_{i,j}< x^{(s)}_{i+1,j}, \\; \\mathrm{if} \\;\\; i+11$ as $x$ increases\nfrom $0$.\\\\\nThe point $x^{(s)}_{i,j}$ is the unique point $x$ at which this\nratio has the value $1.$ Rearranging (\\ref{x1}), and using the\ninequality $\\ln x 1$, gives\n\\begin{equation}\\label{x5}x^{(s)}_{i,j} = 2s\\ds\\left[\n\\frac{\\ln(\\frac{1}{\\sqrt{\\eps_i}})-\n\\ln(\\frac{1}{\\sqrt{\\eps_j}})}{\\sqrt{\\alpha}(\\frac{1}{\\sqrt{\\eps_i}}-\n\\frac{1}{\\sqrt{\\eps_j}})}\\right]=\n\\frac{2s\\;\\ln(\\frac{\\sqrt{\\eps_j}}{\\sqrt{\\eps_i}})}{\\sqrt{\\alpha}(\\frac{1}{\\sqrt{\\eps_i}}-\n\\frac{1}{\\sqrt{\\eps_j}})}<\n2s\\sqrt{\\frac{\\eps_j}{\\alpha}},\\end{equation} which\nis the first part of (\\ref{newbound}). The second part follows immediately from this and (\\ref{a3}).\\\\\nTo prove (\\ref{x4}), writing $\\sqrt{\\eps_k} = \\exp(-p_k)$, for some\n$p_k\n> 0$ and all $k$, it follows that\n\\[x^{(s)}_{i,j}=\\frac{2s(p_i -p_j)}{\\sqrt{\\alpha}(\\exp{p_i} -\\exp{p_j})}.\\] The\ninequality $x^{(s)}_{i,j}< x^{(s)}_{i+1,j}$ is equivalent to\n\\[\\frac{p_i -p_j}{\\exp{p_i} -\\exp{p_j}}<\\frac{p_{i+1} -p_j}{\\exp{p_{i+1}} -\\exp{p_j}}, \\]\nwhich can be written in the form\n\\[(p_{i+1}-p_j)\\exp(p_i-p_j)+(p_{i}-p_{i+1})-(p_{i}-p_j)\\exp(p_{i+1}-p_j)>0. \\]\nWith $a=p_i-p_j$ and $b=p_{i+1}-p_j$ it is not hard to see that\n$a>b>0$ and $a-b=p_i-p_{i+1}$. Moreover, the previous inequality is\nthen equivalent to\n\\[\\frac{\\exp{a}-1}{a}>\\frac{\\exp{b}-1}{b}, \\] which is true because $a>b$ and proves\nthe first part of (\\ref{x4}). The second part is proved by a similar\nargument.\\\\ The analogous results for the $B^r_i$, $B^r_j$ and the\npoints $1-x^{(s)}_{i,j}$ are proved by a similar argument.\\eop\n\\end{proof}\nIn the following lemma sharper estimates of the smooth component are\npresented.\n\\begin{lemma}\\label{lsmooth2}\nLet $\\;A(x)\\;$ satisfy (\\ref{a1}) and (\\ref{a2}). Then the smooth\ncomponent $\\;\\vec v\\;$ of the solution $\\;\\vec u\\;$ of \\eqref{BVP}\nsatisfies for $\\;i=1,\\cdots,n, \\;k=0,1,2,3\\;$ and $\\;x\\in\\oln\\Omega$\n\\[|v_i^{(k)}(x)|\\;\\le\\;C\\;\\ds\\left(1+\\sum_{q=i}^{n}\\frac{B_q(x)}{\\eps_q ^{\\frac{k}{2}-1}}\\right).\\]\n\\end{lemma}\n\n\\begin{proof}Define a barrier function\n\\[\\vec\\psi^\\pm(x)\\;=\\;C[1+B_n(x)]\\vec e\\;\\pm\\;\\vec v^{(k)}(x),\\;\\;k=0,1,2\\;\\;\\;\\text{and}\\;\\;\\;x\\in\\oln\\Omega.\\]\nUsing Lemma \\ref{max}, we find that $\\;\\vec L\\vec\\psi^\\pm(x)\\;\\ge\\;\\vec 0\\;$ and $\\;\\vec\\psi^\\pm(0)\\;\\ge\\;\\vec 0,\\;\\vec\\psi^\\pm(1)\\;\\ge\\;\\vec 0\\;$ for proper choices of the constant $\\;C.\\;$\\\\\nThus using Lemma \\ref{lsmooth} we conclude that for $\\;k=0,1,2,\\;$\n\\begin{equation}\\label{4e}\n|v_i^{(k)}(x)|\\;\\le\\;C[1+B_n(x)],\\;\\;x\\in\\oln\\Omega.\n\\end{equation}\nConsider the system of equations \\eqref{4a}, \\eqref{4b} satisfied by $\\;\\vec v^{\\prime\\prime},$\nand note that $\\;\\parallel\\vec g^{\\prime}\\parallel\\;\\le\\;C\\;$ from Lemma \\ref{lsmooth}.\\\\\nFor convenience let $\\;\\vec p\\;$ denote $\\;\\vec v^{\\prime\\prime}\\;$ then\n\\begin{equation}\\label{4c}\n\\vec L\\vec p\\;=\\;\\vec g,\\;\\;\\vec p(0)\\;=\\;\\vec 0,\\;\\vec p(1)\\;=\\;\\vec 0.\n\\end{equation}\nLet $\\;\\vec q\\;$ and $\\;\\vec r\\;$ be the smooth and singular components of $\\;\\vec p\\;$ given by\n\\[\\vec L\\vec q\\;=\\;\\vec g,\\;\\;\\vec q(0)\\;=\\;A(0)^{-1}\\vec g(0),\\;\\vec q(1)\\;=\\;A(1)^{-1}\\vec g(1)\\]\nand\n\\[\\vec L\\vec r\\;=\\;\\vec 0,\\;\\;\\vec r(0)\\;=\\;-\\vec q(0),\\;\\vec r(1)\\;=\\;-\\vec q(1).\\]\nUsing Lemmas \\ref{lsmooth} and \\ref{lsingular} we have, for\n$\\;i=1,\\cdots,n\\;$ and $\\;x\\in\\oln\\Omega,\\;$\n\\[\\begin{array}{lcl}\n|q_i^{\\prime}(x)|&\\le&C,\\\\\n|r_i^{\\prime}(x)|&\\le&C\\ds\\left[\\dfrac{B_i(x)}{\\sqrt\\vr_i}+\\cdots+\\dfrac{B_n(x)}{\\sqrt\\vr_n}\\right].\n\\end{array}\\]\nHence, for $\\;x\\in\\oln\\Omega\\;$ and $\\;i=1,\\cdots,n,\\;$\n\\begin{equation}\\label{4d}\n|v_i^{\\prime\\prime\\prime}(x)|\\;\\le\\;|p_i^{\\prime}(x)| \\leq\nC\\ds\\left[1+\\dfrac{B_i(x)}{\\sqrt\\vr_i}+\\cdots+\\dfrac{B_n(x)}{\\sqrt\\vr_n}\\right].\n\\end{equation}\nFrom \\eqref{4e} and \\eqref{4d}, we find that for $\\;k=0,1,2,3\\;$ and $\\;x\\in\\oln\\Omega,\\;$\n\\[|v_i^{(k)}(x)|\n\\;\\le\\;C\\;\\ds\\left[1+\\eps_i^{1-\\frac{k}{2}}B_i(x)+\\cdots+\\eps_n^{1-\\frac{k}{2}}B_n(x)\\right]. \\;\\eop\\]. \\end{proof}\n{\\bf Remark :} It is interesting to note that the above estimate\nreduces to the estimate of the smooth component of the solution of\nthe scalar problem given in \\cite{MORS} when $\\;n=1.\\;$\\\\\nBounds on the singular components $\\vec{w}^l,\\; \\vec{w}^r$ of\n$\\vec{u}$ and their derivatives are contained in\n\\begin{lemma}\\label{lsingular} Let $A(x)$ satisfy (\\ref{a1}) and\n(\\ref{a2}).Then there exists a constant $C,$ such that, for each $x\n\\in [0,1]$ and $i=1,\\; \\dots , \\; n$,\n\\[\\left|w^l_i(x)\\right| \\;\\le\\; C B^l_{n}(x),\\;\\;\n\\left|w_i^{l,\\prime}(x)\\right| \\;\\le\\; C\\sum_{q=i}^n\n\\frac{B^l_{q}(x)}{\\sqrt{\\eps_q}},\\]\n\\[\\left|w_i^{l,\\prime\\prime}(x)\\right| \\;\\le\\; C\\sum_{q=i}^n\n\\frac{B^l_{q}(x)}{\\eps_q},\\;\\; \\left|w_i^{l,(3)}(x)\\right| \\;\\le\\;\nC\\sum_{q=1}^n \\frac{B^l_{q}(x)}{\\eps_q^{3\/2}},\\]\n\\[\\left|\\eps_i w_i^{l,(4)}(x)\\right|\\;\\le\\; C\\sum_{q=1}^n \\frac{B^l_{q}(x)}{\\eps_q}.\\]\nAnalogous results hold for $w^r_i$ and its derivatives.\n\\end{lemma}\n\n\\begin{proof}First we obtain the bound on $\\vec{w}^l$. We define the two\nfunctions $\\vec{\\theta}^{\\pm}=CB^l_n\\vec{e} \\pm \\vec{w}^l$. Then\nclearly $\\vec{\\theta}^{\\pm}(0) \\geq \\vec{0}, \\;\\;\n\\vec{\\theta}^{\\pm}(1) \\geq \\vec{0}$ and\n$L\\vec{\\theta}^{\\pm}=CL(B^l_n\\vec{e})$. Then, for $i=1,\\dots, n$,\n$(L\\vec{\\theta}^{\\pm})_i\n=C(\\sum_{j=1}^{n}a_{i,j}-\\alpha\\frac{\\eps_i}{\\eps_n})B^l_n >0$. By\nLemma \\ref{max}, $\\vec{\\theta}^{\\pm}\\geq \\vec{0}$, which leads to\nthe required bound on $\\vec{w}^l$.\n\nAssuming, for the moment, the bounds on the first and second\nderivatives $w_i^{l,\\prime}$ and $w_i^{l,\\prime\\prime}$, the system\nof differential equations satisfied by $\\vec{w}^l$ is differentiated\ntwice to get\n\\[-E\\vec{w}^{l,(4)}+A\\vec{w}^{l,\\prime\\prime}+2A^{\\prime}\\vec{w}^{l,\\prime}+A^{\\prime\\prime}\\vec{w}^l =\\vec{0}.\\] The required bounds on the $w_i^{l,(4)}$ follow from those on $w^l_i$, $w_i^{l,\\prime}$ and\n$w_i^{l,\\prime\\prime}$. It remains therefore to establish the bounds\non $w_i^{l,\\prime}$,$w_i^{l,\\prime\\prime}$ and\n$w_i^{l,\\prime\\prime\\prime}$, for which the following mathematical\ninduction argument is used. It is assumed that the bounds hold for\nall systems up to order $n-1$. It is then shown that the bounds hold\nfor order $n$. The induction argument is completed by observing that\nthe bounds for the scalar case $n=1$ are proved in \\cite{MORS}.\n\nIt is now shown that under the induction hypothesis the required\nbounds hold for $w_i^{l,\\prime}$,$w_i^{l,\\prime\\prime}$ and\n$w_i^{l,\\prime\\prime\\prime}$. The bounds when $i=n$ are established\nfirst.The differential equation for $w^l_n$ gives $\\eps_n\nw_n^{l,\\prime\\prime}=(A\\vec{w}^l)_n$ and the required bound on\n$w_n^{l,\\prime\\prime}$ follows at once from that for $\\vec{w}^l$.\nFor $w_n^{l,\\prime}$ it is seen from the bounds in Lemma\n\\ref{lexact}, applied to the system satisfied by $\\vec{w}^l$, that\n$|w_i^{l,\\prime}(x)| \\leq C\\eps_i^{-\\frac{1}{2}}$. In particular,\n$|w_n^{l,\\prime}(0)| \\leq C\\eps_n^{-\\frac{1}{2}}$ and\n$|w_n^{l,\\prime}(1)| \\leq C\\eps_n^{-\\frac{1}{2}}$. It is also not\nhard to verify that\n$\\vec{L}\\vec{w}^{l,\\prime}=-A^{\\prime}\\vec{w}^l$. Using these\nresults, the inequalities $\\eps_i < \\eps_n, \\; i\\;0.\\;$ It follows that\n\\[\\ds\\left(\\vec{L}^N\\vec{\\Psi}(x_{j*})\\right)_{i*}\\;=\n\\;-\\eps_{i*}\\delta^2\\Psi_{i*}(x_{j*})+\\ds{\\sum_{k=1}^n}\na_{i*,\\;k}(x_{j*})\\Psi_{k}(x_{j*})\\;<\\;0,\\] which is a\ncontradiction, as required. \\eop \\end{proof}\n\nAn immediate consequence of this is the following discrete stability\nresult.\n\\begin{lemma}\\label{dstab} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2}).\nThen, for any mesh function $\\vec \\Psi $,\n\\[\\parallel\\vec \\Psi(x_j)\\parallel\\;\\le\\;\\max\\left\\{||\\vec \\Psi(0)||, ||\\vec \\Psi(1)||, \\frac{1}{\\alpha}||\n\\vec{L}^N\\vec \\Psi||\\right\\}, \\; 0\\leq j \\leq N. \\]\n\\end{lemma}\n\\begin{proof} Define the two functions\n\\[\\vec{\\Theta}^{\\pm}(x_j)=\\max\\{||\\vec{\\Psi}(0)||,||\\vec \\Psi(1)||,\\frac{1}{\\alpha}||\\vec{L^N}\\vec{\\Psi}||\\}\\vec{e}\\pm\n\\vec{\\Psi}(x_j)\\]where $\\vec{e}=(1,\\;\\dots \\;,1)$ is the unit\nvector. Using the properties of $A$ it is not hard to verify that\n$\\vec{\\Theta}^{\\pm}(0)\\geq \\vec{0}, \\; \\vec{\\Theta}^{\\pm}(1)\\geq\n\\vec{0}$ and $\\vec{L^N}\\vec{\\Theta}^{\\pm}(x_j)\\geq \\vec{0}$. It\nfollows from Lemma \\ref{dmax} that $\\vec{\\Theta}^{\\pm}(x_j)\\geq\n\\vec{0}$ for all $0\\leq j \\leq N$.\\eop\n\\end{proof}\nThe following comparison result will be used in the proof of the\nerror estimate.\n\\begin{lemma}\\label{comparison} Assume that the mesh functions $\\vec{\\Phi}$ and $\\vec{Z}$ satisfy, for\n$j=1\\;\\dots \\; N-1$,\n\\[||\\vec{Z}(0)|| \\leq \\vec{\\Phi}(0),\\;\\; ||\\vec{Z}(1)|| \\leq \\vec{\\Phi}(1),\\;\\;\n||\\vec(L^N)(\\vec{Z}(x_j))|| \\leq \\vec(L^N)(\\vec{\\Phi}(x_j)).\\] Then,\nfor $j=0\\;\\dots \\; N$,\n\\[||\\vec{Z}(x_j)||\\vec{e}\\leq \\vec{\\Phi}(x_j).\\]\n\\end{lemma}\n\\begin{proof} Define the two mesh functions $\\vec{\\Psi}^{\\pm}$ by\n\\[\\vec{\\Psi}^{\\pm}=\\vec{\\Phi} \\pm \\vec{Z}.\\] Then $\\vec{\\Psi}^{\\pm}$\nsatisfies, for $j=1\\;\\dots \\; N-1$,\n \\[\\vec{\\Psi}^{\\pm}(0)=\\vec{\\Psi}^{\\pm}(1)=0, \\qquad\n \\vec(L)^N (\\vec{\\Psi}^{\\pm})(x_j)\\ge \\vec{0}.\\] The result follows\n from an application of Lemma \\ref{dmax}.\n\\eop \\end{proof}\n\\section{The local truncation error}\nFrom Lemma \\ref{dstab}, it is seen that in order to bound the\nerror $||\\vec{U}-\\vec{u}||$ it suffices to bound\n$\\vec{L}^N(\\vec{U}-\\vec{u})$. But this expression satisfies\n\\[\n\\vec{L}^N(\\vec{U}-\\vec{u})=\\vec{L}^N(\\vec{U})-\\vec{L}^N(\\vec{u})=\n\\vec{f}-\\vec{L}^N(\\vec{u})=\\vec{L}(\\vec{u})-\\vec{L}^N(\\vec{u})\\]\n\\[=(\\vec{L}-\\vec{L}^N)\\vec{u}\n=-E(\\delta^2-D^2)\\vec{u}\\] which is the local truncation of the\nsecond derivative. Let $\\vec{V}, \\vec{W}$ be the discrete\nanalogues of $\\vec{v}, \\vec{w}$ respectively. Then, similarly,\n\\[\\vec{L}^N(\\vec{V}-\\vec{v})=-E(\\delta^2-D^2)\\vec{v},\\;\\;\\vec{L}^N(\\vec{W}-\\vec{w})=-E(\\delta^2-D^2)\\vec{w}.\\]\nBy the triangle inequality,\n\\begin{equation}\\label{triangleinequality}\n\\parallel \\vec L^N(\\vec{U}-\\vec{u})\\parallel\\;\\leq\\;\\parallel\n\\vec{L}^N(\\vec{V}-\\vec{v})\\parallel+\\parallel\n\\vec{L}^N(\\vec{W}-\\vec{w})\\parallel.\n\\end{equation} Thus, the smooth and singular components\nof the local truncation error can be treated separately. In view of\nthis it is noted that, for any smooth function $\\psi$, the following\nthree distinct estimates of the local truncation error of its second\nderivative hold:\\\\\nfor $x_j \\in M_{\\vec{b}}$\n\\begin{equation}\\label{lte1}\n|(\\delta^2-D^2)\\psi(x_j)|\\;\\le\\; C\\max_{s\\;\\in\\;I_j}|\\psi^{\\prime\\prime}(s)|,\n\\end{equation}\nand\n\\begin{equation}\\label{lte2}\n|(\\delta^2-D^2)\\psi(x_j)|\\;\\le\\;C\\delta_j \\max_{s\\in I_j}|\\psi^{(3)}(s)|, \\end{equation}\nfor $x_j \\notin J_{\\vec{b}}$\n\\begin{equation}\\label{lte3}\n|(\\delta^2-D^2)\\psi(x_j)|\\;\\le\\;C\\delta^2_j \\max_{s\\in\nI_j}|\\psi^{(4)}(s)|,\n\\end{equation}\nfor $\\tau_k \\in J_{\\vec{b}}$\n\\begin{equation}\\label{lte4}\n|(\\delta^2-D^2)\\psi(\\tau_k)|\\;\\le\\;C(\\;|H_k\n-h_k|.|\\psi^{(3)}(\\tau_k)|+\\delta^2_k \\max_{s\\in\nI_k}|\\psi^{(4)}(s)|\\;).\n\\end{equation}\n\\section{Error estimate}\nThe proof of the error estimate is broken into two parts. In the\nfirst a theorem concerning the smooth part of the error is proved.\nThen the singular part of the error is considered. A barrier\nfunction is now constructed, which is used in both parts of the\nproof.\\\\\nFor each $k \\in I_{\\vec{b}}$, introduce the piecewise\nlinear polynomial \\begin{equation*} \\theta_k(x)=\n\\left\\{ \\begin{array}{l}\\;\\; \\dfrac{x}{\\tau_k}, \\;\\; 0 \\leq x \\leq \\tau_k. \\\\\n\\;\\; 1, \\;\\; \\tau_k < x < 1-\\tau_k.\n\\\\ \\;\\; \\dfrac{1-x}{\\tau_k}, \\;\\; 1-\\tau_k \\leq x \\leq 1. \\end{array}\\right .\\end{equation*}\\\\\nIt is not hard to verify that, for each $k \\in I_{\\vec{b}}$,\n\\begin{equation*} L^N(\\theta_k(x_j)\\vec{e})_i \\ge\n\\left\\{ \\begin{array}{l}\\;\\; \\alpha+\\dfrac{2\\eps_i}{ \\tau_k\n(H_k+h_k)},\n\\;\\;\\mathrm{if}\\;\\; x_j=\\tau_k \\in J_{\\vec{b}} \\\\\n\\;\\; \\alpha \\theta_{k}(x_j), \\;\\; \\mathrm{if} \\;\\;x_j \\notin J_{\\vec{b}}. \\end{array}\\right .\\end{equation*}\\\\\nOn the Shishkin mesh $M_{\\vec{b}}$ define the barrier function\n$\\vec{\\Phi}$ by\n\\begin{equation}\\label{barrier}\\vec{\\Phi}(x_j)=C\\,N^{-2}(\\ln N)^3 [1+\\ds\\sum_{k\\in\nI_{\\vec{b}}}\\theta_k (x_j)]\\vec{e},\\end{equation} where $C$ is any\nsufficiently large constant.\\\\ Then $\\vec{\\Phi}$ satisfies\n\\begin{equation}\\label{barrierbound1}0 \\leq \\Phi_{i}(x_j) \\leq\nC\\,N^{-2}(\\ln N)^{3},\\;\\; 1 \\leq i \\leq n.\\end{equation} Also, , for\n$x_j \\notin J_{\\vec{b}}$,\n\\begin{equation}\\label{barrierbound3}\n(L^N\\vec{\\Phi}(x_j))_i \\ge CN^{-2}(\\ln N)^{3}\\end{equation} and, for\n$\\tau_k \\in J_{\\vec{b}}$,\n\\begin{equation*}\\label{barrierbound2}(L^N\\vec{\\Phi}(\\tau_k))_i \\ge\nC(1+\\dfrac{\\eps_i}{\\sqrt{\\eps_k}(H_k+h_k)})(N^{-1}\\ln N)^{2},\n\\end{equation*} from which it follows that, for $\\tau_k \\in J_{\\vec{b}}$ and $H_k \\ge h_k$,\n\\begin{equation}\\label{LHgeh}\n(L^N\\vec{\\Phi}(\\tau_k))_i \\ge C(N^{-2}+\\frac{\\eps_i}{\\sqrt{\\eps_k\n\\eps_{k+1}}} N^{-1} \\ln N)\n\\end{equation}\nand, for $\\tau_k \\in J_{\\vec{b}}$ and $H_k \\le h_k$,\n\\begin{equation} \\label{LHleh}\n(L^N\\vec{\\Phi}(\\tau_k))_i \\ge C(N^{-2}+\\frac{\\eps_i}{\\eps_k} N^{-1}\n\\ln N).\n\\end{equation}\nThe following theorem gives the error estimate for the smooth\ncomponent.\n\\begin{theorem}\\label{smootherrorthm} Let $A(x)$ satisfy (\\ref{a1}) and\n(\\ref{a2}). Let $\\vec v$ denote the smooth component of the exact\nsolution from (\\ref{BVP}) and $\\vec V$ the smooth component of the\n discrete solution from (\\ref{discreteBVP}). Then\n\\begin{equation}\\;\\; ||\\vec{V-v}|| \\leq C\\,N^{-2}(\\ln N)^3. \\end{equation}\n\\end{theorem}\n\\begin{proof} An application of Lemma \\ref{comparison} is made, using the above\nbarrier function. To prove the theorem it suffices to show that the\nratio\n\\begin{equation*}\nR(v_i (x_j))= \\frac{|\\eps_{i}(\\delta^2\n-D^2)v_{i}(x_j)|}{|(L^N\\vec{\\Phi}(x_j))_i|}, \\;\\; x_j \\in\nM_{\\vec{b}}\n\\end{equation*}\nsatisfies\n\\begin{equation}\\label{ratio1}\nR(v_i(x_j))\\leq C.\n\\end{equation}\nFor $x_j \\notin J_{\\vec{b}}$ the bound \\eqref{ratio1} follows\nimmediately from Lemma \\ref{lsmooth}, (\\ref{lte3})and\n(\\ref{geom1}).\\\\\nNow assume that $x_j = \\tau_k \\in J_{\\vec{b}}$. The required\nestimates of the denominator of $R(v_i(\\tau_k))$ are \\eqref{LHgeh}\nand \\eqref{LHleh}. The numerator is bounded above using Lemma\n\\ref{lsmooth2} and \\eqref{lte2}. The cases $b_k =1$ and $b_k=0$ are\ntreated separately and the inequalities \\eqref{geom2},\n\\eqref{geom7}, \\eqref{geom1},\n\\eqref{geom0} and \\eqref{geom3} are used systematically.\\\\\nSuppose first that $b_k=1$, then there are four possible subcases:\\\\\n\\begin{equation}\n\\begin{array}{lll}\n i \\leq k,\\;& H_k \\ge h_k, \\;&\nR(v_i(\\tau_k)) \\le\nC\\eps_{k+1}.\\\\\n & H_k \\leq h_k, \\; & R(v_i(\\tau_k)) \\le\n C\\eps_k.\\\\\n i > k,\\;& H_k \\ge h_k, \\;&\nR(v_i(\\tau_k)) \\le\nC\\eps_{k+1} \\sqrt{\\frac{\\eps_k}{\\eps_i}}.\\\\\n & H_k \\leq h_k, \\; & R(v_i(\\tau_k)) \\le\n C\\eps_k\\sqrt{\\frac{\\eps_k}{\\eps_i}}.\\\\\n\\end{array}\n\\end{equation}\nSecondly, if $b_k=0$, then $b_{k-1}=1$, because otherwise $\\tau_k\n\\notin J_b$, and furthermore $H_k \\le h_k$. There are two possible\nsubcases:\n\\begin{equation}\n\\begin{array}{lll}\n i \\leq k-1,\\;& H_k \\leq h_k,\n\\;& R(v_i(\\tau_k)) \\le\nC\\eps_k(\\frac{\\eps_i}{\\sqrt{\\eps_{k-1}\\eps_k}}+1).\\\\\n i > k-1,\\;& H_k \\leq h_k, \\;&\nR(v_i(\\tau_k)) \\le\nC\\eps_k\\sqrt{\\frac{\\eps_k}{\\eps_i}}.\\\\\n\\end{array}\n\\end{equation}\nIn all six subcases, because of the ordering of the $\\eps_i$, it is\nclear that condition \\eqref{ratio1} is fulfilled. This concludes the\nproof. \\eop\n\\end{proof}\nBefore the singular part of the error is estimated the following\nlemmas are established.\n\\begin{lemma}\\label{est1} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2}). Then, on each mesh $M_{\\vec{b}}$,\nfor $1 \\leq i \\leq n$ and $1 \\leq j \\leq N$, the following estimates\nhold\n\\begin{equation} |\\eps_i(\\delta^2-D^2)w^l_i(x_j)|\\;\\leq\\;\nC\\frac{\\delta^2 _j}{\\eps_1}\\;\\; \\mathrm{for}\\;\\; x_j \\notin\nJ_{\\vec{b}}.\\end{equation} An analogous result holds for the\n$w^r_i$.\n\\end{lemma}\n\\begin{proof} When $x_j \\notin J_{\\vec{b}}$, from (\\ref{lte3}) and Lemma \\ref{lsingular}, it follows that\n\\[\n|\\eps_i(\\delta^2-D^2)w^l _i(x_j)|\\leq C\\delta^2\n_j\\;\\ds\\max_{s\\;\\in\\;I_j}|\\eps_{i} w_i^{l,(4)}(s)|\\] \\[ \\leq\nC\\delta^2 _j\\;\\ds\\max_{s\\;\\in\\;I_j}\\ds\\sum_{q\\;=\\;1}^n\n\\dfrac{B^{l}_{q}(s)}{\\eps_q} \\leq \\dfrac{C\\delta^2 _j}{\\eps_1}\\] as\nrequired.\\eop\\end{proof}\n\nIn what follows fourth degree polynomials of the form\n\\[p_{i;\\theta}(x)=\\sum_{k=0}^4\n\\frac{(x-x_{\\theta})^k}{k!}w_{i}^{l,(k)}(x_{\\theta})\\] are used, where $\\theta$ denotes a pair of integers separated by a comma.\n\n\\begin{lemma}\\label{general} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2}) and assume that $M_{\\vec{b}}$ is\nsuch that $b_k =1$ for some $k$,\\;\\;$1 \\leq k \\leq n-1$. Then, for\neach $i,\\;j$, $1 \\leq i \\leq n$, $1 \\leq j \\leq N$ there exists a\ndecomposition\n\\[ w^l_i=\\sum_{q=1}^{k+1}w_{i,q}, \\] for which the following estimates hold for each $q$ and $r$, $1 \\le q \\le k$, $0 \\leq r \\leq 2,$\n\\[|\\eps_iw_{i,q}^{(r+2)}(x_j)| \\leq C \\eps^{-\\frac{r}{2}}_q B^l_q(x_j)\\]\nand\n\\[|\\eps_i\nw_{i,k+1}^{(3)}(x_j)| \\leq\nC\\sum_{q=k+1}^{n}\\frac{B^l_{q}(x_j)}{\\sqrt{\\eps_q}},\\;\\; |\\eps_i\nw_{i,k+1}^{(4)}(x_j)| \\leq\nC\\sum_{q=k+1}^{n}\\frac{B^l_{q}(x_j)}{\\eps_q}.\\]\nFurthermore, for $x_j \\notin J_{\\vec{b}}$,\n\\begin{equation} \\label{ltew1}\n|\\eps_{i}(\\delta^2 -D^2)w^l _{i}(x_j)| \\leq C(B^l_{k}(x_{j-1})+\\frac{\\delta_j^2 }{\\eps_{k+1}})\\end{equation} and, for $\\tau_k \\in J_{\\vec{b}}$,\n\\begin{equation} \\label{ltew2}\n|\\eps_{i}(\\delta^2 -D^2)w^l _{i}(\\tau_k)| \\leq C(\\; B^l _k (\\tau_k\n-h_k)+\\frac{\\delta_k}{\\sqrt{\\eps_{k+1}}}).\\end{equation} Analogous\nresults hold for the $w^r_i$ and their derivatives.\n\\end{lemma}\n\\begin{proof}\nConsider the decomposition\n\\[w^l_i=\\sum_{m=1}^{k+1}w_{i,m},\\] where the components\nare defined by\n\\[w_{i,k+1}=\\left\\{ \\begin{array}{ll} p_{i;k,k+1} & {\\rm on}\\;\\;[0,x^{(1)}_{k,k+1})\\\\\n w^l_i & {\\rm otherwise} \\end{array}\\right. \\]\nand for each $m$, $k \\ge m \\ge 2$,\n\\[w_{i,m}=\\left\\{ \\begin{array}{ll} p_{i;m-1,m} & \\rm{on} \\;\\; [0,x^{(1)}_{m-1,m})\\\\\nw^l_i-\\sum_{q=m+1}^{k+1} w_{i,q} & {\\rm otherwise}\n\\end{array}\\right. \\]\nand\n\\[w_{i,1}=w^l_i-\\sum_{q=2}^{k+1} w_{i,q}\\;\\; \\rm{on} \\;\\; [0,1]. \\]\nFrom the above definitions it follows that, for each $m$, $1 \\leq m \\leq k$,\n$w_{i,m}=0 \\;\\; \\rm{on} \\;\\; [x^{(1)}_{m,m+1},1]$.\\\\\nTo establish the bounds on the fourth derivatives it is seen that:\n\n\nfor $x \\in [x^{(1)}_{k,k+1},1]$, Lemma \\ref{lsingular} and $x \\geq\nx^{(1)}_{k,k+1}$ imply that\n\\[|\\eps_i w_{i,k+1}^{(4)}(x)| =|\\eps_i w_{i}^{l,(4)}(x)| \\leq\nC\\sum_{q=1}^n \\frac{B^l_q(x)}{\\eps_q} \\leq C\\sum_{q=k+1}^n\n\\frac{B^l_q(x)}{\\eps_q};\\]\n\n\nfor $x \\in [0, x^{(1)}_{k,k+1}]$, Lemma \\ref{lsingular} and $x \\leq\nx^{(1)}_{k,k+1}$ imply that\n\\[|\\eps_i w_{i,k+1}^{(4)}(x)| =|\\eps_i w_{i}^{l, (4)}(x^{(1)}_{k,k+1})|\n\\leq \\sum_{q=1}^{n} \\frac{B^l_q(x^{(1)}_{k,k+1})}{\\eps_q} \\leq\nC\\sum_{q=k+1}^{n} \\frac{B^l_q(x^{(1)}_{k,k+1})}{\\eps_q} \\leq\nC\\sum_{q=k+1}^{n} \\frac{B^l_q(x)}{\\eps_q};\\]\n\nand for each $m=k, \\;\\; \\dots \\;\\;,2$, it follows that\\\\\n\nfor $x \\in [x^{(1)}_{m,m+1},1]$,\\;\\; $w_{i,m}^{(4)}=0;$\n\nfor $x \\in [x^{(1)}_{m-1,m},x^{(1)}_{m,m+1}]$, Lemma \\ref{lsingular}\nimplies that\n\\[|\\eps_i w_{i,m}^{(4)}(x)| \\leq |\\eps_i w_{i}^{l,(4)}(x)|+\\sum_{q=m+1}^{k+1}|\\eps_i w_{i,q}^{(4)}(x)|\n\\leq C\\sum_{q=1}^n \\frac{B^l_q(x)}{\\eps_q} \\leq C\\frac{B^l_m(x)}{\\eps_m};\\]\n\nfor $x \\in [0, x^{(1)}_{m-1,m}]$, Lemma \\ref{lsingular} and $x \\leq\nx^{(1)}_{m-1,m}$ imply that\n\\[|\\eps_i w_{i,m}^{(4)}(x)| =\n|\\eps_i w_{i}^{l,(4)}(x^{(1)}_{m-1,m})| \\leq C\\sum_{q=1}^n\n\\frac{B^l_q(x^{(1)}_{m-1,m})}{\\eps_q} \\leq\nC\\frac{B^l_m(x^{(1)}_{m-1,m})}{\\eps_m} \\leq\nC\\frac{B^l_m(x)}{\\eps_m};\n\\]\n\nfor $x \\in [x^{(1)}_{1,2},1],\\;\\; w_{i,1}^{(4)}=0;$\n\nfor $x \\in [0, x^{(1)}_{1,2}]$, Lemma \\ref{lsingular} implies that\n\\[|\\eps_i w_{i,1}^{(4)}(x)| \\leq |\\eps_i\nw_{i}^{l,(4)}(x)|+\\sum_{q=2}^{k+1}|\\eps_i w_{i,q}^{(4)}(x)|\\leq\nC\\sum_{q=1}^n \\frac{B^l_q(x)}{\\eps_q} \\leq\nC\\frac{B^l_1(x)}{\\eps_1}.\\]\n\n\nFor the bounds on the second and third derivatives note that, for\neach $m$, $1 \\leq m \\leq k $ :\n\n\nfor $x \\in [x^{(1)}_{m,m+1},1],\\;\\;\nw_{i,m}^{\\prime\\prime}=0=w_{i,m}^{(3)};$\n\nfor $x \\in [0, x^{(1)}_{m,m+1}],\\;\\;\n\\ds\\int_x^{x^{(1)}_{m,m+1}}\\eps_i w_{i,m}^{(4)}(s)ds= \\eps_i\nw_{i,m}^{(3)}(x^{(1)}_{m,m+1})- \\eps_i\nw_{i,m}^{(3)}(x)= -\\eps_i w_{i,m}^{(3)}(x)$ \\\\\nand so\n\\[|\\eps_i w_{i,m}^{(3)}(x)| \\leq \\int_x^{x^{(1)}_{m,m+1}}|\\eps_i\nw_{i,m}^{(4)}(s)|ds \\leq \\frac{C}{\\eps_m}\\int_{x}^{x^{(1)}_{m,m+1}}\nB^l_m(s)ds \\leq C\\frac{B^l_m(x)}{\\sqrt\\eps_m}.\\] In a similar way,\nit can be shown that\n\\[|\\eps_i w_{i,m}^{\\prime\\prime}(x)| \\leq C B^l_m(x).\\]\nUsing the above decomposition yields\n\\begin{equation*}|\\eps_i(\\delta^{2}-D^2)w^l_i(x_j)| \\leq\n\\sum_{q=1}^{k}|\\eps_i(\\delta^{2}-D^2)w_{i,q}(x_j)|+\n|\\eps_i(\\delta^{2}-D^2)w_{i,k+1}(x_j)|.\\end{equation*} For $x_j\n\\notin J_{\\vec{b}}$, applying \\eqref{lte3} to the last term and\n\\eqref{lte1} to all other terms on the right hand side, it follows\nthat\n\\begin{equation*}\n |\\eps_i(\\delta^{2}-D^2)w^l_i(x_j)|\\leq C(\\sum_{q=1}^{k}\\max_{s \\in\nI_j}|\\eps_iw_{i,q}^{\\prime\\prime}(s)|+\\delta^2 _j\\max_{s \\in\nI_j}|\\eps_iw_{i,k+1}^{(4)}(s)|).\n\\end{equation*}\nThen \\eqref{ltew1} is obtained by using the bounds on the\nderivatives obtained in the first part of the lemma.\\\\ On the other\nhand, for $x_j=\\tau_k \\in J_{\\vec{b}}$, applying \\eqref{lte2} to the\nlast term and \\eqref{lte1} to the other terms, \\eqref{ltew2} is\nobtained by a similar argument. The proof for the $w^r_i$ and their\nderivatives is similar. \\eop\n\\end{proof}\n\nIn what follows third degree polynomials of the form\n\\[p^*_{i;\\theta}(x)=\\sum_{k=0}^3 \\frac{(x-y_{\\theta})^k}{k!}w_{i}^{l,(k)}(y_{\\theta})\\] are used, where $\\theta$ denotes a pair of integers separated by a comma.\n\n\\begin{lemma}\\label{general1} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2}) and assume that $M_{\\vec{b}}$ is\nsuch that $b_k =1$ for some $k$,\\;\\;$1 \\leq k \\leq n-1$. Then, for\neach $i,\\;j$, $1 \\leq i \\leq n$, $1 \\leq j \\leq N$ there exists a\ndecomposition\n\\[ w^l_i=\\sum_{m=1}^{k+1}w_{i,m}, \\] for which the following\nestimates hold for each $m$, $1 \\le m \\le k$,\n\\[|w_{i,m}^{\\prime\\prime}(x_j)| \\leq C\\frac{B^l_m(x_j)}{\\eps_m},\n \\;\\;|w_{i,m}^{(3)}(x_j)| \\leq\nC\\frac{B^l_{m}(x_j)}{\\eps_m^{3\/2}}\\] and\n\\[|w_{i,k+1}^{(3)}(x_j)| \\leq\nC\\sum_{q=k+1}^{n}\\frac{B^l_{q}(x_j)}{\\eps_q^{3\/2}}.\\] Furthermore\n\\begin{equation} |\\eps_i(\\delta^2-D^2)w^l_i(x_j)| \\leq C\\eps_i\\ds\\left(\n\\frac{B^l_{k}(x_{j-1})}{\\eps_k}+\\frac{\\delta_j\n}{\\eps^{3\/2}_{k+1}}\\right).\\end{equation} Analogous results hold for\nthe $w^r_i$ and their derivatives.\n\\end{lemma}\n\\begin{proof} The proof is similar to that of Lemma \\ref{general}\nwith the points $x^{(1)}_{i,j}$ replaced by the points\n$x^{(3\/2)}_{i,j}$. Consider the decomposition\n\\[w^l_i=\\sum_{m=1}^{k+1}w_{i,m},\\] where the components\nare defined by\n\\[w_{i,k+1}=\\left\\{ \\begin{array}{ll} p^*_{i;k,k+1} & {\\rm on}\\;\\;[0,x^{(3\/2)}_{k,k+1})\\\\\n w^l_i & {\\rm otherwise} \\end{array}\\right. \\]\nand for each $m$, $k \\ge m \\ge 2$,\n\\[w_{i,m}=\\left\\{ \\begin{array}{ll} p^*_{i;m-1,m} & \\rm{on} \\;\\; [0,x^{(3\/2)}_{m-1,m})\\\\\nw^l_i-\\ds\\sum_{q=m+1}^{k+1} w_{i,q} & {\\rm otherwise}\n\\end{array}\\right. \\]\nand\n\\[w_{i,1}=w^l_i-\\sum_{q=2}^{k+1} w_{i,q}\\;\\; \\rm{on} \\;\\; [0,1]. \\]\nFrom the above definitions it follows that, for each $m$, $1 \\leq m\n\\leq k$,\n$w_{i,m}=0 \\;\\; \\rm{on} \\;\\; [x^{(3\/2)}_{m,m+1},1]$.\\\\\nTo establish the bounds on the third derivatives it is seen that:\n\n\nfor $x \\in [x^{(3\/2)}_{k,k+1},1]$, Lemma \\ref{lsingular} and $x \\geq\nx^{(3\/2)}_{k,k+1}$ imply that\n\\[|w_{i,k+1}^{(3)}(x)| = |w_{i}^{l,(3)}(x)| \\leq\nC\\sum_{q=1}^n \\frac{B^l_q(x)}{\\eps_q^{3\/2}} \\leq C\\sum_{q=k+1}^n\n\\frac{B^l_q(x)}{\\eps_q^{3\/2}};\\]\n\n\nfor $x \\in [0, x^{(3\/2)}_{k,k+1}]$, Lemma \\ref{lsingular} and $x\n\\leq x^{(3\/2)}_{k,k+1}$ imply that\n\\[|w_{i,k+1}^{(3)}(x)| = |w_{i}^{l, (3)}(x^{(3\/2)}_{k,k+1})| \\leq \\sum_{q=1}^{n}\n\\frac{B^l_q(x^{(3\/2)}_{k,k+1})}{\\eps_q^{3\/2}} \\leq \\sum_{q=k+1}^{n}\n\\frac{B^l_q(x^{(3\/2)}_{k,k+1})}{\\eps_q^{3\/2}} \\leq \\sum_{q=k+1}^{n}\n\\frac{B^l_q(x)}{\\eps_q^{3\/2}};\\]\n\nand for each $m=k, \\;\\; \\dots \\;\\;,2$, it follows that\\\\\n\nfor $x \\in [x^{(3\/2)}_{m,m+1},1]$, $w_{i,m}^{(3)}=0;$\n\nfor $x \\in [x^{(3\/2)}_{m-1,m},x^{(3\/2)}_{m,m+1}]$, Lemma\n\\ref{lsingular} implies that\n\\[|w_{i,m}^{(3)}(x)| \\leq |w_{i}^{l,(3)}(x)|+\\sum_{q=m+1}^{k+1}|w_{i,q}^{(3)}(x)|\n\\leq C\\sum_{q=1}^n \\frac{B^l_q(x)}{\\eps_q^{3\/2}} \\leq\nC\\frac{B^l_m(x)}{\\eps_m^{3\/2}};\\]\n\nfor $x \\in [0, x^{(3\/2)}_{m-1,m}]$, Lemma \\ref{lsingular} and $x\n\\leq x^{(3\/2)}_{m-1,m}$ imply that\n\\[|w_{i,m}^{(3)}(x)| =|w_{i}^{l,(3)}(x^{(3\/2)}_{m-1,m})| \\leq C\\sum_{q=1}^n\n\\frac{B^l_q(x^{(3\/2)}_{m-1,m})}{\\eps_q^{3\/2}} \\leq\nC\\frac{B^l_m(x^{(3\/2)}_{m-1,m})}{\\eps_m^{3\/2}} \\leq\nC\\frac{B^l_m(x)}{\\eps_m^{3\/2}};\n\\]\n\nfor $x \\in [x^{(3\/2)}_{1,2},1],\\;\\; w_{i,1}^{(3)}=0;$\n\nfor $x \\in [0, x^{(3\/2)}_{1,2}]$, Lemma \\ref{lsingular} implies that\n\\[|w_{i,1}^{(3)}(x)| \\leq |w_{i}^{l,(3)}(x)|+\\sum_{q=2}^{k+1}|w_{i,q}^{(3)}(x)|\n\\leq C\\sum_{q=1}^n \\frac{B^l_q(x)}{\\eps_q^{3\/2}} \\leq\nC\\frac{B^l_1(x)}{\\eps_1^{3\/2}}.\\]\n\n\nFor the bounds on the second derivatives note that, for each $m$, $1\n\\leq m \\leq k $ :\n\n\nfor $x \\in [x^{(3\/2)}_{m,m+1},1],\\;\\; w_{i,m}^{\\prime\\prime}=0;$\n\nfor $x \\in [0, x^{(3\/2)}_{m,m+1}],\\;\\; \\int_x^{x^{(3\/2)}_{m,m+1}}\nw_{i,m}^{(3)}(s)ds = w_{i,m}^{\\prime\\prime}(x^{(3\/2)}_{m,m+1})-\nw_{i,m}^{\\prime\\prime}(x)= -w_{i,m}^{\\prime\\prime}(x)$ \\\\\nand so\n\\[|w_{i,m}^{\\prime\\prime}(x)|\n\\leq \\int_x^{x^{(3\/2)}_{m,m+1}}|w_{i,m}^{(3)}(s)|ds \\leq\n\\frac{C}{\\eps_m^{3\/2}}\\int_{x}^{x^{(3\/2)}_{m,m+1}} B^l_m(s)ds \\leq\nC\\frac{B^l_m(x)}{\\eps_m}.\\] Finally, since\n\\[|\\eps_i(\\delta^{2}-D^2)w^l_i(x_j)| \\leq \\sum_{m=1}^{k}|\\eps_i(\\delta^{2}-D^2)w_{i,m}(x_j)|+ |\\eps_i(\\delta^{2}-D^2)w_{i,k+1}(x_j)|,\\] using (\\ref{lte2}) on the\nlast term and (\\ref{lte1}) on all other terms on the right hand\nside, it follows that\n\\[|\\eps_i(\\delta^{2}-D^2)w^l_i(x_j)| \\leq C(\\sum_{m=1}^{k}\\max_{s \\in\nI_j}|\\eps_iw_{i,m}^{\\prime\\prime}(s)| +\\delta_j\\max_{s \\in\nI_j}|\\eps_iw_{i,k+1}^{(3)}(s)|).\\] The desired result follows by\napplying the bounds on the derivatives obtained in the first part of\nthe lemma. The proof for the $w^r_i$ and their derivatives is\nsimilar.\\eop \\end{proof}\n\n\\begin{lemma}\\label{est3} Let $A(x)$ satisfy (\\ref{a1}) and (\\ref{a2}).\nThen, on each mesh $M_{\\vec{b}}$, the following estimate holds for\n$i=1,\\; \\dots ,\\; n$ and each $j=1, \\;\\dots,\\; N$, \\[\n|\\eps_i(\\delta^2-D^2)w^l_i(x_j)| \\leq CB^l_n(x_{j-1}).\\] An\nanalogous result holds for the $w^r_i$.\n\\end{lemma}\n\\begin{proof}\nFrom $\\;\\eqref{lte1}\\;$ and Lemma \\ref{lsingular}, for each $\\;i=1,\\dots,n\\;$ and $\\;j=1,\\dots,N,\\;$ it follows that\n\\[|\\eps_i(\\delta^2-D^2)w^l_i(x_j)|\\;\\leq\\;\nC\\;\\ds\\max_{s\\in I_j}|\\eps_i w_i^{l,\\prime\\prime}(s)|\\;\\]\\[\\le\\;\nC\\;\\eps_i\\ds\\sum_{q=i}^{n}\\dfrac{B^l_q(x_{j-1})}{\\eps_q}\\;\\le\\;CB^l_n(x_{j-1}).\\]\nThe proof for the $w^r_i$ and their derivatives is similar.\\eop\n\\end{proof}\n\nThe following theorem provides the error estimate for the singular\ncomponent.\n\\begin{theorem} \\label{singularerrorthm}Let $A(x)$ satisfy (\\ref{a1}) and\n(\\ref{a2}). Let $\\vec w$ denote the singular component of the exact\nsolution from (\\ref{BVP}) and $\\vec W$ the singular component of the\n discrete solution from (\\ref{discreteBVP}). Then\n\\begin{equation}\\;\\; ||\\vec{W-w}|| \\leq C\\,N^{-2}(\\ln N)^{3}. \\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nSince $\\vec{w}=\\vec{w}^l+\\vec{w}^r$, it suffices to prove the result\nfor $\\vec{w}^l$ and $\\vec{w}^r$ separately. Here it is proved for\n$\\vec{w}^l$ by an application of Lemma \\ref{comparison}. A similar proof holds for $\\vec{w}^r$.\\\\\nThe proof is in two parts.\\\\ First assume that $x_j \\notin\nJ_{\\vec{b}}$.\nEach open subinterval $(\\tau_k,\\tau_{k+1})$ is treated separately.\\\\\nFirst, consider $x_j \\in (0,\\tau_1)$. Then, on each mesh\n$M_{\\vec{b}}$, $\\delta_j \\leq CN^{-1}\\tau_1$ and the result follows\n from (\\ref{geom2}) and Lemma \\ref{est1}.\\\\\nSecondly, consider $x_j \\in (\\tau_1,\\tau_2)$, then $\\tau_1 \\leq\nx_{j-1}$ and $\\delta_j \\leq CN^{-1}\\tau_2$. The $2^{n+1}$ possible\nmeshes are divided into subclasses of two types. On the meshes\n$M_{\\vec b}$ with $b_1=0$ the result follows from (\\ref{geom2}),\n(\\ref{geom-1}) and Lemma \\ref{est1}. On the meshes $M_{\\vec b}$ with\n$b_1=1$ the result follows from (\\ref{geom2}), (\\ref{geom0}) and Lemma \\ref{general}.\\\\\nThirdly, in the general case $x_j \\in (\\tau_m,\\tau_{m+1})$ for $2\n\\leq m \\leq n-1$, it follows that $\\tau_m \\leq x_{j-1}$ and\n$\\delta_j \\leq CN^{-1}\\tau_{m+1}$. Then $M_{\\vec b}$ is divided into\nsubclasses of three types: $M_{\\vec b}^0=\\{M_{\\vec b}: b_1= \\dots\n=b_m =0\\},\\; M_{\\vec b}^{r}=\\{M_{\\vec b}: b_r=1, \\; b_{r+1}= \\dots\n=b_m =0 \\; \\mathrm{for \\; some}\\; 1 \\leq r \\leq m-1\\}$ and $M_{\\vec\nb}^m=\\{M_{\\vec b}: b_m=1\\}.$ On $M_{\\vec b}^0$ the result follows\nfrom (\\ref{geom2}), (\\ref{geom-1}) and Lemma \\ref{est1}; on $M_{\\vec\nb}^r$ from (\\ref{geom2}), (\\ref{geom-1}), (\\ref{geom0}) and Lemma\n\\ref{general}; on $M_{\\vec b}^m$ from (\\ref{geom2}), (\\ref{geom0})\nand Lemma \\ref{general}. \\\\\nFinally, for $x_j \\in (\\tau_n,1)$, $\\tau_n \\leq x_{j-1}$ and\n$\\delta_j \\leq CN^{-1}$. Then $M_{\\vec b}$ is divided into\nsubclasses of three types: $M_{\\vec b}^0=\\{M_{\\vec b}: b_1= \\dots\n=b_n =0\\},\\; M_{\\vec b}^{r}=\\{M_{\\vec b}: b_r=1, \\; b_{r+1}= \\dots\n=b_n =0 \\; \\mathrm{for \\; some}\\; 1 \\leq r \\leq n-1\\}$ and $M_{\\vec\nb}^n=\\{M_{\\vec b}: b_n=1\\}.$ On $M_{\\vec b}^0$ the result follows\nfrom (\\ref{geom2}), (\\ref{geom-1}) and Lemma \\ref{est1}; on $M_{\\vec\nb}^r$ from (\\ref{geom2}), (\\ref{geom-1}), (\\ref{geom0}) and Lemma\n\\ref{general}; on $M_{\\vec b}^n$ from (\\ref{geom0}) and Lemma\n\\ref{est3}.\\\\\\\\\nNow assume that $x_j = \\tau_k \\in J_{\\vec{b}}$. Analogously to the\nproof of Theorem \\ref{smootherrorthm} the ratio $R(w_i(\\tau_k))$ is\nintroduced in order to facilitate the use of Lemma \\ref{comparison}.\nTo complete the proof it suffices to establish in all cases that\n\\begin{equation} \\label{ratio2} R_i(w(\\tau_k)) \\le C.\n\\end{equation}\nThe required estimates of the denominator of $R(w_i(\\tau_k))$ are\n\\eqref{LHgeh} and \\eqref{LHleh}. The numerator is bounded above\nusing Lemmas \\ref{general} and \\ref{general1}. The cases $b_k =1$\nand $b_k=0$ are treated separately and the inequalities\n\\eqref{geom2}, \\eqref{geom7}, \\eqref{geom1},\n\\eqref{geom0} and \\eqref{geom3} are used systematically.\\\\\nSuppose first that $b_k=1$, then there are four possible subcases:\\\\\n\\begin{equation}\n\\begin{array}{llll}\n\\mathrm{Lemma \\;\\ref{general1}}, \\;\\;& i \\leq k,\\;& H_k \\ge h_k, \\;&\nR(w_i(\\tau_k)) \\le\nC(\\frac{\\eps_i}{\\eps_k}+\\frac{\\sqrt{\\eps_k \\eps_{k+1}}}{\\eps_{k+1}}).\\\\\n & & H_k \\leq h_k, \\; & R(w_i(\\tau_k)) \\le\n C(\\frac{\\eps_i}{\\eps_k}+(\\frac{\\eps_k\n }{\\eps_{k+1}})^{3\/2}).\\\\\n\\mathrm{Lemma \\;\\ref{general}}, \\;\\;& i > k,\\;& H_k \\ge h_k, \\;&\nR(w_i(\\tau_k)) \\le\nC(1+\\frac{\\sqrt{\\eps_k \\eps_{k+1}}}{\\eps_i}).\\\\\n & & H_k \\leq h_k, \\; & R(w_i(\\tau_k)) \\le\n C(1+\\frac{\\eps^{3\/2}_k\n }{\\eps_i \\sqrt{\\eps_{k+1}}}).\\\\\n\n\\end{array}\n\\end{equation}\nSecondly, if $b_k=0$, then $b_{k-1}=1$, because otherwise $\\tau_k\n\\notin J_b$, and furthermore $H_k \\le h_k$. There are two possible\nsubcases:\n\\begin{equation}\n\\begin{array}{llll}\n\\mathrm{Lemma \\;\\ref{general1}}, \\;\\;& i \\leq k-1,\\;& H_k \\leq h_k,\n\\;& R(w_i(\\tau_k)) \\le\nC(\\frac{\\eps_i}{\\eps_{k-1}}+1).\\\\\n\\mathrm{Lemma \\;\\ref{general}}, \\;\\;& i > k-1,\\;& H_k \\leq h_k, \\;&\nR(w_i(\\tau_k)) \\le\nC(1+\\frac{\\eps_k}{\\eps_i}).\\\\\n\\end{array}\n\\end{equation}\nIn all six subcases, because of the ordering of the $\\eps_i$, it is\nclear that condition \\eqref{ratio2} is fulfilled. This concludes the\nproof. \\eop \\end{proof}\n\nThe following theorem gives the required essentially second order\nparameter-uniform error estimate.\n\\begin{theorem}Let $A(x)$ satisfy (\\ref{a1}) and\n(\\ref{a2}). Let $\\vec u$ denote the exact solution from (\\ref{BVP})\nand $\\vec U$ the discrete solution from (\\ref{discreteBVP}). Then\n\\begin{equation}\\;\\; ||\\vec{U-u}|| \\leq C\\,N^{-2}(\\ln N)^3. \\end{equation}\n\\end{theorem}\n\\begin{proof}\nAn application of the triangle inequality and the results of\nTheorems \\ref{smootherrorthm} and \\ref{singularerrorthm} leads\nimmediately to the required result.\\eop\n \\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}