data_all_eng_slimpj/shuffled/split2/finalzzaloa

{"text":"\\subsection{Gauge fields}\n\nWe will consider $SU(N)$\ngauge fields \nin a four dimensional torus of size $L^4$. The twisted boundary\nconditions are implemented by requiring the field to gauge-transform\nunder the displacement of a period\n\\begin{equation}\n  A_\\mu(x+L\\hat \\nu) = \\Omega_\\nu(x)A_\\mu(x)\\Omega_\\nu^+(x) + \n  \\Omega_\\nu(x)\\partial_\\mu\\Omega_\\nu^+(x) \\,,\n\\end{equation}\nwhere $\\Omega_\\mu(x)$ are known as the twist matrices. The uniqueness\nof $A_\\mu(x+L\\hat\\mu+L\\hat\\nu)$ requires that the twist matrices have\nto obey the relation \n\\begin{equation}\n  \\label{eq:consistency}\n  \\Omega_\\mu(x+L\\hat \\nu)\\Omega_\\nu(x) = e^{2\\pi\\imath n_{\\mu\\nu}\/N}\n  \\Omega_\\nu(x+L\\hat \\mu)\\Omega_\\mu(x)\\,,\n\\end{equation}\nwhere $n_{\\mu\\nu}$ is an anti-symmetric tensor of integers modulo $N$\ncalled the twist tensor. It is easy to check that under a gauge\ntransformation, $\\Lambda(x)$, the twist matrices change according to\n\\begin{equation}\n  \\Omega_\\nu(x) \\longrightarrow \\Omega'_\\nu(x) = \\Lambda(x+L\\hat \\nu)\n  \\Omega_\\nu(x)\\Lambda^+(x)\\,,\n\\end{equation}\nbut the twist tensor $n_{\\mu\\nu}$ remains unchanged. Therefore all the\nphysics of the twisted boundary conditions is contained in the twist\ntensor, and the particular choice of twist matrices is \nirrelevant. One can restrict the gauge transformations to those that\nleave the twist matrices unchanged. It is easy to check that the\nnecessary and sufficient condition for the gauge transformations is to\nobey the periodicity condition\n\\begin{equation}\n  \\label{eq:gauge}\n  \\Lambda(x+L\\hat\\nu) = \\Omega_\\nu(x)\\Lambda(x)\\Omega_\\nu^+(x)\\,.\n\\end{equation}\n\nThe reader interested in knowing more about the twisted boundary\nconditions is invited to consult the review~\\cite{ga:torus}. Here we\nwill use a particular setup: we choose to twist only one plane\n(the $x_1-x_2$ plane) by choosing $n_{12} = -n_{21} = 1$, while the\nrest of the components of the twist tensor will be zero. This means\nthat our gauge connections will still be periodic in the $x_3$ and\n$x_4$ directions. As we will\nsee, this choice guarantees that the action has a unique minimum\n(modulo gauge transformations), and therefore it turns out to be a\nconvenient choice for perturbative studies. This is the reason why the very\nsame choice has been made before to define the Twisted Polyakov Loop\nrunning coupling scheme~\\cite{deDivitiis:1994yp}, or for other\nperturbative studies~\\cite{Luscher:1985wf}. We will closely follow the\nnotation and steps \npresented in~\\cite{Perez:2013dra}, a reference that the reader\ninterested in more details should consult.\n\nA convenient implementation of twisted boundary conditions consists in\nusing space-time independent twist matrices. In particular for the\nperiodic directions we set the twist matrices to one\n\\begin{subequations}\n\\begin{eqnarray}\n  \\Omega_{1,2}(x) &=& \\Omega_{1,2} \\\\\n  \\Omega_{3,4}(x) &=& 1\\,.\n\\end{eqnarray}\n\\end{subequations}\n\nWe will use latin indexes ($i,j,\\dots=1,2$) to run over the directions in the\ntwisted plane, while greek indexes ($\\mu,\\nu,\\dots=0,\\dots,3$) will\nrun over the four space time directions. The consistency relation\nEq.~(\\ref{eq:consistency}) implies the \nfollowing condition for the twist matrices\n\\begin{equation}\n  \\Omega_1\\Omega_2 = e^{2\\pi\\imath \/N}\n  \\Omega_2\\Omega_1.\n\\end{equation}\nNotice that the boundary conditions for the gauge\nfield with this choice of the twist matrices are\n\\begin{equation}\n  \\label{eq:twalg}\n  A_\\mu(x+L\\hat k) = \\Omega_kA_\\mu(x)\\Omega_k^+\\,,\n\\end{equation}\nand $A_\\mu=0$ is a valid connection. In fact we\nwill show that it is the only connection compatible with the boundary\nconditions that does not depend on $x$, and therefore it is the\nunique minimum of the action modulo gauge transformations. \n\nEq.~(\\ref{eq:twalg}) defines a generalization of the Dirac algebra. It\ncan be shown~\\cite{ga:torus} that there is a\nunique solution for the matrices $\\Omega_i$ modulo similarity\ntransformations. Introducing the \\emph{color momentum}, $\\tilde p_i =\n\\frac{2\\pi\\tilde n_i}{NL}$ with $n_i=0,\\dots,N-1$\nit is easy to check that the $N^2$ matrices\n\\begin{equation}\n  \\label{eq:defG}\n  \\Gamma(\\tilde p) = \\frac{\\imath}{\\sqrt{2N}}e^{\\imath \\alpha(\\tilde\n    p)} \\Omega_1^{-\\tilde \n    n_2}\\Omega_2^{\\tilde n_1}\\,,\n\\end{equation}\nwhere $\\alpha(\\tilde p)$ are arbitrary phases, are linearly\nindependent and obey the relation \n\\begin{equation}\n  \\Omega_i \\Gamma(\\tilde p) \\Omega_i^+ = e^{\\imath L\\tilde p_i}\n  \\Gamma(\\tilde p)\\,.\n\\end{equation}\nMoreover all but\n$\\Gamma(\\tilde p=0)$ are traceless, and therefore they can be used as\na basis of the Lie algebra of the gauge group. This means that any\ngauge connection can be expanded as\n\\begin{equation}\n  A_\\mu^a(x)T^a = \\sum_{\\tilde p}'\\hat A_\\mu(x,\\tilde p)e^{\\imath\n    \\tilde px}\\Gamma(\\tilde p).\n\\end{equation}\nThe prime over the sum means that the term $\\tilde p_i=0$ is\nabsent in the sum, as required for a $SU(N)$ gauge group. Notice that\nthe coefficients $\\hat A_\\mu(x,\\tilde p)$ are functions (not \nmatrices) periodic in $x$. Therefore one can do an usual Fourier\nexpansion and obtain\n\\begin{equation}\n  A_\\mu^a(x)T^a = \\frac{1}{L^4} \\sum_{p,\\tilde p}'\\tilde A_\\mu(p,\\tilde\n  p)e^{\\imath \n    (p+\\tilde p)x}\\Gamma(\\tilde p)\\,,\n\\end{equation}\nwith the usual spatial momentum \n\\begin{equation}\n  p_\\mu = \\frac{2\\pi n_\\mu}{L}\\quad (n_\\mu\\in \\mathbb Z)\\,.\n\\end{equation}\nFinally we define the \\emph{total} momentum as the sum of the color\nand space momentum $P_i = p_i+\\tilde p_i$, $P_{3,4} = p_{3,4}$. Noting\nthat any $P_\\mu$ can be \nuniquely decomposed in the space momentum and color momentum degrees\nof freedom we can safely write $\\Gamma(P)$ instead of $\\Gamma(\\tilde\np)$. Our main conclusion is that any gauge connection compatible\nwith our choice of boundary conditions can be written as \n\\begin{equation}\n  \\label{eq:gaugetw}\n  A_\\mu^a(x)T^a = \\frac{1}{L^4} \\sum_{P}'\\tilde A_\\mu(P)\n  e^{\\imath Px}\\Gamma(P)\\,.\n\\end{equation}\nIn particular the only connection that does not depend on $x$ is given\nby $\\tilde A_\\mu(P) = 0$. In general the matrices $\\Gamma(P)$ are not\nanti-hermitian, but one can choose the phases \n$\\alpha(P)$ of equation~(\\ref{eq:defG}) so that this condition is\nenforced\n\\begin{equation}\n  \\alpha(P) = \\frac{\\theta}{2}P_1P_2\\qquad \n  \\left(\n    \\theta = \\frac{NL^2}{2\\pi}\n  \\right)\\,.\n\\end{equation}\nIn this case, the Fourier coefficients $\\tilde A_\\mu(P)$ satisfy the\nusual relation\n\\begin{equation}\n  \\tilde A_\\mu(P)^* = \\tilde A_\\mu(-P)\\,,\n\\end{equation}\nand the $\\Gamma$ matrices are normalized according to\n\\begin{equation}\n  {\\rm Tr}\\left\\{ \\Gamma(P)\\Gamma(-P)\\right\\} = -\\frac{1}{2}\\,.\n\\end{equation}\n\nWe finally note that a simlar expansion is possible on the lattice,\nwith the only difference that the space momentum will be restricted\nto the Brillouin zone. \n\n\\subsection{Matter fields}\n\\label{sc:fermions}\n\nThe inclusion of matter fields interacting with gauge fields with\ntwisted boundary conditions is not completely straightforward. To\nunderstand why it is better first to consider how to include \nfermion fields in the fundamental representation. Since the twist\nmatrices tell us how fields change under translations, one naively\nexpects \n\\begin{equation}\n  \\psi(x+L\\hat i) = \\Omega_i\\psi(x)\\,,\n\\end{equation}\nbut one can easily see that this choice is not consistent, \nsince the value of the field $\\psi(x+L\\hat i+L\\hat j)$ depends on the\norder in which we perform the translations due to the\nnon-commutativity of the twist matrices. This difficulty can be\navoided by introducing more fermions, or what usually is called a\n``smell'' degree of freedom~\\cite{Parisi:1984cy}. If\n$\\alpha,\\beta=1,\\dots,N_s$ are indices that run over the $N_s$ smells of\nfermions, and $a,b=1,\\dots,N$ run over the color degrees of freedom,\nthe boundary conditions of the fermions read\n\\begin{equation}\n  \\psi^a_\\alpha(x+L\\hat i) =\n  e^{\\imath\\theta_i}(\\Omega_i)_{ab}(\\Omega^*_i)_{\\alpha\\beta} \n  \\psi^b_\\beta(x) \\,.\n\\end{equation}\nThis means that a fermion smell becomes a linear combination of the\ngauge transformed fermion smells under a translation. $\\theta_i$\nare in principle arbitrary, but introduced for \nconvenience to remove the zero-momentum modes of the Dirac\noperator. These phases have to be chosen\nsuch that they are not elements of the gauge group\n(i.e. $e^{\\imath\\theta} \\not\\in SU(N)$).\nThis choice of boundary conditions for the  \nfermion fields is consistent, but they require the number of\nsmells to be equal to the number of colors. One can easily extend the\nconstruction to the case when the ratio $N_s\/N$ is an integer, but in\ngeneral one can not have an arbitrary number of fermions in the\nfundamental representation.\n\nOn the other hand fermions in the adjoint representation transform in\nthe same way as the gauge fields, and therefore any number of fermions\nwould be compatible with the twisted boundary conditions. \n\nRegardless of the representation but assuming that the matter fields\nare compatible with the twisted boundary conditions, $\\mathcal O(a)$\nimprovement for massless Wilson quarks is automatically\nsatisfied since fields live on a torus, and the boundary conditions do\nnot break chiral symmetry (see~\\cite{Sint:2010eh,Frezzotti:2003ni}). \n\n\n\n\n\\subsection{Cutoff effects in the twisted running coupling}\n\nThe comparison of the lattice and the continuum computations of\n$\\mathcal E(t)$ can give us an idea of the size of cutoff effects (to\nleading order in $g_0^2$) of the twisted gradient flow coupling. We\nare going to study in detail the case of lattice simulations using the\nWilson action, the Wilson flow, and the clover definition for the\nobservable. If we\ndefine \n\\begin{equation}\n  \\hat{\\mathcal N}_T(c,a\/L) = \n  \\frac{c^4}{128}\\sum_P' e^{-\\frac{c^2L^2}{4}\\hat P^2}\n  \\frac{\\mathring P^2 C^2 - (\\mathring P_\\mu C_\\mu)^2}{\\hat P^2}\\,,\n\\end{equation}\nthe quantity\n\\begin{equation}\n  Q(c, a\/L) = \\left|\\frac{\\hat{\\mathcal N}_T(c,a\/L) - \\mathcal N_T(c)}\n{{\\mathcal N}_T(c)}\\right|\\,,\n\\end{equation}\nquantifies to leading order the size of cutoff effects as a function of the\nlattice size and the scheme parameter $c$. A global picture of cutoff\neffects for the groups $SU(2)$ and $SU(3)$ \ncan be seen in the figure~\\ref{fig:cut1}. \n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\textwidth]{fig\/cutoff}\n  \\caption{Cutoff effects to leading order of perturbation theory in\n    the twisted gradient flow coupling. As we \n  see, for $c\\in[0.3-0.5]$ cutoff effects are below the 7\\% for an\n  $L\/a=8$ lattice.}\n  \\label{fig:cut1}\n\\end{figure}\n\nThese figures may lead to the conclusion that a large value of $c$ is\noptimal. But from the point of view of lattice simulations, it is\nknown~\\cite{Fritzsch:2013je} that larger values of $c$ lead to larger\nstatistical errors when computing the coupling via lattice\nsimulations. For the typical lattice sizes ($L\/a\\sim 10-20$) that one\nuses in step scaling studies the values $c\\in[0.3,0.5]$ seem reasonable. \n\n\\subsection{Improved coupling definition}\n\nIf one is computing $t^2\\langle E(t)\\rangle$ non-perturbatively via\nlattice simulations, and one is using the Wilson action, the Wilson\nflow and the clover observable for the evaluation of the energy\ndensity observable, one can alternatively define the coupling via\n\\begin{equation}\n\\label{eq:latcou}\n  g_T^2(L) = \\hat{\\mathcal N}_T^{-1}(c,a\/L)t^2\\langle E(t) \\rangle\n  \\Big|_{t=c^2L^2\/8} \\,.\n\\end{equation}\nThis last coupling definition has the same properties, but one expects\nan improved scaling towards the continuum limit, since the leading\norder cutoff effects $\\propto g_0^2$ have been removed thanks to\nthe lattice factor $\\hat{\\mathcal N}_T(c,a\/L)$.\n\nIn a similar way, any choice of discretizations that define a coupling\ncan be normalized with a factor computed on the lattice\n(cf. section~\\ref{sc:disc}), leading to an improved scaling towards\nthe continuum. \n\n\n\n\n\\subsection{Perturbative behavior of the gradient flow in a twisted box: continuum}\n\n\\subsubsection{Generalities and gauge fixing}\n\nWe are interested in the perturbative expression for  $\\langle E(t)\n\\rangle$, and in order to avoid some difficulties in the definition of\npropagators, it turns out to be convenient to fix the gauge of the\nflow field $B_\\mu(x,t)$. This can be achieved by studying the modified\nflow equation\n\\begin{equation}\n    \\frac{{\\rm d} B_\\mu^{(\\alpha)}(x,t)}{{\\rm d}t} = D_\\nu^{(\\alpha)}\n    G_{\\nu\\mu}^{(\\alpha)}(x,t) +  \n  \\alpha D_\\mu^{(\\alpha)}\\partial_\\nu B_\\nu^{(\\alpha)}(x,t) \\,.\n\\end{equation}\nThe superscript ${(\\alpha)}$ recalls that covariant derivatives and field\nstrength are made of the modified flow field $B_\\mu^{(\\alpha)}(x,t)$,\nsolution of the previous equation. A solution of this modified flow\nequation $B_\\mu^{(\\alpha)}(x,t)$ can be transformed in a solution of the\noriginal flow equation~(\\ref{eq:flow}) by a time dependent gauge\ntransformation~\\cite{Luscher:2011bx}\n\\begin{equation}\n  B_\\mu = \\Lambda B_\\mu^{(\\alpha)}\\Lambda^{-1} + \n  \\Lambda \\partial_\\mu\n  \\Lambda^{-1} \\,,\n\\end{equation}\nwhere \n\\begin{equation}\n  \\frac{{\\rm d} \\Lambda}{{\\rm d}t} =\n  \\alpha \\Lambda \\partial_\\mu B_\\mu \\,;\\quad\n  \\Lambda\\big|_{t=0} = 1\\,.\n\\end{equation}\n\nTherefore gauge invariant quantities are independent of $\\alpha$. Note\nthat the previously defined gauge transformation \n$\\Lambda(x)$ belongs to the restricted set of gauge transformations\nthat leave the twist matrices invariant (see\nequation~(\\ref{eq:gauge})), and the boundary conditions of\n$B_\\mu^{(\\alpha)}$ are also independent of $\\alpha$. \n\n\\subsubsection{Flow field and energy density to leading order}\n\nThe particular choice $\\alpha=1$ simplify the computations, and we\nwill use it for the rest of this section. The modified flow equation\nreads in this case\n\\begin{equation}\n  \\label{eq:flowmd}\n  \\frac{{\\rm d} B_\\mu}{{\\rm d}t} = D_\\nu G_{\\nu\\mu} +\n  D_\\mu\\partial_\\nu B_\\nu \\,.\n\\end{equation}\nIn perturbation theory one re-scales the gauge potential with the bare\ncoupling $A_\\mu \\rightarrow g_0A_\\mu$, and the flow field has an\nasymptotic expansion in the bare coupling\n\\begin{equation}\n  \\label{eq:flowfg0}\n  B_\\mu(x, t) = \\sum_{n=1}^{\\infty} B_{\\mu,n}(x, t) g_0^n \\,.\n\\end{equation}\nTo leading order our flow equation~(\\ref{eq:flowmd}) is just the heat\nequation\n\\begin{eqnarray}\n  \\label{eq:flowlo}\n   \\frac{{\\rm d} B_{\\mu,1}(x,t)}{{\\rm d}t} &=& \\partial_\\nu^2\n   B_{\\mu,1}(x,t) \\\\\n   B_{\\mu,1}(x,0) &=& A_{\\mu}(x)\\, ,\n\\end{eqnarray}\nexpanding $B_{\\mu,1}(x,t)$ in our preferred basis~(\\ref{eq:gaugetw}) one\ncan easily solve~(\\ref{eq:flowlo}) and obtain\n\\begin{equation}\n  B_{\\mu,1}(x,t) = \\frac{1}{L^4}\n  \\sum_P' e^{-P^2t} \\tilde A_\\mu(P) e^{\\imath Px}\n  \\Gamma(P)\\,.\n\\end{equation}\n\nFinally our observable of interest also has an expansion in powers of\n$g_0$\n\\begin{equation}\n  \\langle E(t)\\rangle = -\\frac{1}{2}\\langle\n  {\\rm Tr}\\{G_{\\mu\\nu}(x, t)G_{\\mu\\nu}(x,t)\\}\\rangle = \\mathcal E(t) + \\mathcal\n  O(g_0^4)\\,.\n\\end{equation}\nOne can easily obtain \n\\begin{eqnarray}\n  \\mathcal E(t) &=& \\frac{g_0^2}{2}\\langle \n  \\partial_\\mu B_{\\nu,1}\\partial_\\mu B_{\\nu,1} - \\partial_\\mu\n  B_{\\nu,1}\\partial_\\nu B_{\\mu,1} \n  \\rangle  \\\\\n  \\nonumber\n  &=& \\frac{-g_0}{2L^8}\\sum_{P,Q}'e^{-(P^2+Q^2)t}e^{\\imath (P+Q)x} \n  \\left( P_\\alpha Q_\\alpha\\delta_{\\mu\\nu} -\n    P_\\mu Q_\\nu\\right) \\langle \\tilde A_\\mu(P)\\tilde A_\\nu(Q)\\rangle \n  {\\rm Tr}(\\hat\\Gamma(P)\\hat\\Gamma(Q))\\,.\n\\end{eqnarray}\nFinally using the expression for the gluon propagator\n\\begin{equation}\n  \\langle \\tilde A_\\mu(P)\\tilde A_\\nu(Q) \\rangle =\n  L^4 \\delta_{P_\\alpha, -Q_\\alpha} \\frac{1}{P^2}\n  \\left[ \\delta_{\\mu\\nu} - (1-\\lambda^{-1})\\frac{P_\\mu P_\\nu}{P^2}\\right]\n  \\frac{1}{{\\rm Tr}({\\Gamma(-P)\\Gamma(P)})} + \\mathcal O(g_0^2)\n\\end{equation}\none gets\n\\begin{equation}\n  \\label{eq:et}\n  \\mathcal E(t) = \n  \\frac{3g_0^2}{2L^4}\\sum_P' e^{-2P^2t}\\,.\n\\end{equation}\n\n\\subsection{Perturbative behavior of the gradient flow in a twisted box: lattice}\n\nWhen defining the gradient flow in the lattice one has to make several\nchoices. These basically correspond to the particular discretizations\nof the action whose gradient is used to define the flow, as well as\nthe discretization of the energy density and the choice of action that\none simulates (i.e. Wilson\/improved actions). \n\nFirst we will analyze the popular case where the Wilson action is\nsimulated, and one uses the same action to define the flow (in this\ncase is called the Wilson flow). The clover definition of the\nobservable has been a typical choice~\\cite{Luscher:2010iy} for a \ndiscretization of\nthe energy density. Later we will comment on the general case. \n\n\\subsubsection{Generalities and gauge fixing}\n\nOn the lattice the gradient flow is substituted by a discretized\nversion. There are several possibilities: one can use the Wilson\naction  \n\\begin{equation}\n  S_w(V) = \\frac{1}{g_0^2} \\sum_{\\rm p} {\\rm Re}\\{{\\rm Tr}(1-U_{\\rm p})\\}\n\\end{equation}\nwhere the sum runs over the oriented plaquettes, and define the flow\nequation by equating the time derivative of the links with\nthe gradient of the Wilson action\n\\begin{equation}\n  \\label{eq:flowlat}\n  a^2\\partial_t V_\\mu(x,t) = -g_0^2 \\{T^a\\partial_{x,\\mu}^a S_w(V)\\}\n  V_\\mu(x,t) \\,,  \\qquad V_\\mu(x,0) = U_\\mu(x)  \\,.\n\\end{equation}\nIn this case the gradient flow is usually referred as the Wilson\nflow. Some explanations of our notation are in order. \nIf\n$f(U_\\mu(x))$ is an arbitrary function of the link variable\n$U_\\mu(x)$, the components of its Lie-algebra valued derivative\n$\\partial_{x,\\mu}^a $  \nare defined as \n\\begin{equation}\n   \\partial_{x,\\mu}^a f(U_\\mu(x)) = \\left.\\frac{ {\\rm d} f(e^{\\epsilon\n            T^a}U_\\mu(x))}{ {\\rm d}\\epsilon} \\right|_{\\epsilon=0}\\,. \n\\end{equation}\nIn perturbation theory one is interested in a neighborhood of the\nclassical vacuum configuration. In this neighborhood the lattice  \nfields $U_\\mu(x)$ and $V_\\mu(x,t)$ are parametrized as follows:\n\\begin{align}\n  U_\\mu(x)   &= \\exp\\{ag_0 A_\\mu(x)\\}   \\;, &\n  V_\\mu(x,t) &= \\exp\\{ag_0 B_\\mu(x,t)\\} \\;.\n\\end{align}\n\nAgain it is convenient to study a modified flow equation where the\ngauge degrees of freedom are damped. We will consider\n\\begin{equation}\n  \\label{eq:flowlatmd}\n  a^2\\partial_t V_\\mu^\\Lambda(x,t) = g_0^2 \\left\\{ \n        -\\big[ T^a\\partial_{x,\\mu}^a S_w(V^\\Lambda) \\big] \n        + a^2\\hat D_\\mu^{\\Lambda}\\big[\\Lambda^{-1}(x,t)\\dot \\Lambda(x,t)\\big] \n                                           \\right\\} V_\\mu^\\Lambda(x,t) \\,,\n\\end{equation}\nwith $V_\\mu^\\Lambda(x,0) = U_\\mu(x)$ and the forward lattice covariant\nderivative \n$\\hat D_\\mu^{\\Lambda}$ acting on Lie-algebra valued functions according to\n\\begin{equation}\n  \\hat D_\\mu f(x) = \\frac{1}{a}\\left[\n    V_\\mu(x,t)f(x+\\hat\\mu)V_\\mu^{-1}(x,t) - f(x)\n  \\right] \\,.\n\\end{equation}\n\nSolutions of the modified and original flow equations are related by a\ngauge transformation\n\\begin{equation}\n  V_\\mu(x,t) = \\Lambda(x,t)V_\\mu^\\Lambda(x,t)\\Lambda^{-1} (x+\\hat\\mu,t)\\,.\n\\end{equation}\nThe most natural choice for $ \\Lambda(x,t)$ is the same functional\nused for gauge fixing \n\\begin{equation}\n  \\label{eq:lam}\n  \\Lambda^{-1}\\frac{{\\rm d} \\Lambda}{{\\rm d}t} = \\alpha\n  \\hat\\partial^*_\\mu B_\\mu(x,t)\\,,\\qquad \n  \\Lambda\\big|_{t=0} = 1\\,.\n\\end{equation}\nwhere $\\hat \\partial, \\hat \\partial^*$ denote the forward\/backward \nfinite differences. We again note that the boundary conditions of\n$V_\\mu^\\Lambda(x,t)$ are independent of $\\alpha$, since $\\Lambda(x,t)$\nbelongs to the restricted class of gauge transformations that leave\nthe twist matrices unchanged.\n\n\n\\subsubsection{Flow field and energy density to leading order}\n\nAgain the choice $\\alpha=1$ turns out to be convenient and we\nwill stick to it from now on.\n\nThe modified flow equation reads\n\\begin{equation}\n  a^2\\partial_t V_\\mu(x,t) = g_0^2 \\left\\{ -[T^a\\partial_{x,\\mu}^a\n      S_w(V)] + a^2\\hat D_\\mu(\\hat\\partial_\\nu^* B_\\nu ) \n      \\right\\}\n  V_\\mu(x,t) \\,,  \\qquad V_\\mu(x,0) = U_\\mu(x) \\,.\n\\end{equation}\nThe flow field can be expanded in powers of $g_0$\n(equation~(\\ref{eq:flowfg0})) and to first order in $g_0$ we have\n\\begin{equation}\n  \\label{eq:wflowlato1}\n  \\partial_t B_{\\mu,1}(x,t) = \\hat \\partial_\\nu\\hat\\partial_\\nu^*\n  B_{\\mu,1}(x,t) \\,.\n\\end{equation}\nExpanding the flow field in our favorite Lie-algebra basis\n(equation~(\\ref{eq:gaugetw})) one can write the solution to the\nprevious equation\n\\begin{equation}\n  B_{\\mu,1}(x,t) = \\frac{1}{L^4}\\sum_P' e^{-\\hat P^2t} \\tilde A_\\mu(P)\n  e^{\\imath Px} \\Gamma(P)\\,,\n\\end{equation}\nwhere \n\\begin{equation}\n  \\hat P_\\mu = \\frac{2}{a}\\sin\\left(a\\frac{P_\\mu}{2}\\right)\n\\end{equation}\nis the usual lattice momentum.\n\nWe can choose among different discretizations for the energy\ndensity. The most popular one consists in using the clover definition\nfor $G_{\\mu\\nu}(x,t)$~\\cite{Luscher:2010iy}. To leading order we have\n\\begin{eqnarray}\n  \\nonumber\n  \\hat G_{\\mu\\nu}(x,t) &=& \\frac{g_0}{2}\\,\\mathring\\partial_\\mu\\left[B_{\\nu,1}(x,t) + \n    B_{\\nu,1}(x-\\hat \\nu,t)\\right] \\\\\n  &-&\n  \\frac{g_0}{2}\\,\\mathring\\partial_\\nu\\left[B_{\\mu,1}(x,t) + \n    B_{\\mu,1}(x-\\hat \\mu,t)\\right] + \\mathcal O(g_0^2) \\,,\n\\end{eqnarray}\nwhere $\\mathring\\partial_{\\mu} = \\tfrac{1}{2}(\\hat \\partial_\\mu +\n\\hat \\partial^*_\\mu)$ is the symmetric finite difference. The energy\ndensity computed with the clover definition for the field strength\ntensor reads \n\\begin{equation}\n  \\langle E^{\\rm cl}(t)\\rangle = -\\frac{1}{2}\\langle {\\rm Tr}\\{ \\hat\n  G_{\\mu\\nu} \\hat G_{\\mu\\nu}\\} \\rangle = \\mathcal E^{\\rm cl}(t, a\/L) +\n  \\mathcal O(g_0^2) \n\\end{equation}\nUsing the definitions\n\\begin{subequations}\n\\begin{eqnarray}\n  \\mathring P_\\mu &=& \\frac{1}{a}\\sin\\left(aP_\\mu\\right)\\,,\\\\\n  C_\\mu &=& \\cos\\left(a\\frac{P_\\mu}{2}\\right)\\,,\n\\end{eqnarray}\n\\end{subequations}\nand the lattice gluon propagator, one can easily obtain\n\\begin{equation}\n  \\label{eq:etcl}\n  \\hat{\\mathcal E}^{\\rm cl}(t, a\/L) = \n  \\frac{g_0^2}{2L^4}\\sum_{P}' e^{-2\\hat P^2t}\n  \\frac{\\mathring P^2 C^2 - (\\mathring P_\\mu C_\\mu)^2}{\\hat P^2}\\,.\n\\end{equation}\n\n\n\\subsubsection[Some comments on different\n  discretizations]{Some comments on different\n  discretizations\\protect\\footnote{The author wants to thank S. Sint for his\n    help in understanding the points discussed in this section.}}\n\\label{sc:disc}\n\nIn general the lattice computation of the leading order behavior of\nthe energy density involves several choices of discretization: the\naction that one simulates (labelled (a)), the action whose gradient\ndefines the flow evolution (labelled (f)), and finally the\ndiscretization used to compute the observable (labelled (O)). To\nleading order, these three choices can be expressed as \nchoice of ``actions'' \n\\begin{subequations}\n  \\begin{eqnarray}\n    S_a[\\tilde A_{\\mu}] &=& \\frac{1}{4L^4}\\sum_{P}' \\tilde A_\\mu(-P)\n    K_{\\mu\\nu}^{(a)}(P) \\tilde \n    A_\\nu(P) + \\mathcal O(g_0^2)\\,,\\\\\n    S_f[\\tilde A_{\\mu}] &=& \\frac{1}{4L^4}\\sum_{P}' \\tilde A_\\mu(-P)\n    K_{\\mu\\nu}^{(f)}(P) \\tilde \n    A_\\nu(P) + \\mathcal O(g_0^2)\\,, \\\\\n    S_O[\\tilde A_{\\mu}] &=& \\frac{1}{4L^4}\\sum_{P}' \\tilde A_\\mu(-P)\n    K_{\\mu\\nu}^{(O)}(P) \\tilde \n    A_\\nu(P) + \\mathcal O(g_0^2) \\,.\n  \\end{eqnarray}\n\\end{subequations}\n\nThe matrices $K^{(a)}$ and $K^{(f)}$ may (and should) contain a gauge\nfixing part, but not the one corresponding to the observable\n$K^{(O)}$. In this way final results will be independent of the\nchoices of gauge.\nThe inverse of the $K_{\\mu\\nu}^{(a)}$ defines the lattice gluon propagator\n\\begin{eqnarray}\n  \\langle A_\\mu(-P)A_\\nu(P)\\rangle &=& D_{\\mu\\nu}(P)\\,, \\\\\n  K_{\\mu\\alpha}^{(a)}(P)D_{\\alpha\\nu}(P) &=& \\delta_{\\mu\\nu}\\,.\n\\end{eqnarray}\n\nUsing this notation it is trivial to obtain the form of the flow field\nto leading order\n\\begin{equation}\n  \\tilde B_{\\mu,1}(P) = \\left( \\exp\\{-t K^{(f)}(P)\\}\\right)_{\\mu\\nu}\n  \\tilde A_\\nu(P) = H_{\\mu\\nu}(t,P) \\tilde A_\\nu(P) \\,, \n\\end{equation}\nand noting that the reality of the action requires that $H^+(t,P) =\nH(t,-P)$,  we can write the expression of the energy density to\nleading order as \n\\begin{eqnarray}\n  \\mathcal E(t,a\/L) &=& g_0^2 \\langle S_O[\\tilde\n  B_{\\mu,1}]\\rangle \\\\ \n  &=& \\frac{g_0^2}{2L^4} \\sum_P' {\\rm Tr}\\{ H^+(t,P)K^{(O)}(P)H(t,P) \n  D(P)\\}\\,.\n\\end{eqnarray}\n\nThis formula allows an easy evaluation of the energy density, to\nleading order in perturbation theory, for any choice of\ndiscretizations. One general point that one can make is that if one\nuses the Wilson flow the matrix $H(t,P)$ can be chosen to be\nproportional to the identity (by an appropriate gauge choice), and\ntherefore commutes with any other matrix. Moreover if the \naction that one simulates is the same as the one that we use to\ncompute the observable, the product of matrices $DK^{O}$ together with\nthe trace simply result in a factor 3, and therefore one obtains\n\\begin{equation}\n  \\mathcal E(t,a\/L) = \\frac{3g_0^2}{2L^4} \\sum_P' e^{-2t\\hat P^2}\\,.\n\\end{equation}\n\nThis means that without changing the flow, improving\nthe action and the observable leads to exactly the same cutoff effects\nthan if one does not improve anything (to leading order). \n\n\\subsection{Tests}\n\nIn order to check the previous computations one can perform several\nconsistency checks. First it is obvious that the continuum result\n(equation~(\\ref{eq:et})) is recovered from the lattice one \n(equation~(\\ref{eq:etcl})) if one takes the limit $a\/L \\rightarrow\n0$. In the infinite volume limit boundary conditions are irrelevant,\nand therefore for $L\\rightarrow \\infty$ one should recover the result\nof~\\cite{Luscher:2010iy} that reads\n\\begin{equation}\n  \\mathcal E^{(L=\\infty)}(t) = \\frac{3g_0^2(N^2-1)}{128\\pi^2 t^2}\\,.\n\\end{equation}\n\nThis result is reproduced from our expression\nequation~(\\ref{eq:et}) by simply noting that \n\\begin{equation}\n  P_\\mu = \\frac{2\\pi}{L}\\left( n_\\mu + \\frac{\\tilde n_\\mu}{N}\\right)\\,,\n\\end{equation}\nand therefore \n\\begin{equation}\n  \\frac{1}{L^4} \\sum_P' \\xrightarrow[L\\rightarrow \\infty]{}\n  \\frac{1}{(2\\pi)^4}\\sum_{\\tilde p_i}' \\int_{-\\infty}^\\infty {\\rm d}^4P\\,.\n\\end{equation}\nFinally recalling that there are $N^2-1$ terms in the sum over $\\tilde\np_i$ (the term $\\tilde p_i=0$ is explicitly excluded) one obtains\n\\begin{equation}\n  \\mathcal E(t)\\xrightarrow[L\\rightarrow \\infty]{}\n  \\frac{3g_0^2}{32\\pi^4}\\sum_{\\tilde p_i}' \\int_{-\\infty}^\\infty {\\rm d}^4P\n  e^{-2P^2 t} = \\frac{3g_0^2(N^2-1)}{128\\pi^2 t^2}\\,.\n\\end{equation}\n\nTo check the lattice computations we have performed some dedicated\npure gauge lattice \nsimulations. We use the plaquette action of an $SU(2)$ gauge theory in\ntwo different volumes $L\/a=4^4$ and $L\/a= \n6^4$. We collect $10,000$ measurements of $\\langle E^{\\rm\n  cl}(t)\\rangle$ for different values of $t$ and $\\beta = 2\/g_0^2 =\n40, 80,  120,  200,  400,  560,  800,  960,  1120, 1280$. In these\nlarge-$\\beta$ simulations the \nmeasured $\\langle E^{\\rm cl}(t)\\rangle$ should\nreproduce the perturbative expression\n(equation~(\\ref{eq:etcl})). Being more precise, we will study\nnumerically the\nquantity\n\\begin{equation}\n  R(g_0, t) = \\frac{\\langle E^{\\rm cl}(t)\\rangle - \\mathcal E^{\\rm\n      cl}(t)}{\\mathcal E^{\\rm cl}(t)}\\,.\n\\end{equation}\nWe expect that $R(g_0,t) = \\mathcal O(g_0^2)$, and therefore by fitting\nthe data from the simulations to a linear behavior \n\\begin{equation}\n  R(g_0, t)  = m(t)g_0^2 + n(t)\n\\end{equation}\none should obtain an intercept $n(t)$ compatible with zero within\nerrors. Indeed this is the case, for different values of $t$ and $L$,\nas the reader can check in table~(\\ref{tab:fit}). A couple of\nrepresentative fits are shown in the figure~\\ref{fig:fit}.\n\n\\input{fit-table.tex}\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/L4c030}\n    \\caption{Fit for $L=4^4$ and $c=0.3$.}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/L6c050}\n    \\caption{Fit for $L=6^4$ and $c=0.5$}\n  \\end{subfigure}\n\n  \\caption{Some representative fits to the large-$\\beta$\n    simulations. The plots show the function $R(g_0, t)$ at fixed\n    $t=c^2L^2\/8$ versus $g_0^2$.} \n  \\label{fig:fit}\n\\end{figure}\n\n\n\n\n\n\n\\section{Introduction}\n\n\\input{intro.tex}\n\n\\section{Twisted boundary conditions}\n\\label{sc:tw}\n\\input{bc.tex}\n\n\\section{The gradient flow in a twisted box}\n\\label{sc:flow}\n\\input{flow.tex}\n\n\\section{Running coupling definition}\n\\label{sc:coupling}\n\\input{coupling.tex}\n\n\\section{$SU(2)$ running coupling}\n\\label{sc:run}\n\\input{running.tex}\n\n\n\\section{Conclusions}\n\\input{conclusions.tex}\n\n\n\\section*{Acknowledgments}\n\nThis work has a large debt with\nM. Garc\\'ia Perez and A. Gonz\\'alez-Arroyo for sharing some of their results\nand notes before publication and for the many illuminating\ndiscussions. The help and advice of R. Sommer and U. Wolff was\ninvaluable in many of the steps of this work. \n\nI also want to thank my colleagues at DESY\/HU, specially\nP. Korcyl, P. Fritzsch, S. Sint and R. Sommer for the very many interesting\ndiscussions and their careful reading of the manuscript. D. Lin was very\nkind reading and helping to improve a manuscript of this work. \n\n\n\n\\subsection{Numerical computation of the step scaling function and\n  running coupling}\n\n\\subsubsection{Simulation details}\n\nWe will simulate $SU(2)$ YM theory using the Wilson action\n\\begin{equation}\n  S = \\frac{\\beta}{4}\\sum_{\\rm p} {\\rm Tr}\\left\\{ 1-U_{\\rm p}\\right\\}\n\\end{equation}\nwhere the sum runs over all oriented plaquettes. We simulate lattices\nof size $L\/a=20, 24, 30, 36$, and in order \nto compute the step scaling function also lattices of half this\nsize ($L\/a=10, 12, 15, 18$). The range of values of $\\beta$ (between 2.75\nand 12.0) translate to renormalized couplings\n$g_{\\rm TGF}^2(L)$ between 7.5 and 0.6 (for $c=0.3$), enough to cover both the\nnon-perturbative and perturbative regions of the \ntheory. Appendix~\\ref{ap:values} collects the values of the $g^2_{\\rm\n  TGF}(L)$ of our simulations. \n\nWe will use a combination of heatbath~\\cite{Creutz:1980zw,Fabricius:1984wp,Kennedy:1985nu} and\noverrelaxation~\\cite{Creutz:1987xi} as suggested\nin~\\cite{Wolff:1992nq}. In particular we \nchoose to do one heatbath sweep followed by $L\/a$ overrelaxation\nsweeps. Since measuring the coupling (i.e. integrating the flow\nequations) is numerically more expensive \nthan the Monte Carlo updates, we repeat this process 50 times between\nmeasurements. \n\nIn total we collect 2048 measurements of the coupling for each lattice\nsize, each value of $\\beta$, and several values of\n$c\\in[0.3,0.5]$. These measurements are collected in $N_r$ \nparallel runs (replica) of length $N_{\\rm MC}$ each so that $N_r\\times\nN_{\\rm MC} = 2048$. \nWe check that there are no autocorrelation between\nmeasurements (i.e. $\\tau_{\\rm int}=0.5$ within errors), even for our\nlarger lattices and larger values of $c$. We conclude that we can\nsafely consider the measurements independent. \n\nThe Wilson flow equations are integrated using the adaptive step size\nintegrator described in appendix D of~\\cite{Fritzsch:2013je}. With\nthis scheme we \nmake sure that the integration error in each step is not larger than\n$10^{-6}$. \n\n\\subsubsection{Data analysis}\n\nFor each $L\/a$ we have computed the value of the twisted gradient flow\ncoupling at different values of\n$\\beta$ (we call it $g^2_{\\rm TGF}(\\beta;L\/a)$). These data are fitted to a\nPad\\'e-like ansatz\n\\begin{equation}\n  \\label{eq:pade}\n  g^2_{\\rm TGF}(\\beta;L\/a) = \\frac{4}{\\beta} \\quad\n  \\frac{\\sum_{n=0}^{M-1} a_n\\beta^n + \\beta^M}\n  {\\sum_{n=0}^{M-1} b_n\\beta^n + \\beta^M}\\,.\n\\end{equation}\nThis fit imposes the one-loop constraint to the data (i.e. $g^2_{\\rm\n  TGF}(\\beta;L\/a) \\rightarrow 4\/\\beta$ at large $\\beta$), and has\na total of $2M$ free fit parameters. \n\nAlternatively, and to estimate the dependence of our results on the \nchoice of functional form used to fit the data, we use a different Pad\\'e\ninspired functional form\n\\begin{equation}\n  \\label{eq:taylor}\n  g^2_{\\rm TGF}(\\beta;L\/a) = \\frac{4}{\\beta} \\quad\n  \\frac{1}\n  {1 + \\sum_{n=1}^{M} c_n\/\\beta^n}\\,,\n\\end{equation}\nthat also ensures the correct one-loop behavior at large $\\beta$.\n\nWe obtain good fits ($\\chi^2\/{\\rm\nndof}\\sim 0.6-1.9$) with $M=2$ when using the functional form of\nEq.~\\ref{eq:pade} to fit the lattice data (i.e. 4 fitting\nparameters). When using the functional form of Eq.\\ref{eq:taylor} we\nneed $M=4$ to accurately describe the data on the small lattices\n($L\/a=10,12$) and $M=6$ for the larger ones\n($L\/a=15,18,20,24,30,36$). It is important to \nstress that the data are statistically uncorrelated, since they\ncorrespond to different simulations. \n\nIn the figures~\\ref{fig:fit_l24} we show a couple of these fits. Our\nworst fit corresponds to the $L\/a=24$ lattice and \nthe Pade fit gives a $\\chi^2\/{\\rm ndof}=1.69$, while the Taylor fit\nresults in a fit quality of $\\chi^2\/{\\rm ndof}=1.9$. We see how\nin this case the two different functional forms interpolate\ndifferently between the data, giving us confidence that if one\nestimates the error of the interpolation using both functional forms,\none will be on the safe side\\footnote{We point that probably a more\nsophisticated analysis technique (or simply, simulating an additional\nlattice to avoid having large gaps in the data), might result in a\nmore precise result.}. \n\\begin{figure}\n  \\centering\n        \\begin{subfigure}[t]{0.45\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/fit_l24}\n          \\caption{}\n        \\end{subfigure}\n        \\begin{subfigure}[t]{0.45\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/fit_l36}\n          \\caption{}\n        \\end{subfigure}\n  \\caption{Some examples of our fits to interpolate the values of the\n    renormalized coupling for different values of $\\beta$. \n    (a): Our worst fits corresponds to the $L\/a=24$. As we\n            can see there is a difference between the different\n            interpolating functions between the data points. We\n            stress that this systematic effects is taken into account\n            in our analysis by using both functional forms to estimate\n          the error of the interpolations (see the text for more details).\n    (b): Fits to the data of the $L\/a=36$ lattice. As we can\n            see, in this case both interpolating functions agree\n            within errors, although the polynomial fits tends to have\n            larger errors.\n}\n  \\label{fig:fit_l24}\n\\end{figure}\n\nWe use resampling methods to propagate errors by using 4000 bootstrap\nsamples. All fitting parameters derived from our original data are\ncomputed for each bootstrap sample. Interpolation points are computed\nfor each bootstrap sample and each functional form. The final error of\nthe interpolated point is computed using \\emph{both} functional forms\nand \\emph{all} bootstrap samples,\nand therefore takes into account not only the statistical uncertainty,\nbut also the systematic effect due to the dependence of the\ninterpolating functional form. \n\n\\subsubsection{Step scaling function}\n\nWe will first show the continuum extrapolations of the step scaling\nfunction $\\Sigma(u,a\/L)$ at some representative values of\n$u=7.5, 3.75, 1.5$. Figure~\\ref{fig:ss} shows that these\nextrapolations are mild. We have used the value $c=0.3$ that gives a\nprecision in the data for the renormalized coupling between $0.15\\%$\nand $0.25\\%$.\n\nOne of the advantages of the use of the\ntwisted boundary conditions is the absence of $\\mathcal O(a)$ cutoff\neffects, that are present for example in the Schr\\\"odinger functional\ndue to boundary effects. Here the invariance under\ntranslations guarantee that the continuum limit can be safely taken by\na linear extrapolation in $(a\/L)^2$.\n\n\n\\begin{figure}\n  \\centering\n        \\begin{subfigure}{\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/extra_75}\n        \\end{subfigure}\n        \\begin{subfigure}[b]{\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/extra_375}\n        \\end{subfigure}\n        \\begin{subfigure}[b]{\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/extra_15}\n        \\end{subfigure}\n  \\caption{Examples of the continuum extrapolation of the step scaling\n  function. The three figures corresponds (from top to bottom) to the\n  values $u=7.5, 3.75, 1.5$. We recall that we use a scale factor\n  $s=1\/2$, and the scheme is defined by the parameter $c=0.3$.} \n  \\label{fig:ss}\n\\end{figure}\n\n\n\\subsubsection{Running coupling}\n\nAs a final application, we will compute the running coupling.\nWe will fix the scheme by setting \n$c=0.3$. We start our recursion in a volume $L_{\\rm max}$ defined by the\ncondition\n\\begin{equation}\n  g^2_{\\rm TGF}(L_{\\rm max})\\Big|_{c=0.3} = 7.5\\,.\n\\end{equation}\nThe lattice step scaling function and its continuum limit is computed\nas described in the previous sections. As figure~\\ref{fig:ss}\nshows, the extrapolations towards the \ncontinuum are rather flat. The continuum limit values are used to\nfurther compute the values of the step scaling function at larger\nrenormalization scales (smaller volumes), up to $L_{\\rm min} = L_{\\rm\n  max}\/2^{26}$, \nwhere $g^2_{\\rm TGF}(L_{\\rm min})|_{c=0.3}=0.5324(84)$.\n\nSince the same functional form (fitting parameters) are used\nrecursively to compute the values of the coupling at different scales,\none has to propagate errors taking into account the correlations\ncorrectly. This is done in the spirit of the resampling methods in the\nmost naive way: one uses as input for the coupling at a scale $L$ all\nthe bootstrap samples of the coupling from the scale $2L$. We recall\nhere that these bootstrap samples carry the information not only of\nthe statistical uncertainties, but also of the dependence of our\nresults on the functional form chosen to fit the data.\nOur results have carefully taken into account the two sources of\nsystematic uncertainty: the continuum extrapolation and the choice of\nfitting function for our lattice data.\n\nFigure~\\ref{fig:gvsL} shows the running of the coupling from the low\nenergies to the high energies, over a factor $2^{26}$ change in scale,\nwhile table~\\ref{tab:gvsL} contains the numerical values of the\ncoupling at different renormalization scales. The fact that the\nabsolute error in the renormalized coupling tends to be constant a\nlarge energies (small volumes), is a consequence of the error\npropagation, that dominates for large energies the error budget. \n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{fig\/gvsL}\n  \\caption{$g^2_{\\rm TGF}(L)$ as a function of the renormalization\n    scale $\\log(L\/L_{\\rm min})$, and a comparison with the two loop\n    perturbative prediction. Errors are plotted, but compatible with\n    the size of the points.}\n  \\label{fig:gvsL}\n\\end{figure}\n\nAs a further consistency test, we have repeated the full running of\nthe coupling using as scale factor to define the step scaling function\n$s=2$ (i.e. we run from high to low energies), obtaining\nconsistent results. \n\n\\begin{table}\n  \\centering\n  \\begin{tabular}{l|llllll}\n    \\toprule\n    $L=L_{\\rm max}\/2^k$ & $k=0$ & $k=1$ & $k=2$ & $k=3$ & $k=4$ & $k=5$ \\\\\n    $g^2_{\\rm TGF}(L)$ & 7.5 & 4.824(17) & 3.581(15) & 2.858(12) &\n    2.383(10) & 2.0464(95) \\\\\n    \\midrule\n    $L=L_{\\rm max}\/2^k$ & $k=6$ & $k=7$ & $k=8$ & $k=9$ & $k=10$ & $k=11$ \\\\\n    $g^2_{\\rm TGF}(L)$ & 1.7949(94) & 1.5995(94) & 1.4432(93) &\n    1.3153(92) & 1.2085(90) & 1.1181(89) \\\\\n    \\midrule\n    $L=L_{\\rm max}\/2^k$ & $k=12$ & $k=13$ & $k=14$ & $k=15$ & $k=16$ & $k=17$ \\\\\n    $g^2_{\\rm TGF}(L)$ & 1.0405(87) & 0.9732(86) & 0.9143(84) &\n    0.8621(83) & 0.8158(83) & 0.7742(82) \\\\\n    \\midrule\n    $L=L_{\\rm max}\/2^k$ & $k=18$ & $k=19$ & $k=20$ & $k=21$ & $k=22$ & $k=23$ \\\\\n    $g^2_{\\rm TGF}(L)$ & 0.7368(82) & 0.7028(82) & 0.6720(82) &\n    0.6437(82) & 0.6178(82) & 0.5939(83) \\\\\n    \\midrule\n    $L=L_{\\rm max}\/2^k$ & $k=24$ & $k=25$ & $k=26$ &  &  &  \\\\\n    $g^2_{\\rm TGF}(L)$ & 0.5718(83) & 0.5514(84) & 0.5324(84) &&& \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Values of the renormalized twisted gradient flow coupling\n    as a funtion of the renormalization scale $\\mu = 1\/cL$ for\n    $c=0.3$. The final error at large scales\n    (small volumes) is dominated by the error propagation.}\n  \\label{tab:gvsL}\n\\end{table}\n\nThe $\\Lambda$ parameter can be extracted, in units of $L_{\\rm max}$\nvia\n\\begin{equation}\n  \\Lambda = \\mu(\\beta_0g^2(\\mu))^{-\\beta_1\/2\\beta_0^2} e^{-1\/2\\beta_0g^2(\\mu)}\n  e^{-\\int_0^{g^2(\\mu)}\\left\\{\\frac{1}{\\beta(x)}+\\frac{1}{\\beta_0x^3}-\\frac{\\beta_1}{\\beta_0^2x}\\right\\}}\\,,\n\\end{equation}\nusing that $\\mu = 1\/cL$. The previous formula is exact, but the last\nexponential is essentially unknown analytically. Nevertheless if one\nuses a value of $g^2_{\\rm TGF}(L)$ where the difference between the two loop\nand the non-perturbative results are negligible, the effect of the\nlast exponential is also negligible. Of course this is\nmore certain the smaller the coupling, but since the relative error of\nthe coupling grows as the coupling decreases, this would translate in\na larger error for the $\\Lambda$ parameter. Below we quote a couple of\nvalues as example. \n\\begin{eqnarray*}\n  \\Lambda L_{\\rm max} = 1.509(44)\\quad (@ g_{TGF}^2(L) = 1.7949(94))\\,,\\\\\n  \\Lambda L_{\\rm max} = 1.57(13)\\quad  (@ g_{TGF}^2(L) = 1.0405(87))\\,.\n\\end{eqnarray*}\n\nWe want to end this section with a small comment on the use of\ndifferent values of $c$. The main point has already been\nraised in~\\cite{Fritzsch:2013je}: the larger the value of $c$, the larger\nthe (relative) statistical error of the coupling, but the scaling\ntowards the continuum seems better. This general behavior is\nconsistent with the leading order in perturbation theory as we have\nseen. We will simply say that the relative error in the raw data\nincreases with $c$, and roughly one can say that for $c=0.4$ the\nrelative error is two times larger than for $c=0.3$, while for\n$c=0.5$ the error is three times larger. This statement seem to hold\ntrue independently of the volume (i.e. of the value of $g^2_{\\rm TGF}$). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{The criterion illustrated by the cuprate example}\n\\label{app:example}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=.7]{fig4a.eps}\n\\includegraphics[scale=.7]{fig4b.eps}\n\\caption{(a)~The 2D nodal band structure and its projections on $(1\\bar{1})$ and $(01)$\nsurfaces for a model $d$-wave superconductor. The first BZ and one extended zone are drawn. The solid black curves denote the Fermi surface and the shaded region is filled in the normal state.\nRed arrows are the unit vectors in \\Eq{n}. The blue dots with $\\pm$ signs represent nodes with\nvorticity $\\pm 2$ respectively. Top left: the slanted thin black line segment is the surface\nBZ of the $(1\\bar{1})$ edge. The black arrow with letter ``P'' indicates the direction of projection.\nThick red line segments mark the surface momenta with zero energy bound states. Top right:\nthe thin black horizontal line segment is the BZ of the $(01)$ edge.\nThe $\\pm$ vorticity nodes overlap after projection.\nThe  vorticity of the node enclosed by the parallelogram is equal to the winding number difference along the two side blue segments because the winding along the top and bottom segments cancel due to the periodicity in momentum space. This can be explicitly seen by following the turning of the red arrows (the actual winding number is twice of the winding shown by the arrows, due to the spin degeneracy). (b) The (11) edge bandstructure. The surface flat bands are marked red.  In constructing this figure we have used $\\epsilon(\\v k)=-\\cos k_x-\\cos k_y-\\mu$ and $\\Delta(\\v k)=\\Delta_0 (\\cos k_x-\\cos k_y)$ in \\Eq{dwave}. Here  $\\mu=0.45,\\Delta_0=0.1$.}\n\\label{fig:d-wave}\n\\end{center}\n\\end{figure}\n\nThe idea behind the criterion presented in the main text is \nbest illustrated by using the cuprate superconductor as an example.\nThe Bogoliubov-de Gennes (BdG) Hamiltonian of the cuprate superconductor read\n\\be\nH_{\\rm cuprates}(\\v k)=\\epsilon(\\v k)\\s_0\\otimes\\tau_3+\\Delta(\\v k)\\s_0\\otimes\\tau_1,\\label{dwave}\n\\ee\nwhere $\\s_0$ is the identity matrix acting in the spin space and $\\tau_{1,3}$ are $2\\times 2$ Pauli matrices in the Nambu space, $\\epsilon(\\v k)$ is the normal state dispersion satisfying $\\e(-\\v k)=\\e(\\v k)$ and $\\Delta(\\v k)$ is the d-wave gap function. (Since the cuprates are quasi two dimensional materials, we shall use two dimensional notations in the following discussions.) The Fermi surface and the gap nodes are shown in \\Fig{fig:d-wave}a, therefore $d=2,q=0$. In the same figure, the normalized vector  \\be \\hat{n}(\\v k)=(\\epsilon(\\v k),\\Delta(\\v k))\/\\sqrt{\\epsilon(\\v k)^2+\\Delta(\\v k)^2}\\label{n}\\ee is plotted as a function of $\\v k$ over two BZs (see red arrows). Inspecting these arrows one notices each node is a ``vortex'' in $\\hat{n}(\\v k)$. Around each vortex the arrows exhibit non-zero winding. The total winding number associated with each node is given by \n\\be\nw={2\\over 2\\pi}\\oint d\\v p\\cdot [n_1(\\v k)\\gr_{\\v k} n_2(\\v k)-n_2(\\v k)\\gr_{\\v k} n_1(\\v k)].\n\\label{w}\n\\ee (The extra factor of 2 is due to spin degeneracy). Clearly each node is characterized by an even integer winding number. The BdG Hamiltonian defined on all one ($=d-q-1$) dimensional loop enclosing the node are topologically nontrivial.\n\nNow consider the bandstructure projected along the $(1\\bar{1})$ direction. For each transverse momentum $k$ along $(11)$ we have a 1D chain running in $(1\\bar{1})$.\nSo long as $k$ does not coincide with the projection of the nodes the spectrum is fully gapped and characterized by the winding number defined in \\Eq{w}. Any two chains whose $k$ straddle the projection of a node their winding numbers must differ by $\\pm 2$ (see \\Fig{fig:d-wave}a captions), hence at least one of them is topologically non-trivial and possess $E=0$ end states\nwhen the boundary condition along $(1\\bar{1})$ changes from closed to open.\n~This implies  E=0 bound states exists for {\\em intervals} of $k$. Therefore $d_{E=0}$ is indeed $q+1=1$. An example of the (11) boundary bandstructure is shown in \\Fig{fig:d-wave}b. The $k$ intervals showing the flat bands are represented by the thick red line segments in the top left corner of \\Fig{fig:d-wave}a.  The only edges which do not possess ZBABS are the $\\{10\\}$ (Miller's notation is used) edges where the projection of positive and negative nodes overlap (see top right corner of \\Fig{fig:d-wave}a).\nFor the real material $d=3, q=1$ and the only modification is $d_{E=0}$ changes from 1 to 2.\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{\\label{intro}Introduction}\nHawks and Doves, also known as Chicken, or the Snowdrift game, is a two-person, symmetric\ngame with the following payoff bi-matrix:\n\n\\begin{table}[!ht]\n\\begin{center}\n{\\normalsize\n\\begin{tabular}{c|cc}\n & H & D\\\\\n\\hline\n{\\rule[-3mm]{0mm}{8mm}}\nH & ($\\frac{G-C}{2},\\frac{G-C}{2}$) & ($G,0$)\\\\\n{\\rule[-3mm]{0mm}{4mm}}\nD & ($0,G$) & ($\\frac{G}{2},\\frac{G}{2}$)\\\\\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\\vspace{-0.4cm}\n\n\n\\noindent In this matrix, H stands for ``hawk'', and D stands for ``dove''.\nMetaphorically, a hawkish behavior means a strategy of fighting, while a dove, when facing a confrontation, will always yield.\nAs in the \n\\textit{Prisoner's Dilemma} \\cite{axe84}, this game, for all its simplicity, appears to\ncapture some important features of  social interactions. In this sense, it applies\nin many situations in which ``parading'', ``retreating'', and ``escalading'' are common.\nOne striking example of a situation\nthat has been thought to lead to a Hawk-Dove dilemma is the Cuban missile crisis in 1962\n \\cite{poundstone92}.\nIn the payoff matrix above, $G > 0$ is the gain that a hawk obtains when it meets a dove; the dove retreats and looses nothing.\nIf a dove meets another dove, one of them, or both, will retreat and they will gain half of\nthe price each ($G\/2$) in the average. Finally, when a hawk meets another hawk, they \nboth fight and each obtains an average payoff of $(G-C)\/2$, where $C$ is the cost of any\ninjury that might occur in the fight. It is assumed that $C > G$, i.e. the cost of injury\nalways exceeds the prize of the fight.\nThe game has the same structure as the Prisoner's Dilemma in that if both players\ncooperate (i.e. they play dove), they both gain something, although there is a strong motivation\nto act aggressively (i.e. to play the hawk strategy). However, in this game one makes the\nassumption that one player is willing to cooperate, even if the other does not, and that mutual defection, i.e. result (H,H), is detrimental to both players. \n\nIn contrast to the Prisoner's Dilemma which has a unique Nash equilibrium that corresponds to\nboth players defecting, the Hawk-Dove game has two Nash equilibria in pure strategies\n(H,D) and (D,H), and a third equilibrium in mixed strategies where strategy H is played\nwith probability $G\/C$, and strategy D with probability $1 - G\/C$. Note that we only consider\none-shot games in this work; repeated games are not taken into account.\n\nConsidering now not just two players but rather a large, mixing population of identical players,\n\\textit{evolutionary\ngame theory} \\cite{hofb-sigm-book-98} prescribes that the only evolutionarily\nstable strategy (ESS) of the population is the mixed strategy, giving rise, at equilibrium,\nto a frequency of hawks in the population equal to $G\/C$.\nIn the case of the Prisoner's Dilemma, one finds a unique ESS with all the individuals defecting.\nHowever,\nin 1992, Nowak and May \\cite{nowakmay92} showed\nthat cooperation in the population is sustainable in the Prisoner's Dilemma under certain conditions,\nprovided that the network of the interactions between players has a lattice spatial structure. Killingback and Doebeli \\cite{KD-96} extended the\nspatial approach to the Hawk-Dove game and found that a planar lattice structure\nwith only nearest-neighbor interactions may favor cooperation, i.e. the fraction of doves in\nthe population is often higher than what is predicted by evolutionary game theory. In addition,\ncomplex dynamics resembling phase transitions were observed, which is not the case in the\nmixing population.\nIn a more recent work however, Hauert and Doebeli  \\cite{hauer-doeb-2004} were led to a different conclusion, namely that\nspatial structure does not seem to favor cooperation in the Hawk-Dove game. \nAdditional\nresults on the Hawk-Dove game on a two-dimensional lattice have been recently obtained by Sysi-Aho et al.\n\\cite{myopic-hd-05} using a simple local decision rule for the players that does not reduce to the customary\n\\textit{replicator} or \\textit{imitation} dynamics \\cite{hofb-sigm-book-98}. They concluded that,\nwith their player's decision rule, cooperation persists, giving results different from those obtained\nwith the replicator dynamics. \nThese apparently\ncontradictory results aroused our curiosity, and motivated us to study the situation in a more general\nsetting, in which the mixing population and the lattice are special cases. \n\nFollowing pioneering work by sociologists in the sixties such as that of Milgram\n\\cite{milgram67}, in the last few years it has become apparent that the topological structures of social\ninteractions networks have particular, and partly unexpected, properties that are a consequence\nof their \\textit{small-world} characteristics. Roughly speaking, small-world networks are\ngraphs that have a short \\textit{average path length}, i.e. any node is relatively close to any other\nnode, like random graphs and unlike regular lattices.\nHowever, in contrast with random graphs, they also have a certain amount of local structure,\nas measured, for instance, by a quantity called the \\textit{clustering coefficient} (an excellent\nreview of the subject is \\cite{newman-03}).\nIn the same vein, many real conflicting situations in economy and sociology are not well \ndescribed neither by a fixed\ngeographical position of the players in a regular lattice, nor by a mixing population or a\nrandom graph. Starting from the two limiting cases of a random-graph and the two-dimensional lattice, our objective\nhere is to study the Hawk-Dove game on small-world networks in order to cover the ``middle ground''\nbetween them. Although the Watts--Strogatz networks \\cite{watts-strogatz-98} used here are not faithful representations\nof the structure of real social networks, they are a useful first step toward a better understanding\nof evolutionary games on networks. While we study here the Hawk-Dove game, this class of networks has been previously used for the Prisoner's \nDilemma in \\cite{social-pd-kup-01,pd-dyn-sw-02,watts99}. The work of \\cite{social-pd-kup-01} is especially relevant\nfor our present study, while the two others deal either with special features such as\n``influential individuals''\\cite{pd-dyn-sw-02}, or refer to iterated versions of the game \\cite{watts99}.\n\nRecently, Santos and Pacheco \\cite{santos-pach-05} have investigated both the Prisoner's\nDilemma and Hawk-Dove games on\nfixed scale-free networks. The main observation from their simulations is that, at least on preferential attachment\nnetworks, the amount of cooperative behavior is much higher than in either mixing or \nlattice-structured populations. In the abstract, and in some particular social situation\nin which some individuals have an unusually high number of contacts than the rest, this is an interesting result.\nHowever, scale-free graphs, which characterize the web and Internet among others, are not a realistic \nmodel of most observed social networks for various reasons (see \\cite{jin-gir-newman-01,ebel-dav-born-03}),\nwhich is why we do not comment further on the issue.\n\n\\section{\\label{sect:model}The Model}\nIn this section we present our network models and their dynamical properties.\n\n\\subsection{\\label{pop-topo}Population Topologies}\nWe consider a population $P$ of $N$ players where\neach individual $i$ in $P$ is represented as a vertex $v_i$  of a graph $G(V,E)$,\nwith $v_i \\in V, \\; \\forall i \\in P$. An interaction between two players $i$ and\n$j$ is represented by the undirected edge $e_{ij} \\in E, \\; e_{ij} \\equiv e_{ij}$.\nThe number of neighbors of player $i$ is the degree $k_i$ of vertex $v_i$. The average\ndegree of the network will be called $\\bar k$.\n\nWe shall use three main graph population structures: regular lattices, random graphs, and small-worlds.\nIn fact, our goal is to explore significant population network structures that somehow fall\nbetween the regular lattice and random graph limits, including the bounding cases.\n\nOur regular lattices are two-dimensional with $k_i=8, \\; \\forall v_i \\in V$  and periodic boundary conditions. \nThis neighborhood is usually called the Moore neighborhood and comprises nine individuals, including the central node.\nWe would like to stress that we believe regular lattice structures are not realistic representations\nof most actual population structures, especially human, except when mobility and dispersal ability\nof the individuals are limited as, for example, in plant ecology and territorial animals. \nThe main reasons why lattices have been so heavily used is that they are more amenable to mathematical analysis and are easier to simulate.\nWe include them here for two reasons: as an interesting bounding case, and to allow comparison with previous work.\n \nThe small-world networks used here are similar to the graphs proposed by Watts and Strogatz\n\\cite{watts-strogatz-98}. However, there are two main differences (see \\cite{boccara-04}). First, we start from a \ntwo-dimensional regular lattice substrate, instead of a one-dimensional lattice. This does not\nmodify the main features of the resulting graphs, as observed in \\cite{watts99}, and as measured by us.\nThe reason for starting from a two-dimensional lattice is to keep with the customary ordered population\ntopology that is used in structured evolutionary games.  Although\nthey have been used as a starting point for the Prisoner's Dilemma by Abramson and Kuperman \\cite{social-pd-kup-01}, one-dimensional lattices do not\nmake much sense in a social or biological setting, although after some rewiring the effect of\nthe substrate becomes almost negligible. \n\nThe second difference is in the rewiring process. The\nalgorithm used here comes from \\cite{boccara-04} and works as follows: starting from a regular two-dimensional lattice with\nperiodic boundary conditions, visit each edge and, with probability $p$, replace it  by\nan edge between two randomly selected vertices, with the constraint that two vertices are\nnot allowed to be connected by more than one edge. As in the original Watts--Strogatz model,\nthe average vertex degree $\\bar k$ does not change, and the process may produce disconnected graphs, \nwhich have been avoided in our simulations.\nThe advantage of this construction is that, for $p \\rightarrow 1$,\nthe graph approaches a classical Erd\\\"os--R\\'enyi random graph, while this is not the case for\nthe original Watts--Strogatz construction, since in the latter, the degree of any vertex is \nalways larger than or\nequal to $k\/2$, $k$ being the degree of a vertex in the original lattice. \n\nWe would like to point out that it is known that Watts--Strogatz small worlds \nare not adequate representations of social networks. Although they share some common statistical\nproperties with the latter, i.e. high clustering and short average path length, they lack other features that characterize\nreal social networks such as clusters, and dynamical self-organization \\cite{ebel-dav-born-03}. In \nspite of these shortcomings, they are\na convenient first approximation for studying the behavior of agents in situations where the interaction\nnetwork is neither regular, nor random. Note also that once fixed, the interaction network\ndoes not change during the system evolution in our study, only the strategies may evolve. Evolutionary games\non dynamic networks have been studied, for instance in \\cite{zimm-et-al-04,luthi-giac-tom-05,games-ecal-05}.\n\n\\subsection{\\label{dyn}Population Dynamics}\n\\subsubsection{Local Dynamics}\nThe local dynamics of a player $i$ only depends on its own strategy $s_i \\in \\{H,D\\}$, and on\nthe strategies of the $k_i$ players in its neighborhood $N(i)$. Let us call $M$ the payoff matrix\nof the game (see section \\ref{intro}). The quantity\n$$G_i(t) = \\frac{1}{k_i} \\sum _{j \\in N(i)} s_i(t)\\; M\\; s_{j}^T(t)$$\n\\noindent is the average payoff collected by player $i$ at time step $t$.\nNote that in our study, $i \\notin N(i)$ meaning that self-interaction is not considered when\ncalculating the average payoff of an individual.\nSelf-interaction has  traditionally been taken into account in some previous work on the\nPrisoner's Dilemma\ngame on grids \\cite{nowakmay92,nowaketal94} on the grounds that, in biological applications,\nseveral entities may occupy a single patch in the network. Nowak et al. find that\nself-interaction does not qualitatively change the results in the Prisoner's Dilemma game. In the Hawk-Dove game\nself-interaction is usually not considered; moreover, in this work we wish to compare results\nwith those of \\cite{KD-96,hauer-doeb-2004}, where self-interaction is not included.\n\n\nWe use three types of rules to update a player's strategy. The rules are among those employed by\nHauert et al. \\cite{hauer-doeb-2004} to allow for comparison of the results in regular lattices\nand in small-world networks. Decision rules based on the player's satisfaction degree, \nsuch as those used in \\cite{zimm-et-al-04,games-ecal-05,luthi-giac-tom-05,myopic-hd-05} are not examined here.\nThe rules are the following:\n\\begin{enumerate}\n\\item replicator dynamics;\n\\item proportional updating;\n\\item best-takes-over.\n\\end{enumerate}\n\n\nThe \\textit{replicator dynamics} rule aims at maximal consistency with the original\nevolutionary game theory equations. Player $i$ is updated by drawing\nanother player $j$ at random from the neighborhood $N(i)$\nand replacing $s_i$ by $s_j$ with probability $p_j = \\phi(G_j - G_i)$ \\cite{hofb-sigm-book-98}.\n\n\nThe \\textit{proportional updating} rule is also a stochastic rule. All the players in the neighborhood $N(i)$,\nplus the player $i$ itself compete for the strategy $i$ will take at the next time step, each with a probability $p_j$ given by\n$$ p_j = \\frac{G_j}{\\sum_l G_l}, \\;\\; l,j \\in \\{N(i) \\cup i \\}.$$\n\\noindent Negative payoffs cannot be used with this rule, since the probabilities of replication\nmust be $p_j \\ge 0$. In order to avoid negative, or zero, values, the payoffs have been\nshifted by an amount equal to the cost $C$ which, of course, leaves the game's Nash equilibria invariant.\n\n\n\nIn \\textit{best-takes-over}, the strategy $s_i(t)$ of individual $i$ at time step $t$ will\nbe\n$$s_i(t) = s_j(t-1),$$\nwhere\n$$j \\in \\{N(i) \\cup i\\} \\;s.t.\\; G_j = \\max \\{G_k(t-1)\\}, \\; \\forall k \\in \\{N(i) \\cup i\\}.$$\n\\noindent That is, individual $i$ will adopt the strategy of the player with the highest\npayoff among its neighbors.\nIf there is a tie, the winner individual is chosen uniformly at random between the best, and its strategy\nreplaces the current strategy of player $i$, otherwise the rule is deterministic. It should be\nnoted that this rule does not fit to the usual continuous evolutionary game theory which\nleads to replicator dynamics, since the update\ndecision is a step function.\n\n\\subsubsection{Global Dynamics}\nCalling $C(t) = (s_1(t), s_2(t), \\ldots , s_N(t))$ a \\textit{configuration} of the population\nstrategies at time\nstep $t$, the global \\textit{synchronous} system dynamics is implicitly given by:\n$$C(t) = F(C(t-1)), \\;\\; t =1,2, \\ldots $$\n\\noindent where $F$ is the evolution operator.\n\nSynchronous update, with its idealization of a global clock, is customary \nin spatial evolutionary games, and most results have been obtained using this model \n\\cite{nowakmay92,KD-96}.\nHowever, perfect synchronicity is only an abstraction. Indeed, in some biological\nand, particularly, sociological environments, agents normally act at different and possibly uncorrelated\ntimes, which seems to preclude a faithful globally synchronous simulation in most\ncases of interest \\cite{hubglance93}. In spite of this, it has been shown that the\nupdate mode does not fundamentally alter the results, as far\nas evolutionary games are concerned \\cite{nowaketal94,hauer-doeb-2004}. In this paper we\npresent results for both synchronous and asynchronous dynamics.\n\nAsynchronous dynamics must nevertheless be further qualified, since there \nare many ways for serially updating the strategies\nof the agents. Here we use the discrete update dynamics that makes the least assumption\nabout the update sequence: the next cell to be updated is chosen\nat random with uniform probability and with replacement.\nThis corresponds to a binomial distribution of the updating probability and is a good approximation of a continuous-time Poisson\nprocess. This asynchronous update is analogous to the one used by Hauert et al. \n\\cite{hauer-doeb-2004}, which will allow us to make meaningful comparisons.\n\n\\section{\\label{sim}Simulation Results}\nIn order to analyze the influence of the structure of the network\non the proportion of cooperation (i.e. dove behavior),\n2500 players were organized into 5 different networks:\na 50 by 50 toroidal lattice where every cell is connected to its 8 nearest neighbors,\nthree different small-world networks, and the random graph.\nThe three categories of small worlds are obtained by rewiring each edge\nwith a certain probability $p$ using the technique described under \\ref{pop-topo}.\nThe values used are $p \\in \\{0.01,0.05,0.1\\}$.\nThe random graph is generated by first creating the lattice\nand then rewiring each link, in the same manner used to construct\nsmall worlds, but with probability $p=1$. Although our population size is smaller\nthan that used in \\cite{hauer-doeb-2004}, which is $10000$, results turn out to be\n qualitatively similar and comparable.\nFor each of the 5 networks mentioned above and for all update policies,\n50 runs of 5000 time steps each were executed.\nIn the following figures, the curves indicating the proportion of doves in the population\nwere obtained by averaging over the last 10 time steps of each run, well after all transients\nhave decayed.\nAt the beginning of each run, we generate a new network of the type being studied\nand randomly initialize it with 50\\% doves and 50\\% hawks. For completeness, we mention\nthat experiments with 10\\% and 90\\% initial cooperators respectively, give results that\nare qualitatively indistinguishable from the 50\\% case in the long run. Therefore, we do not include the \ncorresponding graphs for reasons of space.\n\nIn the following figures, the dashed diagonal line going from a fraction of cooperators\nof $1$ for $r=0$, to a fraction of $0$ for $r=1$, represents the equation $1-G\/C = 1-r$,\nwhich is the equilibrium fraction of cooperators as a function of $r$ given by the standard replicator-dynamics equations \\cite{hofb-sigm-book-98},\nand it is reported here for the sake of comparison. \nIt should be noted, however, that the simulations are not expected to fit this line. The reason is\nthat the analytic solution is obtained under two main hypotheses: the population size is very\nlarge, and individuals are matched pairwise randomly. These conditions are not satisfied by\nthe finite-size, discrete systems used for the simulations, and thus one should not expect strict\nadherence to the mean-field equations. On the other hand, the type of \\textit{mesoscopic} system\nsimulated here is probably closer to reality, where finiteness and discreteness are the rule.\nAnother reason why we do not expect the results of the simulations to closely fit the\ntheoretical solution is that two of the local update rules\n(best-takes-over and proportional updating) do not reduce to the standard replicator dynamics.\n\nThis section is subdivided into three separate parts, one per decision rule previously\nmentioned under \\ref{dyn}.\n\n\\subsection{Replicator Dynamics}\nTo determine the probability $p_j$ for replacing an individual $i$, having a gain $G_i$, \nby one of its randomly chosen neighbors $j$, whose gain is $G_j$,\nwe use the the previously introduced function $\\phi(G_j - G_i)$ as follows:\n\\begin{eqnarray}\np_j = \\phi(G_j - G_x) =\n\\begin{cases} \\textrm{\\large{$\\frac{G_j - G_i}{{d}_{max}}$}} & \\textrm{if $G_j - G_i > 0$}\\\\\\\\\n0 & \\textrm{otherwise}\n\\end{cases}\n\\label{repl_dyn_eq}\n\\end{eqnarray}\nwhere ${d}_{max} = \\frac{G+C}{2}$ is the largest difference in gain\nthere can be between two players.\n\nWith this definition of $\\phi$, individual $i$ imitates neighbor $j$'s strategy\nwith a certain probability proportional to the difference of their average payoffs\nand only if $j$ has a higher gain than $i$.\nNotice that if $i$ and $j$ have the same average payoffs, $i$'s strategy is left untouched, while\nif $G_j -G_i = {d}_{max}$, $i$ necessarily adopts $j$'s strategy.\n\nNow taking a look at figures \\ref{repl_dyn_async} and \\ref{repl_dyn_sync},\nwe clearly observe that for both synchronous and asynchronous dynamics,\ncooperation is globally inhibited by spatial structure, confirming the results of \\cite{hauer-doeb-2004}.\nEven the case of the random graph generates higher rates of hawks.\nFurther details as to why this may occur can be found in section \\ref{disc}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{repl_dyn_async_uc_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{repl_dyn_async_uc_dPercentage}}\\\\\n(a) & &(b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{repl_dyn_async}(Color online) asynchronous replicator dynamics updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{repl_dyn_sync_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{repl_dyn_async_uc_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{repl_dyn_sync}(Color online) synchronous replicator dynamics updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\nWe note in passing that the experimental curve corresponding to the random graph limit appears\nto be close to the curve corresponding to the pair approximation calculation in Hauert and Doebeli's\nwork \\cite{hauer-doeb-2004}. This is not surprising, given that pair approximation works\nbetter in random graphs than in regular lattices, unless higher-order effects are taken into\naccount \\cite{van-baalen-00}. Since the curves for the random graphs in \nfigures \\ref{repl_dyn_async} and \\ref{repl_dyn_sync} are averages over many graph realizations,\neach pair has some probability to contribute in the simulation,\nwhich explains the resemblance between our experimental curves and the calculations of \\cite{hauer-doeb-2004}.\n\n\\subsection{Proportional Updating}\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{prop_async_uc_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{prop_async_uc_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{prop_async}(Color online) asynchronous proportional udpating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{prop_sync_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{prop_sync_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{prop_sync}(Color online) synchronous proportional updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\nFigures \\ref{prop_async} and \\ref{prop_sync} show that,\nwhen using the proportional updating rule,\nspatial structure neither favors nor inhibits dove-like behavior\ncontrary to what \\cite{KD-96} and \\cite{hauer-doeb-2004} seem to suggest.\nIndeed, for low values of $r$, the more the network is structured,\nthe higher the proportion of doves.\nHowever as $r$ increases, the tendency is reversed,\nthus giving a lower percentage of doves in the lattice and small-world networks\nthan present in the random graph topology.\nThis phenomenon is even more marked when using the asynchronous update.\n\nThus when using the proportional updating rule, if spatial structure should favor one strategy over the other for a given value of $r$,\nit would be the one that is already present in greater numbers\nwhen the topology is a random graph.\n \n\nAnother interesting aspect observed is the higher percentage of doves\nwhen updating asynchronously compared to the synchronous equivalent.\nThis will be discussed in more detail in section \\ref{disc}.\n\n\n\\subsection{Best-takes-over}\nAs pointed out by Hauert and Doebeli \\cite{hauer-doeb-2004}, the best-takes-over rule lacks stochasticity,\nwhich in figures \\ref{bto_async} and \\ref{bto_sync}, translates into discontinuous jumps.\n\nNote that when updating synchronously, best-takes-over is the only rule,\nout of the three studied here, where spatial structure actually favors cooperation, as remarked\nin \\cite{KD-96}, where this was the local update rule used. In fact, the same qualitative results\nwere found in \\cite{hauer-doeb-2004}; however, they appear in the \"supplementary material\" section,\nnot in the main text.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_async_uc_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_async_uc_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{bto_async}(Color online) asynchronous best-takes-over updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{bto_sync}(Color online) synchronous best-takes-over updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\n\\subsection{Time Evolution}\n\\label{tev}\n\nWhile the figures in the previous subsections summarize the results at system stability,\nhere we describe the dynamical behavior of populations through the first $100$ time\nsteps, where fluctuations might influence the system dynamics.\n\nWe have studied both asynchronous and synchronous dynamics for\nthe three update rules in three topologies each: lattice, random graph, and a small\nworld with $p=0.05$. This was done for $r=0.7$, where defection predominates. The results are \nrelatively uninteresting for the replicator and proportional updates in all topologies. One\nobserves in the average a monotone decrease of cooperation starting with $50\\%$ at time $0$ until\nthe curve flattens out at the values reported in figures \\ref{repl_dyn_async} to \\ref{prop_sync}.\nThe only difference is that\nthe variance is more pronounced in the proportional case, as one would expect looking\nat standard deviations in figures \\ref{repl_dyn_async} to \\ref{prop_sync}.\n\nThe situation is different, and more interesting, in the case of best-takes-over\nupdate whose determinism causes stronger variations. The most striking feature is a \nsudden drop of cooperation at the beginning\nof the simulation, followed by an increase and by fluctuations whose amplitude diminishes\nover time. The effect is much more pronounced with synchronous dynamics, shown in Fig. \\ref{time-ev}, than\nwith the asynchronous one. The behavior appears in all three topologies but the drop is stronger\nin lattices and small worlds with respect to the random graph at earlier times. As time\ngoes by, fluctuations remain larger in the random graph case. Nevertheless, no experiment\nled to total extinction of cooperators at $r=0.7$.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_07_0_dPercentage_time_ev}}\\protect &\n\\hspace*{0.3cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_07_005_dPercentage_time_ev}}\\protect &\n\\hspace*{0.3cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_07_1_dPercentage_time_ev}}\\\\\n(a) & & (b) & & (c)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{time-ev} time evolution (first $100$ steps) of the proportion of doves\nfor best-takes-over update; synchronous evolution with $r=0.7$. (a) lattice structure;\n(b) small world with $p=0.05$; (c) random graph. Ten randomly chosen evolutions are shown in\neach case.}\n\\end{figure}\n\n\n\n\\section{\\label{disc}Analysis and Discussion}\nIf we take a closer look when comparing Fig. \\ref{prop_async} and Fig. \\ref{prop_sync},\nwe notice that, for proportional dynamics, asynchronous updating allows for better cooperation than its\nsynchronous counterpart. The reason for this difference can be intuitively understood\nin the following manner:\nwhen updating asynchronously, let us suppose a player $y$ has just imitated the strategy\nof one of its neighbors $x$. Another way of viewing this change, is to say that player $x$ has ``infected''\nindividual $y$ with its strategy. If $x$ is a dove player, making $y$ a dove as well,\nnot only does the percentage of doves increase in the population, but\nthe next time either $x$ or $y$ is evaluated for an update, it will be able to take advantage of the other one's\npresence to help increase its payoff. Hence, the two players mutually reinforce each other.\nMeanwhile, if $y$ is infected by $x$ and turns into a hawk, on the one hand $x$ has successfully propagated\nhis strategy thus increasing the overall amount of hawks in the population,\nbut on the other hand this propagation will cause him to have a lower payoff than he previously had.\nNot only is $x$'s payoff negatively affected, but $x$'s prescence also harms $y$'s payoff.\n\nThe same reasoning cannot be held when updating synchronously.\nIndeed, a player $x$ may change strategies\nat the same time it infects its neighbor $y$.\nSo if $x$'s initial strategy was $D$, it might switch\nto $H$ as it infects its neighbor $y$, in which case\n$x$ will no longer have a positive effect on $y$'s payoff\ncontrary to what happens in asychronous updating.\n\nWhen applying the replicators dynamics rule, the small drop of the percentage of doves  seen\non the very left of figures \\ref{repl_dyn_async} and \\ref{repl_dyn_sync} is due to the fact that for\n$r=0$ the game is somewhat degenerated. Indeed, any cluster of more than one hawk will either reduce to\na single hawk or totally disappear, since a dove, no matter what its neighborhood comprises,\nwill always have a gain of zero while a hawk that interacts with at least one other hawk will have a negative payoff.\nThe remaining lone hawks will however survive but will not be able to propagate (having a gain \nexactly equal to that of their neighboring doves). The system is thus found locked in a configuration\nof a very high proportion of doves with a significant number of isolated hawks.\n\nIf $r > 0$, lone hawks always have a higher payoff than the doves in their surroundings and will thus infect one\nof its neighbors with its strategy. However for $0 < r \\leq 0.1 $, once the pair of hawks is established, their payoff\nis lower than the one of any of the doves connected to either one them. Even a dove that interacts with both\nhawks has an average payoff still greater than what a hawk composing the\npair receives.\nConsequently, when $0 < r \\leq 0.1$, clusters of hawks first start by either disappearing\nor reducing to single hawks like previously explained for the $r=0$ case, but then these lone hawks\nwill become pairs of hawks.\nIf the updates are done synchronously, a pair of hawks will either vanish\nor reduce back to a single hawk. One can clearly see that in the long run, hawks will become extinct.\nNow if the updates are done asynchronously, a pair cannot totally disappear since only one\nplayer is updated at a time. However, this mechanism of  a pair reducing to a single hawk and\nturning back to a pair again\nwill cause the small groups of two hawks to move across the network and ``collide'' with each other\nforming larger groups that reduce back to a single-pair hawk formation. Therefore, after a large\nnumber of time steps, only a very few hawks will survive.  \n\nIf we take another look at figures \\ref{repl_dyn_async} and \\ref{repl_dyn_sync},\nwe notice that when the population of players is constrained to a lattice-like structure,\nthe proportion of doves is reduced to zero for values of\nthe gain-to-cost ratio greater or equal to approximately $0.8$,\nwhile this not the case when the topology is a random graph.\nLet us try to give a qualitative explanation of the two different behaviors:\nthe first thing to be pointed out is that in the case of the replicators dynamics,\nif a dove is surrounded by 8 hawk-neighbors,\nit is condemned to die for values of $r$ greater than $\\frac{7}{9}$ whatever the topology may be.\nHowever, this does not explain why for these same values, doves no longer exist on\nsquare lattices or small worlds but are able to survive on random graphs.\nIf the population were mixing, $r=0.8$ would induce a proportion of doves equal to $20\\%$.\nTherefore, let us suppose that at a certain time step,\nthere is approximately $20\\%$ of doves in our population.\nFurthermore, as pointed out by Hauert and Doebeli \\cite{hauer-doeb-2004}, in the Hawk-Dove\ngame on lattices, the doves are usually spread out and form many small-isolated patches.\nThus, we will also suppose $20\\%$ of doves in the  population\nimplies that in a set of players comprising an individual and its immediate eight neighbors,\nthere are about two doves.\nHence, a D-player has on average one dove and seven hawks in its neighborhood.\nIn the lattice network, this pair of doves can be linked in two different manners\n(see Fig. \\ref{2D_lattice_configs}), having either two or four common neighbors,\nthus an average of three.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=3.5cm, height=3.5cm]{lattice_2D_a}}\\protect\n&  \\hspace{1cm} & \\mbox{\\includegraphics[width=3.5cm, height=3.5cm]{lattice_2D_b}}\\\\\n(a) & \\hspace{1cm} & (b)\\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\label{2D_lattice_configs}(Color online) lattice: two possible configurations.}\n\\end{figure}\n\nMore generally, if we denote $\\Gamma$ the clustering coefficient of the graph and $\\overline{k}$\nthe average degree, a pair of doves will have on average $\\Gamma(\\overline{k} - 1)$\ncommon neighbors.\nLet us denote $x$ one of the two doves composing the pair,\n$H_x$ a hawk linked to $x$ but not to the other dove of the pair and $H_{x,y}$ one that is connected to both doves.\nIf $\\frac{2}{3} < r < \\frac{7}{8}$ and assuming that the hawks surrounding the pair of doves are not interacting with any other doves\n(this gives the pair of doves a maximum chance of survival), we have\n$$G_{H_x} < G_x < G_{H_{x,y}},$$\nwhere $G_\\alpha$ is the average payoff of player $\\alpha$\n\nConsequently, according to Eq. (\\ref{repl_dyn_eq}),\n$x$ can infect $H_x$, and $H_{x,y}$ can infect $x$.\n\nLet us now calculate for what values of $r$ the probability that $x$ invades the site\nof at least one $H_x$ is less than an $H_{x,y}$ infecting $x$.\nTo do so, let us distinguish the case of the asynchronous udating policy from the synchronous one.\n\n\\subsubsection*{Asynchronous Dynamics}\nThe probability that an $H_x$ neighbor is chosen to be updated and adopts strategy $D$ is given by\n\\begin{equation}\n\\underbrace{\\frac{(1 - \\Gamma)(\\overline{k} - 1)}{N}}_{(\\ast)} \\underbrace{\\frac{1}{\\;\\overline{k}\\;}}_{(\\ast\\ast)} \\;\\phi(G_x - G_{H_x}),\n\\label{H2D_async}\n\\end{equation}\nwhere $N$ is the size of the population, $(\\ast)$ the probability an $H_x$ hawk is chosen to be updated (among the $N$ players),\n$(\\ast\\ast)$ the probability the chosen $H_x$ hawk compares its payoff with player $x$,\nand finally $\\phi$ is the function defined in Eq. (\\ref{repl_dyn_eq}).\n\nThe probability that $x$ is chosen to be updated and is infected by one of the $H_{x,y}$ hawks is given by\n\\begin{equation}\n\\underbrace{\\frac{1}{N}}_{(\\ast)} \\underbrace{\\frac{\\Gamma(\\overline{k}-1)}{\\overline{k}}}_{(\\ast\\ast)} \\;\\phi(G_{H_{x,y}} - G_x),\n\\label{D2H_async} \n\\end{equation}\nwhere $(\\ast)$ is the probability $x$ is chosen to be updated,\n$(\\ast\\ast)$ the probability it measures itself against an $H_{x,y}$ neighbor,\nand $\\phi$ the function defined by Eq. (\\ref{repl_dyn_eq}).\n\nFor a square lattice with a Moore neighborhood ($\\Gamma = \\frac{3}{7}$ and $\\overline{k} = 8$),\nexpressions \\ref{H2D_async} and \\ref{D2H_async} give us $r > \\frac{46}{59} \\approx 0.78$,\nwhereas for a random graph, $\\Gamma  = \\frac{\\overline{k}}{N-1} = \\frac{8}{2499} \\simeq 0.003 \\approx 0$ implies that\na pair of doves does not have any common hawk neighbors enabling them to survive\nif $r < \\frac{7}{8}$.\nAs for the small-world cases, the clustering coefficient is very close to that of the lattice, generating a behavior\npratically identical to the latter.\nThis gives a qualitative explanation for the difference observed in Fig. \\ref{repl_dyn_async}.\n\n\\subsubsection*{Synchronous Dynamics}\nThe probability that at least one $H_x$ adopts strategy $D$ is given by\n\\begin{equation}\n1 - \\overbrace{[1 - \\underbrace{\\frac{1}{\\;\\overline{k}\\;}\\;\\phi(G_x - G_{H_x})}_{(\\ast)}]^{(1-\\Gamma)(\\overline{k}-1)}}^{(\\ast\\ast)},\n\\label{H2D_sync}\n\\end{equation}\nwhere $(\\ast)$ is the probability a specific $H_{x}$ turns into a dove and $(\\ast\\ast)$ the probability none of the $H_x$\nadopt strategy $D$.\n\nThe probability that $x$ adopts the hawk strategy is given by\n\\begin{equation}\n\\underbrace{\\frac{\\Gamma(\\overline{k} - 1)}{\\overline{k}}}_{(\\ast)} \\;\\phi(G_{H_{x,y}} - G_x),\n\\label{D2H_sync}\n\\end{equation}\nwhere $(\\ast)$ is the probability player $x$ compares its payoff with one of its $H_{x,y}$ neighbors.\n\nFor a square lattice with a Moore neighborhood ($\\Gamma = \\frac{3}{7}$ and $\\overline{k} = 8$),\nexpressions \\ref{H2D_sync} and \\ref{D2H_sync} yield\n$$\n1 - \\left[1 - \\frac{1}{8}\\left(\\frac{-8G+7C}{G+C}\\right)\\right]^4 < \\frac{3}{8}\\left(\\frac{9G - 6C}{G+C}\\right),\n$$\nand given that $\\frac{G}{C} = r$, we obtain\n$$\n1 - \\left[1 - \\frac{1}{8}\\left(\\frac{-8r+7}{r+1}\\right)\\right]^4 < \\frac{3}{8}\\left(\\frac{9r - 6}{r+1}\\right),\n$$\nwhich is true for about $r > 0.775$. This also holds for the small-world cases, since, once again,\nthey have a $\\Gamma$ close to the one of the lattice. \n\nFor a random graph of $N=2500$ nodes and $\\overline{k}=8$, we have $\\Gamma \\approx 0$.\nTherefore, a pair of doves has a negligible probability of having a hawk neighbor in common and thus cannot be infected by the H strategy if $r < \\frac{7}{8}$.\nThis enables a small percentage of doves to survive on the random graph topology contrary to the lattice and small-world\nnetworks (see Fig. \\ref{repl_dyn_sync}). \n\nIn a few words, whether the update policy is asynchronous or synchronous, as soon as $r > \\frac{7}{9}$,\nisolated doves, as well as pairs of doves surrounded by hawks, will end up disappearing in the\nlattice and small-world cases due to the high clustering coefficient.\nHowever, in the random graph scenario, although isolated doves are also bound to die if $r > \\frac{7}{9}$,\npairs of doves have a more than even chance of surviving (at least as long as $r < \\frac{7}{8}$).\n\n\\section{\\label{concl}Conclusions}\nIn this work we clarify previous partially contradictory results on cooperation in populations playing the\nHawk-Dove game on regular grids. Furthermore, we notably extend the study to Watts--Strogatz small-world graphs, as\nthese population structures lie between the two extreme cases of regular lattices and random graphs,\nand are a first simple step towards real social interaction networks. This allows us to\nunravel the role of network clustering on cooperation in the Hawk-Dove game.\nWe find that, in general, spatial structure on the network of interactions in the  game\neither favors or inhibits cooperation with respect to the perfectly mixed case.\nThe influence it has depends not only on the rule that determines a player's future strategy,\nbut also on the value of the gain-to-cost ratio $G\/C$ and to a lesser degree,\non the synchronous and asynchronous timing of events.\n\nIn the case of the best-takes-over rule, dove-like behavior is advantaged if synchronous\nupdate is used but the rule is noisy due to its discrete nature.\nIn the case of the proportional update\nrule, giving the network a regular structure tends to increase\nthe percentage of the strategy that would already be in majority on a random graph\nconfiguration of the population. The more important the structure, in terms of clustering\ncoefficient, the higher the percentage of the dominant strategy. In fact, cooperation predominates\nfor low to medium $r$ values, while for higher $r$ values cooperation falls below the \nlarge population, mixing case.\nFinally, the replicators dynamics rule tends to favor hawks over doves on\nspatially structured topologies such as small worlds and square lattices, thus\nconfirming previous results for regular lattices and extending them to small-world networks.\nIn the end, although small-world topologies show behaviors that are somewhat in between\nthose of the random graph and the two-dimensional lattice, they usually tend more\ntowards the latter, at least in terms of cooperation level.\n\nIn this work, we have used static\nnetwork structures, which is a useful step but it is not realistic enough as the interactions\nthemselves help shape the network. In future work we shall\nextend the study using more faithful social network structures, including their dynamical aspects.\n\n\n\\begin{small}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Noncommutative spacetime and nonabelian gauge fields}\n\\hspace*{12pt}Between 1921-1948, Kaluza, Klein and Thiry \\cite{KK} had shown that the Hilbert-Einstein action in the spacetime extended with a circle ${\\cal M}^4 \\times S^1$ consisted of gravity, electromagnetism and a Brans-Dicke scalar. In 1968, R.Kerner \\cite{Kerner} had generalized the Kaluza-Klein theory to include nonabelian gauge fields. Today, the multi-dimensional theories are studied widely as candidates of unified theories of interactions. However, these theories have a weakness of containing an infinite tower of massive fields leading to theoretical and observational obstacles.\n\nIn the 1980's, Connes had put forward the new concept of spacetime based on noncommutative geometry(NCG) \\cite{Co}.In 1986, Connes and Lott \\cite{CoLo} applied the idea to the spacetime extended by two discrete points ${\\cal M}^4 \\times Z_2$ and shown that Higgs fields emerged naturally in a gauge theory with a quartic potential. The most attractive feature of NCG with discrete extra dimensions is that it does not contains an infinte tower of massive fields.\n \nIn 1993, Chamsedine, Felder and Fr\\\"ohlich \\cite{CFF} made the first attempt to generalize the Hilbert-Einstein action to NCG, leading to no new physical content. In 1994, G.Landi, N.A.Viet and K.C.Wali \\cite{LVW} had overcome this no-go result and derived the zero mode sector of the Kaluza-Klein theory from the generalized Hilbert-Einstein's action. Viet and Wali \\cite{VW1} have generalized this model further and obtained a full spectum consisting of bigravity, bivector and biscalar. In each pair, one field is massless and the other one is masive.\n\nThe incorporation of the nonabelian gauge fields in Viet-Wali's model is a not trivial task. Recently, Viet and Du \\cite{VietDu} have successfully derived the nonabelian gauge interaction from the Hilbert-Einstein's action. However, it is possible to do so only in two following cases:\n\ni. The gauge vector fields must be abelian on one sheet of spacetime and nonabelian on the other one. This is exactly the case of the electroweak gauge fields on the two copies of Connes-Lott's spacetime of chiral spinors.\n\nii. The gauge vector field must be the same on both copies of spacetime of chiral spinors. This is also the case of QCD of strong interaction.\n\nSo, NCG can \"explain\" the specific gauge symmetry structure of the Standard Model.\n \nIn this article, we propose a new noncommutative spacetime structure ${\\cal M}^4 \\times Z_2 \\times Z_2$, which is the ordinary spacetime extended by two discrete extra dimension, each consists of two discrete points. In other words, this noncommutative spacetime consists of two copies of Connes-Lott's spacetime. The generalized  Hilbert-Einstein action in this new spacetime contains all the known interactions of Nature and the observed Higgs field. In a more general case, this theory can also lead to multigravity, which might be necessary to explain the dark matter and inflationary cosmology related observations. \n\\section{A new model of noncommutative spacetime and gravity}\n\\hspace*{12pt}The noncommutative spacetime ${\\cal M}^4 \\times Z_2 \\times Z_2$ is the usual four dimensional spinor manifold extended by two extra discrete dimensions given by two differential elements $DX^5$ and $DX^6$ in addition to the usual four dimensional ones $dx^\\mu$. Each extra dimension consists of only two points. This structure can also be viewed as four sheeted space-time having a noncommutative differential structure with the following spectral triplet:\n\ni) The Hilbert space ${\\mathcal H}= {\\mathcal H}^v \\oplus {\\cal H}^w $ which is a direct sum of two Hilbert spaces ${\\mathcal H}^u = {\\cal H}^u_L \\oplus {\\cal H}^u_R, u = v,w$, which are direct sums of the Hilbert spaces of left-handed and right-handed spinors. Thus the wave functions $ \\Psi \\in {\\mathcal H}$  can be represented as follows \n\\begin{equation}\n    \\Psi(x) =  \\pmatrix{\n    \\Psi^v(x) \\cr\n    \\Psi^w(x) \\cr\n    } ~~,~~ \n    \\Psi^u(x) = \\pmatrix{\n        \\psi^v_L(x) \\cr\n        \\psi^w_R(x) \\cr\n        }  \\in {\\mathcal H}^u ~;~ u=v,w,\n\\end{equation}\nwhere the functions $\\psi^u_{L,R}(x) \\in {\\cal H}^u_{L,R}$ are defined on the 4-dimensional spin manifold ${\\cal M}^4$.\n\nii) The algebra ${\\cal A}={\\cal A}^v \\oplus {\\cal A}^w ; {\\cal}^u = {\\cal A}^u_L \\oplus {\\cal A}^u_R$ contains the 0-form ${\\cal F}$ \n\\begin{equation}\\label{0form}\n   {\\mathcal  F}(x) =  \\pmatrix{\n    F^v(x) & 0 \\cr\n    0 & F^w(x) \\cr\n    } ~,~ F^u(x) =  \\pmatrix{\n        f^u_L(x) & 0 \\cr\n        0 & f^u_R(x) \\cr\n        } \\in {\\cal A}^u,\n\\end{equation}\nwhere $f^u_{L,R}(x)$ are real valued function operators defined on the ordinary spacetime ${\\cal M}^4$ and acting on the Hilbert spaces ${\\cal H}^u_{L,R}$.\n\niii) The Dirac operator ${\\cal D} = \\Gamma^P D_P = \\Gamma^\\mu \\partial_\\mu + \\Gamma^5 D_5+ \\Gamma^6 D_6, P=0,1,2,3,5,6 $ is defined as follows\n\\begin{eqnarray}\\label{Dirac1}\n{\\cal D} &=& \\pmatrix{\nD & m_1 \\theta_1 \\cr\nm_1 \\theta_1 & D \\cr\n}~,~ D = \\pmatrix{\nd & m_2 \\theta_2 \\cr\nm_2 \\theta_2 & d \\cr} ~,~ d = \\gamma^\\mu \\partial_\\mu \\\\ \nD_\\mu {\\cal F} &=& \\pmatrix{\n\\partial_\\mu F^v(x) & 0 \\cr\n0 & \\partial_\\mu F^w(x) \\cr \n} ~,~\n\\partial_\\mu F^u = \\pmatrix{\n\\partial_\\mu f^u_L(x) & 0 \\cr\n0 & \\partial_\\mu f^u_R(x) \\cr \n} \\\\\nD_{6} {\\cal F} &=& m_1(F^v - F^w){\\bf r} ~,~ D_5 F^u = m_2(f^u_L-f^u_R) {\\bf r} ~,~ {\\bf r} = \\pmatrix{\n1 & ~0\\cr\n0 & -1\\cr\n} \n\\end{eqnarray}\nwhere $\\theta_1, \\theta_2$ are Clifford elements $ \\theta^2_1 = \\theta^2_1=1$, $m_1, m_2$ are parameters with dimension of mass. \n\nThe construction of noncommutative Riemannian geometry in the Cartan formulation is given in \\cite{VW1} in a perfect parallelism with the ordinary one. Here we will use the following flat and curved indices to extend the 4 dimensions with 5-th and 6-th dimensions.\n\\begin{eqnarray}\nE,F,G = A, \\dot{6} ~,~ & A,B,C = a, \\dot{5}&~,~ a,b,c =0,1,2,3 \\\\\nP,Q,R = M,6 ~,~ &M,N,L = \\mu, 5 &~,~\\mu,\\nu, \\rho=0,1,2,3.\n\\end{eqnarray}\n\nThe construction of noncommutative Riemannian geometry \\cite{VW1} is in a perfect parallelism with the ordinary one. The starting point is the locally flat reference frame, which is a linear transformation of the curvilinear one with the vielbein coefficients. For transparency, let us write down the vielbein in 4,5 and 6 dimensions as follows \n\\begin{equation}\ne^a = dx^\\mu e^a_\\mu(x) ~,~ E^A = DX^M E^A_M(x) ~,~{\\cal E}^E = DX^P {\\cal E}^E_P(x),\n\\end{equation}\nwhere $e^a_\\mu(x), E^A_M(x), {\\cal E}^E_P$ are 4,5 and 6-dimensional vielbeins.\nThe Levi-Civita connection 1-forms $\\Omega^\\dagger_{EF} = - \\Omega_{FE}$ are introduced as a direct generalization of the ordinary case. With a condition \\cite{VW1}, which is a generalization of the torsion free condition one can determine the Levi-Civita connection 1-forms and hence the Ricci curvature tensor from the generalized Cartan structure equations    \n\\begin{eqnarray}\n{\\cal T}^E &= &  DE^E + E^F \\Omega^E_F \\label{Torsion}\\\\\n{\\cal R}^{EF} &=& D\\Omega^{EF}+ \\Omega^E_G \\wedge \\Omega^{GF} \\label{Curv}\n\\end{eqnarray}\nThen we can calculate the Ricci scalar curvature $R=\\eta^{EG} \\eta^{FH} R_{EFGH}$.\n\nThe construction of our model is carried out in two subsequent steps. First, we construct the 6-dimensional Ricci curvature with an ansatz containing one 5-dimensional gravity and two 5-dimensional vectors fields, where one is abelian and the other is nonabelian to use Viet-Du's results. Then\n\\begin{equation}\nR_6 = R_5 - {1 \\over 4}  G^{MN} G_{MN} = R_5 + {\\cal L}_g(5)\n\\end{equation}\nwhere $G_{MN}; M,N=0,1,2,3,5$ is the 5-dimensional covariant field streng tensor of the nonabelian $SU(2)\\times U(1)$ gauge fields.\n\nIn the second step, the gravity sector is reduced further to 4-dimensional gravity  nonabelian gauge $SU(3)$ vector of strong interaction \n\\begin{equation}\nR_5 = R_4 - {1 \\over 4} Tr H^{\\mu \\nu} H_{\\mu \\nu} ~~,~~ H_{\\mu \\nu} = \\partial_\\mu B_\\nu - \\partial_\\nu B_\\mu + i g_S [B_\\mu, B_\\nu],\n\\end{equation}\nwhere $B_\\mu = B^i_\\mu(x) \\lambda^i$ are the gluon field and $\\lambda^i, i=1,..,8$ are the GellMann matrices.\n\nConnes-Lott's procedure can be applied now to reduce the 5-dimensional gauge Lagrangian ${\\cal L}_g(5)$ to the 4-dimensional electroweak gauge-Higgs sector of the Standard Model as follows\n\\begin{equation}\n{\\cal L}_g(5) = -{1 \\over 4}( F^{\\mu\\nu} F_{\\mu\\nu}+ G^{\\mu \\nu} G_{\\mu \\nu}) + {1 \\over 2} \\nabla^\\mu \\bar{H} \\nabla_\\mu H + V(\\bar{H}, H),  \n\\end{equation} \nwhere $H$ is a Higgs doublet, $\\nabla_\\mu$ is the gauge covariant derivative and $V(\\bar{H}, H)$ is the usual quartic potential of the Higgs field.\n\\section{Multigravity in noncommutative spacetime}\n\n\\hspace*{12pt}In Section 2, we have presented the minimal ansatz to include the all known interactions and the Higgs fields. New cosmological observations might shed light to more detailed structure of the new commutative spacetime. In principles, in a more general case, our model can adopt up to 4 gravitational fields, one of those is massless while the other ones are massive. \n\nFrom theoretical points of view, this model can provide a geometric construction approach to the massive gravity, which has\nrecently attracted a lot of attention as a candidate theory of modified gravity \\cite{DeRham}. From the viewpoints of modern cosmology, multigravity might give new explanations the existence of dark matter and inflationary  cosmology.\n  \n\\section{Summary and discusions}\nWe have presented a new noncommutative spacetime ${\\cal M}^4 \\times Z_2 \\times Z_2$, which can unify all the known interactions and Higgs field on a geometric foundation. This is very similar to the foundation of Einstein's general relativity.  This model unifies all forces in nature without resorting to infinite tower of massive fields.\n\nIn the most general case, this theory can contain more (but still a finite number) degrees of freedom, including four different massless and massive gravity fields, Brans-Dicke scalars and more gauge fields. The model can provide a geometric foundation to the theories of massive and modified gravity. The reality might be just a special case of the most general theory. The cosmological observations might help us to see more details of this theory.\n\nThere are some issues, at the moment we are not able to answer such as the physical meaning of the sixth dimension and the energy scale of this theory. It is worth to quote the following relation from the work by Viet and Du \\cite{VietDu}\n\\begin{equation}\ng = 8 m \\sqrt{\\pi G_N},\n\\end{equation}\nwhere $g$ the weak coupling constant and $G_N$ is the Newton constant. This relation must hold when the theory becomes valid. One can speculate this might happen at an energy scale, which is million times lower that the Planck scale. That might be the case in some evolution stage of our universe after the Big Bang. All the above perspectives would merit more research.\n\\section*{Acknowledgments}\nThe discussions with Pham Tien Du, Do Van Thanh (College of Natural Sciences, VNU) and Nguyen Van Dat (ITI-VNU) are greatly appreciated. The author would also like to thank Jean Tran Thanh Van for the hospitality at Quy Nhon and supports. The work is partially supported by ITI-VNU and Department of Physics, College of Natural Sciences, VNU.\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\nThe past decades have seen a proliferation of research using evolutionary theory to study social traits, in the fields of biology, animal behavior, and even social science \\cite{Nowak1998,Nowak2004,Hilbe2018a,Ohtsuki2006,Weitz2016,Tilman2020,Allen2017a}.\nMost of this theoretical development has been based on mathematical models that assume either infinite populations \\cite{Taylor1978,Schuster1983,Nowak2006a,Weitz2016,Tilman2020}  or finite populations of constant size \\cite{Hilbe2018a,Ohtsuki2006,Nowak2004}.\nDespite these simplifying assumptions, mathematical models provide rich insights into how exogenous and intrinsic factors drive evolutionary dynamics of social behavior. \nFor example, \nthe literature has produced a rich set of explanations for cooperation based on repeated interactions, the establishment of reputations, and various forms of population structure \\cite{Nowak1992,Ohtsuki2006,Tarnita2009,Allen2017a,McAvoy2021,Su2022,Su2022nhb,Cooney2019,Hilbe2018a,Santos2018,Nowak2005,Nowak1993,Nowak2006fiverule,Stewart2013}. Several key theoretical insights have been validated by controlled experiments on human subjects \\cite{Gachter2009,Yamauchi2011,Yoelia2013,Greiner2005}. This field of research has been so successful that the question of how cooperation can be favored by natural selection, famously posed by Darwin, is now not only resolved, but resolved in several distinct ways applicable in different contexts.\n\nHere we reveal an qualitatively different and pervasive mechanism that can promote cooperation by natural selection or payoff-biased imitation. \nMost mechanisms known to support cooperation boil down to some form of population structure \\cite{Kay2020} -- either physical limitations on social interactions, reproduction, or imitation, or structure imposed by tags or reputations. By contrast, here we describe a much more simple scenario that can favor cooperation in a population that lacks any form of exogenous or endogenous structure. We show that demographic stochasticity, which is \\textit{a priori} a realistic feature of any natural population, can by itself promote social behaviors that would otherwise be suppressed in idealized populations of constant (or infinite) size.\n\nThere is precedent for the idea that demographic stochasticity alters evolutionary dynamics. The fact that mortality, reproduction, and migration are subject to demographic fluctuations in populations -- as well as processes of imitation and innovation -- is known influence the dynamics of competing types under frequency-independent selection \\cite{Parsons2007,Parsons2007a,Parsons2010,McKane2005,Butler2009,Hallatschek2007,Stollmeier2018,Wienand2017,Taitelbaum2020,Chotibut2017} and also frequency-dependent selection \\cite{Constable2016,Houchmandzadeh2012,Houchmandzadeh2015,Huang2015}.\nFor example, when a population contains two types with the same expected number of offspring, one type can be favored when the population size is small, and the other type favored when the population size is near to its carrying capacity \\cite{Parsons2007,Parsons2007a,Parsons2010}.\nAnd a few studies have shown that demographic stochasticity can even reverse the direction of natural selection, promoting a type that would otherwise be disfavored without stochasticity  \\cite{Constable2016,Houchmandzadeh2012,Houchmandzadeh2015}.\n\nNonetheless, prior work on selection with demographic stochasticity has either assumed constant fitness, in which one's fitness is independent of the composition of the population, or assumed different carrying capacities for different phenotypes, e.g.,~producers enjoy a larger carrying capacity than non-producers \\cite{Constable2016,Houchmandzadeh2012,Houchmandzadeh2015}.\nMost models of demographic stochasticity \nalso assume that offspring numbers follow a Poisson distribution \\cite{Constable2016,Huang2015,Parsons2010,Parsons2007,Parsons2007a,Houchmandzadeh2012,Houchmandzadeh2015}, so that the mean and variance in offspring number are identical. But empirical field studies have found that over-dispersion in offspring number (variance exceeding mean) is commonplace across diverse taxa \\cite{Zuur2009,Linden2011,VerHoef2007,Richards2008}.\n\n\n\nIn this paper, we develop a general framework to study evolutionary dynamics with demographic stochasticity, which can capture both frequency-dependent fitness, arising from social interactions, as well as over-dispersion in the number of offspring. We provide a simple analytical condition that governs the long-term outcome of competition between multiple types. Applied to pairwise social interactions involving cooperation or defection, we find that demographic stochasticity can favor cooperators provided the offspring variance is sufficiently large, even without any other mechanisms. For more general pairwise payoff structures, we show that demographic stochasticity can reverse the stability of equilibria, from coexistence to bi-stability and vice versa, or from dominance of one type to dominance of another. Our analysis highlights the profound effects of demographic stochasticity on the evolution of interacting types in a population.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figure_main\/main_figure1.pdf}\n    \\caption{\\textbf{Evolutionary dynamics with demographic stochasticity.} \n    (A) Competition between cooperators (blue circle) and defectors (red circle) in a stochastic population of non-constant size.  Each individual $i$ derives payoff $\\pi_i$ from pairwise game-play with each other individual in the population. The number of offspring produced by an individual within time $\\Delta t$ has mean $(B+s\\pi_i)\\Delta t$ and variance $(\\delta_1B+\\delta_2 s\\pi_i)\\Delta t$, which are both higher for defectors than for cooperators. When selection is weak ($s \\ll \\alpha$), the population quickly reaches carrying capacity (during time period I) while the frequency of cooperators and defectors remains unchanged from its initial value ($p_0=1\/2$ shown here). Thereafter (time period II) the population remains near carrying capacity ($M \\approx 1000$ shown here), while the frequency of cooperators and defectors slowly vary until either cooperators go extinct (example in panel B) or defectors go extinct (panel C).\n    Parameters: $b=3$, $c=1$, $s=0.01$, $\\delta_1=\\delta_2=1$, $x_0=y_0=10$, $\\lambda=1\\times10^{-3}$, $B=2$, $D=1$.}\n    \\label{fig1}\n\\end{figure}\n\n\\section{Model}\n\nWe first consider an evolving population of two types: cooperators (C) and defectors (D).\nEach individual interacts pairwise with each other, in which the cooperator pays a cost $c$ to bring his opponent a benefit $b$ ($b>c$), and the defector pays no cost and provides no benefit. In other words, pairwise interactions follow a simple ``donation game\", which provides a minimal model for studying the evolution of cooperation \\cite{rapoport1965prisoner}.\nFollowing all pairwise interactions, \neach individual obtains an average payoff that will determine their reproductive output (or, equivalently, the number of individuals who copy their type by social contagion). In a population with $x$ cooperators and $y$ defectors, the cooperator's payoff (denoted by $\\pi_C$) and the defector's payoff (denoted by $\\pi_D$) are \n\n\n\\begin{subequations}\n\\begin{align}\n    \\pi_C=&\\frac{x}{x+y}b-c, \\\\ \\pi_D=&\\frac{x}{x+y}b.\n\\end{align}\n\\end{subequations}\n\n\nIn a classic Moran model, each birth event is followed by a death event, and so the population size remains constant. Here we remove this constraint by decoupling the birth and death events. Births are assumed to follow a continuous-time Markov process with independent and stationary increments (see Section S1 in Supplementary Information), such that the expected number of offspring individual $i$ produces per unit time is \\begin{equation}\n     \\mathbb{E}(\\xi_i)=B+s\\pi_i,\n     \\label{eq:mean}\n\\end{equation}\nwhere $B$ is a baseline number of offspring, $\\pi_i$ is individual $i$'s payoff, and the parameter $s>0$ is the intensity of selection. Note that the baseline birth rate is the same for all individuals, regardless of type, and it does not depend upon payoffs from social interactions. The selection intensity $s$ measures to what degree the payoff derived from social interactions affects the offspring number. In this paper we focus on the case of weak selection ($s\\ll 1$), a regime widely adopted in the literature \\cite{Allen2017a,McAvoy2021,Nowak2004,Ohtsuki2006}. Since the defector's payoff $\\pi_D$ is larger than the cooperator's payoff $\\pi_C$ in any population state, defectors always have a greater expected fecundity (Fig.~\\ref{fig1}). \n\nTo fully describe the birth process, we also specify the variance in the number of offspring. We are particularly interested in cases of over-dispersion, which can be modelled in many alternative ways \\cite{Linden2011,VerHoef2007}, such as a quasi-Poisson model (variance proportional to mean), mixed-effects Poisson model, and negative binomial model (variance a quadratic function of mean). Here we study a general class of Markov birth models by stipulating\n\\begin{equation}\n    {\\rm Var}(\\xi_i)=\\delta_1 B+\\delta_2 s\\pi_i,\n    \\label{eq:variance}\n\\end{equation}\nwhere parameters $\\delta_1$ and $\\delta_2$ measure the magnitude of offspring variance ${\\rm Var}(\\xi_i)$ relative to the mean $\\mathbb{E}(\\xi_i)$. The parameter $\\delta_1$ controls how  offspring variance is influenced by the baseline birth rate; and $\\delta_2$ controls how  offspring variance is influenced by payoffs from social interactions. Specific choices of $\\delta_1$ and $\\delta_2$ reduce to well-known classical models, such as a deterministic system ($\\delta_1=\\delta_2=0$) or a Poison birth process ($\\delta_1=\\delta_2=1$). In the regime of weak selection, the number of offspring produced per unit time is over-dispersed whenever  $\\delta_1>1$.\n\n\n\nDeath events are modelled as a Poisson process, arising from two rates that are summed. First, an individual dies at constant baseline rate, $D$. Second, in order to model competition for limited resources, additional deaths occur at rate $\\lambda$ times the current total population size.\n\n\\section{Results}\n\n\n\n\n\n\\subsection{Evolution of cooperation with demographic stochasticity}\n\\label{section3.1}\nLet $x$ and $y$ denote the number of cooperators and defectors respectively, which will change over time. \nGiven the class of models described above for the payoff-dependent birth-process and the population-size dependent death process,  the evolutionary dynamics of $x$ and $y$ can be approximated by a two-dimensional It\\^o stochastic differential equation (see Section S1 in Supplementary Information):\n\\begin{subequations}\n    \\begin{align}\n    {\\rm d}x&=x\\left[\\alpha+s\\pi_{\\rm C}-\\lambda (x+y)\\right] {\\rm d}t+ \\sqrt{x\\left[\\delta_1 B+\\delta_2 s\\pi_{\\rm C}+D+\\lambda(x+y)\\right]}{\\rm d}W^{(1)}_t, \\label{eq:ito_diffusion_a} \\\\\n    {\\rm d}y&=y\\left[\\alpha+s\\pi_{\\rm D}-\\lambda (x+y)\\right]{\\rm d}t+ \\sqrt{y\\left[\\delta_1 B+\\delta_2 s\\pi_{\\rm D}+D+\\lambda(x+y) \\right]}{\\rm d}W^{(2)}_t, \\label{eq:ito_diffusion_b} \n    \\end{align} \\label{eq:ito_diffusion} \n\\end{subequations}\nwhere $\\alpha =B-D>0$ indicates the net growth rate from baseline birth and death events, and $W_t^{(1)}$ and $W_t^{(2)}$ are independent standard Wiener processes. Although the birth process can be over-dispersed in our model (when $\\delta_1>1$), deaths follow a simple Poisson process with variance equal to mean. \n\nTo study how the relative abundance of cooperators and the total population size evolve over time, we make the co-ordinate transformation $(p,n)=(x\/(x+y),x+y)$. Applying It\\^o's lemma in Eq.~\\ref{eq:ito_diffusion}, the system can then be described by the equations\n\\begin{subequations}\n\\begin{align}\n          {\\rm d}p=&scp(1-p)\\left(-1 +\\frac{\\delta_2}{n}\\right){\\rm d}t+\\frac{y}{n^2} \\sqrt{x(\\delta_1 B+D+\\lambda n) }{\\rm d}W^{(1)}_t  \\notag\\\\\n          &-\\frac{x}{n^2}\\sqrt{y(\\delta_1 B + D+\\lambda n)}{\\rm d}W^{(2)}_t, \\label{eq:transformed_a}\\\\\n          {\\rm d}n=&[n\\alpha+s(b-c)pn-\\lambda n^2]{\\rm d}t+ \\sqrt{x(\\delta_1 B +D+\\lambda n)}{\\rm d}W^{(1)}_t   \\notag \\\\\n          & +\\sqrt{y(\\delta_1 B +D+\\lambda n)}{\\rm d}W^{(2)}_t. \\label{eq:transformed_b}\n\\end{align}\n\\label{eq:transformed}\n\\end{subequations}\n\nThe simple case in which stochasticity is absent (i.e., $\\delta_1=\\delta_2=0$ for births, and no variance for deaths) provides a deterministic reference point for comparison to any stochastic system. \nIn the deterministic system, ${\\rm d}p$ is always negative and the abundance of cooperators continuously decreases until cooperators reach extinction. Thus, cooperation is never favored by natural selection in the deterministic limit. Moreover, in this deterministic limit, changes in the total population size $n$ depend on both $p$ and $n$. But for sufficiently weak selection intensity ($s\\ll \\alpha$), changes in the total population size $n$ are much more rapid than changes in the cooperator frequency, $p$.\nIn the regime of weak selection, before $p$ changes its value at all, $n$ has grown logistically to its equilibrium value $(\\alpha+s(b-c)p)\/\\lambda$, which we denote by $M$. $M$ is called carrying capacity, and it describes the maximum number of individuals that the environment can sustain.  When the net growth rate is much larger than selection intensity, $\\alpha \\gg s$, the carrying capacity is well approximated by $M\\approx \\alpha\/\\lambda$. \n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.75]{figure_main\/heat_simu.pdf}\n    \\caption{\\textbf{Demographic stochasticity can favor the evolution of cooperation.} Colors represent the fixation probability of cooperation relative to neutral drift, $\\rho - p_0$, as a function of parameters $\\delta_2$ and $\\delta_1$. We say that selection favors cooperation when cooperators are more likely to fix than under neutrality (blue regions). Panel (A) shows exact solutions sampled from the stochastic differential equation (Eq.~\\ref{eq:ito_diffusion}), whereas panel (B) shows the analytical approximation in the regime of weak selection (Eq.~\\ref{eq:fixation prob}). The dashed line indicates the separation between regimes that favor cooperation (blue) or favor defection (red).\n       \n    Parameters: $B=2$, $D=1$, $s=0.005$, $b=1.1$, $c=1$, $\\lambda=5\\times 10 ^{-3}$, $x_0=y_0=50$.}\n    \\label{fig2}\n\\end{figure}\n\nFor a stochastic system ($\\delta_1 \\ne 0$ and $\\delta_2 \\ne 0$) the trajectories of $p$ and $n$ are not determined by the initial conditions alone, but depend upon chance events. We quantify the evolutionary advantage of cooperators by studying the fixation probability -- namely, the chance of absorption into the full-cooperation state ($p=1$). \nStarting from $x_0$ cooperators and $y_0$ defectors initially (thus $p_0=x_0\/(x_0+y_0)$ and $n_0=x_0+y_0$), the fixation probability, denoted by $\\rho(x_0,y_0)$ or $\\rho(p_0,n_0)$, is the probability that at some time $t$ defectors become extinct while cooperators still exist, that is $y(t)=0$ but $x(t)>0$ \\cite{Czuppon2018}. \nIn the regime $s \\ll \\alpha$ the fixation probability can be calculated by separating the time-scale of changes in $p$ versus changes in $n$ \\cite{Parsons2017}. This analysis is tantamount to assuming that the total population size $n$ rapidly reaches its carrying capacity, while $p$ remains unchanged from $p_0$, and that subsequently $p$ evolves in one dimension while the population size remains near the slow manifold $n=M$ (see Fig.~\\ref{fig1} and Supplementary Fig.~S3). Under this analysis, we can approximate the fixation probability by a simple expression (Section S2.1 in Supplementary Information) \n\\begin{equation}\n    \\rho(p_0,n_0) \\approx p_0+\\frac{sc}{(\\delta_1+1) B}\\left(\\delta_2-M\\right)p_0(1-p_0).\n    \\label{eq:fixation prob}\n\\end{equation}\nWe performed numerical simulations, drawing sample paths from the full SDE system given by  Eq.~\\ref{eq:ito_diffusion}, to verify the accuracy of this analytic approximation for the fixation probability  (Fig.~\\ref{fig2}).\n\nNote that fixation probability does not depend on the initial population size, but rather on the initial frequency of cooperators. In the absence of selection ($s=0$), the fixation probability equals the initial frequency of cooperators, $p_0$. And so we say that cooperation is favored by selection if the fixation probability exceeds $p_0$, which will occur whenever\n\\begin{equation}\n    \\delta_2>M. \\label{eq:condition}\n\\end{equation}\nThis simple condition tells us when demographic stochasticity causes selection to favor cooperators, even though selection disfavors cooperation in a deterministic setting. In particular, demographic stochasticity can favor the fixation of cooperators when the offspring variance is sufficiently large -- in particular, when $\\delta_2$ exceeds the carrying capacity $M$. What matters for the direction of selection, then, is the size of the offspring variance arising from payoffs in social interactions, relative to its mean.\n\n\n\nWe can gain some useful intuition for the forces that govern the fate of cooperators by considering the deterministic part of Eq.~\\ref{eq:transformed_a}. The first term in this equation, $-scp(1-p)$, represents the deterministic contribution to the evolution of cooperator frequency, which always opposes cooperators. Whereas the second term in this equation, $\\delta_2scp(1-p)\/n$, arises from demographic stochasticity and it always favors cooperators. Whether or not cooperation is favored overall depends upon the balance between these two forces -- the deterministic force suppressing cooperation and demographic stochasticity that favors cooperation. For $\\delta_2<M$, the deterministic disadvantage is the stronger force and cooperators are net disfavored (recall that $n$ rapidly reaches carrying capacity $n=M$ before cooperators change frequency). However, if $\\delta_2>M$, the stochastic advantage matters more, so that cooperators are favored, which constitutes an evolutionary reversal compared to a classical model without demographic stochasticity.\n\n\nOther model parameters, $s$, $c$, $p_0$, $\\delta_1$ and $B$, do not produce a reversal in the direction of selection for cooperation, but they nonetheless influence the fixation probability. For example, increasing $\\delta_1$ or increasing the baseline birth rate $B$ moves the fixation probability towards the neutral value, $p_0$. Moreover, in the regime where demographic stochasticity favors cooperation, $\\delta_2>M$, the fixation probability is increased yet further when the selection intensity $s$ is large or when the cost of cooperation $c$ is large (Eq.~\\ref{eq:fixation prob}). Both of these results contravene the classical intuition that selection and the cost of cooperation should disfavor cooperators. \nWe have performed simulations to verify the effects of all these parameters, in comparison to the analytical approximation (Supplementary Fig.~S2).\n\n\n\\subsection{An explicit birth-death process}\n\nOur model of demographic stochasticity is quite general, stipulating only several properties of the Markov birth and death processes for competing types. We have analyzed this class of models by approximation, using a stochastic differential equation. In this section we construct an explicit example of birth and death processes that satisfy our model stipulations, and we compare the predictions of our SDE analysis to individual-based simulations of the discrete stochastic process.\n\nMost prior studies of demographic stochasticity are based on a reproduction process with a single offspring per birth event, which naturally leads to a Poisson birth process \\cite{Feller1950,Huang2015,Constable2016,Parsons2010,Parsons2007,Parsons2007a}. The Poisson process occurs as a special case within our family of models, when $\\delta_1=\\delta_2=1$. In this case, our analysis shows that demographic stochasticity alone cannot favor cooperation, because $\\delta_2<M$. \nWe will therefore consider non-Poisson birth process, in which the offspring produced per unit time is over-dispersed. This is a realistic scenario for many species, especially pelagic organisms, that have heavy-tailed offspring distributions \\cite{Davis1956,Eldon2006}; as well as for social contagion \\cite{brady2017emotion,schroder2021social}.\n\nWe will define a birth process by two factors: the times of birth events and the litter size (offspring number) in each such birth event. A natural way to describe this is through a compound Poisson process \\cite{Last2017}. Specifically, for individual $i$ with payoff $\\pi_i$, the times of birth events obey a Poisson process with intensity $\\theta_i$. In each such birth event, the number of offspring produced (litter size) is also stochastic. We consider two cases: the litter size itself follows a Poisson distribution with mean $\\mu_i$, or the litter size follows a negative binomial distribution with parameters $q_i$ and $m$ ($q_i\\in [0,1]$ and $m\\in \\mathbb{N}^*$). Both of these distributions have been used to model litter sizes in empirical studies \\cite{Sheldon2003,Richards2008,Linden2011}. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.75]{figure_main\/compound_poisson_2.pdf}\n    \\caption{\\textbf{Selection for cooperation in a compound Poisson birth process.}\n    We simulated a compound Poisson birth process with either a Poisson-distributed litter size (A) or a negative binomial litter size (B). The parameters of the birth process ($\\theta_i$ and $\\mu_i$ in panel A; $\\theta_i$ and $q_i$ in panel B) can be chosen to satisfy our general conditions for the mean and variance in total offspring produced per unit time, for any choice of $\\delta_1>1$, $\\delta_2$, and $B$.  \n    Two examples with the parameters that correspond to $(\\delta_1=6,\\delta_2=60)$ and $(\\delta_1=6,\\delta_2=140)$ are shown in each panel. Blue squares indicate the fixation probability of cooperators, starting from an initial population with $x_0=y_0=50$, observed in $5\\times 10^7$ replicate Monte Carlo simulations, with carrying capacity either $M=100$ or $M=200$. Selection favors cooperation if the fixation probability $\\rho$ exceeds the initial fraction of cooperators,  0.5 (horizontal dashed line).\n    The solid lines plot our analytical approximation for the fixation probability (Eq.~\\ref{eq:fixation prob}). As predicted by our analysis, cooperation is favored when $\\delta_2>M$.\n    Parameters: $B=2$, $D=1$, $\\delta_1=6$, $s=0.001$, $b=1.1$, $c=1$, $m=5$ (negative binomial), $x_0=y_0=50$, $\\lambda=1\/100$ ($M=100$) or $\\lambda=1\/200$ ($M=200$).}\n    \\label{fig3}\n\\end{figure}\n\nThe parameters of the compound Poisson process depend upon an individual's payoff and the selection intensity. For the Poisson-Poisson case (the litter size follows a Poisson distribution) the reproductive process of individual $i$ is characterized by parameters $\\theta_i$ and $\\mu_i$, and we assume that the payoff $\\pi_i$ affects both $\\theta_i$ and $\\mu_i$ linearly\n\\begin{subequations}\n\\begin{align}\n    \\theta_i&=\\theta_0+k_\\theta s\\pi_i, \\\\\n    \\mu_i&=\\mu_0+k_\\mu s\\pi_i.\n\\end{align}\n\\end{subequations}\nFor the Poisson-negative binomial case (the litter size follows a negative binomial distribution), we assume that all individuals share the same $m$ and that payoffs affect $q_i$ and $\\theta_i$ as follows:\n\\begin{subequations}\n\\begin{align}\n    \\theta_i&=\\theta_0+k_\\theta s\\pi_i, \\\\\n    q_i&=q_0+k_q s\\pi_i.\n\\end{align}\n\\end{subequations}\n\nGiven these equations, we can always choose parameters of the compound Poisson process\nthat satisfy our general stipulations on the mean and variance in the total offspring produced per unit time (Eq.~\\ref{eq:mean} and Eq.~\\ref{eq:variance}), provided $\\delta_1>1$ and $\\delta_2>0$ (see Section S3 in Supplementary Information). Note that for both of these compound Poisson birth processes (Poisson-Poisson and Poisson-Negative-Binomial) the total number of offspring produced per unit time must be over-dispersed ($\\delta_1>1$).\n\nWe can compare Monte-Carlo simulations of these explicit population processes (discrete state, continuous time) to the analytical prediction for the fixation probability that we derived from a stochastic differential equation (Eq.~\\ref{eq:fixation prob}). We find good agreement between the individual-based simulations and analytic approximations, for carrying capacities as small as $M=100$ or $M=200$  (Fig.~\\ref{fig3}). Note that in both cases shown in Fig.~\\ref{fig3}, for sufficiently large $\\delta_2$ we have $k_\\theta<0$ and $k_m>0$ or $k_q>0$. In other words, higher payoffs reduce the rate of birth events but increase the mean litter size per birth event; and when these effects are strong enough, then selection favors cooperation.\n\n\n\n\n\n\\subsection{Intuition for the effects of demographic stochasticity}\n\nThere is a simple intuition for how demographic stochasticity can favor cooperation in our class of models, even though cooperation is always disfavored in models with constant (or infinite) population size. The key insight has to do with the rapid growth of the total population size to carrying capacity, followed by slow dynamics in the frequency of cooperators near the manifold $n=M$. Importantly, during the slow dynamics there are still small fluctuations that move the population off the manifold $n=M$, followed by a rapid return back to carrying capacity. These small fluctuations have the effect of inducing an advective force pushing the frequency of cooperators $p$ in one direction or another. \n\nTo be more precise, we have already noted that the total population size $n$ equilibrates much more quickly than the frequency of cooperators $p$ (Eq.~\\ref{eq:transformed}), in the regime we study $\\alpha \\gg s$. And so, given an arbitrary initial state $p_0$ and $n_0$, $n$ will quickly converge to the slow manifold\n\\begin{equation}\n    n=\\frac{\\alpha+s(b-c)p_0}{\\lambda}\\approx \\frac{\\alpha}{\\lambda}=M,\n    \\label{eq:slow manifold}\n\\end{equation}\nwhile $p$ does not change from $p_0$ (see example in Supplementary Fig.~S3B). After the population size reaches carrying capacity, trajectories then move along the slow manifold until one type or the other fixes ($p=0$ or $p=1$). We focus on the dynamics on the slow manifold, which simplifies the analysis to a one-dimensional system \\cite{Parsons2017}. \n\nIn the co-ordinate system $(x,y)$, the slow manifold is defined $x+y=M$, and the fast manifolds are lines connecting the origin to points on the slow manifold (see Fig.~\\ref{fig4}A). \nGiven any initial conditions, the trajectory will rapidly approach the slow manifold along one of these lines, and then subsequently move within the slow manifold. However, unlike the case of a strictly constant population size, the system with demographic stochasticity does not lie precisely on the slow manifold at all times. Small fluctuations take the system off the slow manifold briefly, and then the system rapidly  returns to the slow manifold. Critically, the position where the system returns to the slow manifold, after a fluctuation, is not necessarily the same as where it started. In fact, there can be a systematic deviation in the position on the slow manifold that arises from stochastic fluctuations and rapid returns -- which produces an advective force on the frequency $p$ along the slow manifold (see Fig.~\\ref{fig4}B, C, D). It is this systematic deviation, caused by demographic stochasticity, that introduces a force favoring cooperation.\n\nIn particular, fluctuations from $x+y=M$ follow a two-dimensional Gaussian distribution with variance $x(\\delta_1B+\\delta_2s\\pi_C+D+\\lambda n)$ in the $x$-direction and variance $y(\\delta_1B+\\delta_2 s\\pi_D+D+\\lambda n)$ in the $y$-direction. In Fig.~\\ref{fig4}, we illustrate the fluctuation starting from state $x=y=M\/2$ (see Supplementary Information Section S2.2 for the analysis of any other states).\nWhen $\\pi_C=\\pi_D$, the Gaussian fluctuation is isotropic, and so a fluctuation followed by return along a fast-manifold line produces no expected change in the resulting position on the slow manifold (see Fig.~\\ref{fig4}B). However, whenever $\\pi_C \\ne \\pi_D$, the two-dimensional Gaussian fluctuation has an ellipsoid shape, and fluctuation followed by rapid return produces an expected change in the frequency of cooperators, $p$, along the slow manifold. In particular, when $\\pi_C < \\pi_D$, the expected change due to demographic stochastic favors cooperators, whereas if $\\pi_C > \\pi_D$ the expected change favors defectors (Fig.~\\ref{fig4}C,D). In general, we can analytically calculate the adjective force along the slow manifold that arises from these stochastic fluctuations and rapid returns (Section S2.2 in Supplementary Information).  \n\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=0.65]{figure_main\/fig4.pdf}\n    \\caption{\\textbf{How demographic stochasticity can favor cooperation or defection.}  \n    (A) The system features a separation of timescales, where the total number of individuals $n=x+y$ changes much faster than the fraction of cooperators $p=x\/(x+y)$.\n    Starting from $x_0$ and $y_0$ cooperators and defectors, trajectories rapidly converge to the slow manifold ($x+y=M$) along the fast manifold $x\/y=x_0\/y_0$.\n    (B, C, D) Stochastic fluctuations away from the slow manifold, followed by rapid return, can induce an advective force on the frequency of cooperators.\n    For simplicity we consider constant payoffs, where $\\pi_C$ and $\\pi_D$ are independent of the number of cooperators and defectors. The ellipses illustrate the variance-covariance structure  of two-dimensional Gaussian fluctuations around the slow manifold from a given point $x=M\/2$ and $y=M\/2$ (red point $O$). \n    (B) When $\\pi_C=\\pi_D$, fluctuations from point $O$ are isotropic, shown as a circle.\n    We consider four representative fluctuations from point $O$, $X_{-},X_{+},Y_{-},Y_{+}$, and the following points of return $X_{-}',X_{+}',Y_{-}',Y_{+}'$ to the slow manifold.\n    For isotropic fluctuations there is no expected change in $p$ after return to the slow manifold.\n    (C) For $\\pi_C<\\pi_D$, the Gaussian fluctuations are an-isotropic, shown as an ellipse, with larger fluctuations in the number of defectors. This asymmetry leads to an expected increase in cooperator frequency $p$ after return to the slow manifold, as indicated by the blue arrow.\n    (D) For $\\pi_C>\\pi_D$, the larger fluctuation occurs in the number of cooperators, which leads to an expected decrease in cooperator frequency after return to the slow manifold. These effects of an-isotropic noise are similar to those discussed by \\cite{Constable2016}, but they arise here even when both types have the same baseline birth rate and the same carrying capacity, under weak selection.}\n    \\label{fig4}\n\\end{figure}\n\n\nFor the donation game we have studied so far, cooperators always have a lower payoff than defectors regardless of the population state. And so the advective force arising from demographic stochasticity always favors cooperation, regardless of $p$. If this force is large enough relative to the deterministic force favoring defectors, then it can produce a net advantage for cooperators. For other types of pairwise games, however, the direction of deterministic selection  ($\\pi_C$ vs $\\pi_D$) may depends on the current frequency $p$ in the population, and so the noise-induced advection may change sign along the slow manifold, producing complicated effects on long-term dynamics. We investigate these effects of demographic noise on evolutionary dynamics for general two-player games in the next section.  \n\n\n\n\\subsection{General evolutionary game dynamics with demographic stochasticity}\n\nFor an arbitrary two-player game that gives rise to payoffs, the two-dimensional system can be simplified to a one-dimensional system by separation of timescales, provided selection is weak enough, $s \\ll \\alpha$.  Suppose the game has the following payoff structure:\n\\begin{equation}\n    \\begin{array}{cc}\n       & \\begin{array}{cc}\n          {\\rm C} & {\\rm D} \n      \\end{array} \\\\ \\begin{array}{c}\n           {\\rm C}\\\\{\\rm D} \n      \\end{array}\n     & \\left(\\begin{array}{cc}\n          a&b \\\\\n          c&d \n     \\end{array}\\right).\n\\end{array}\n\\end{equation}\nPlayers have two strategies, which we still generically call cooperation (C) or defection (D). When two cooperators interact, both of them receive payoff $a$. When a cooperator interacts with a defector, the cooperator receives $b$ and the defector $c$. Mutual defection brings payoff $d$ to both players. The average payoff for a cooperator or defector in a population are respectively\n\\begin{equation}\n\\begin{split}\n    \\pi_C&=\\frac{xa+yb}{x+y}, \\\\\n    \\pi_D&=\\frac{xc+yd}{x+y}.\n\\end{split}\n\\end{equation}\n\nSimilar to Section \\ref{section3.1}, we can describe the system by a stochastic differential equation:\n\\begin{subequations}\n\\begin{align}\n          {\\rm d}p=&sp(1-p)\\left(1-\\frac{\\delta_2}{n}\\right)(\\pi_{\\rm C}-\\pi_{\\rm D}){\\rm d}t+\\frac{1-p}{n} \\sqrt{x(\\delta_1 B+D+\\lambda n)}{\\rm d}W^{(1)}_t \\notag \\\\\n          &-\\frac{p}{n}\\sqrt{y(\\delta_1 B +D+\\lambda n)}{\\rm d}W^{(2)}_t, \\label{eq:transformed_general_a}\\\\\n          {\\rm d}n=&[n\\alpha+s(p\\pi_{\\rm C}+(1-p)\\pi_{\\rm D})pn-\\lambda n^2]{\\rm d}t+ \\sqrt{x(\\delta_1 B+D+\\lambda n)}{\\rm d}W^{(1)}_t\\notag \\\\\n          &+\\sqrt{y(\\delta_1 B +D+\\lambda n)}{\\rm d}W^{(2)}_t. \\label{eq:transformed_general_b}\n\\end{align}\n\\label{eq:transformed_general}\n\\end{subequations}\nSince the population size quickly equilibrates to the carrying capacity  $M\\approx\\alpha\/\\lambda$, we substitute $n=M$ into Eq.~\\ref{eq:transformed_general_a} which yields a one-dimensional equation for the evolution of $p$ along the slow manifold:\n\\begin{subequations}\n\\begin{align}\n    {\\rm d}p=&sp(1-p)\\left[\\left(1-\\frac{\\delta_2}{M}\\right)\\left(b-d+(a-b-c+d)p\\right) \\right]{\\rm d}t \\notag \\\\\n    &+\\sqrt{\\frac{(\\delta_1+1) B p(1-p)}{M}}\\left(\\sqrt{1-p}{\\rm d}W^{(1)}_t-\\sqrt{p}{\\rm d}W^{(2)}_t\\right).\n\\end{align}\n\\label{eq:general one dimensional}\n\\end{subequations}\n\nIn the case of deterministic births and deaths ($\\delta_1=\\delta_2=0$ and neglecting variance in the death process), this equation simplifies to the classic replicator equation \\cite{Schuster1983,Nowak2006a}. For general games there may be interior equilibrium points, and so the fixation probability is no longer a good measure to describe long-term evolutionary outcomes. Instead, we analyze the dynamics from two perspectives. One is from the perspective of the deterministic behavior on the slow manifold,\nwhich neglects stochasticity altogether in Eq.~\\ref{eq:general one dimensional} and studies the equilibria of the resulting ordinary differential equation. The other, more nuanced perspective accounts for stochasticity. Since $p=0$ and $p=1$ are the only absorbing states, any trajectory will finally reach one of these states and then become invariant. However, we can impose a reflecting condition on the boundary, which is equivalent to assuming that, when the number of one phenotype reaches zero, a new mutant of this phenotype arises instantly. The resulting evolutionary process of $p$ becomes an ergodic Markov process which has a unique stationary distribution $v^*(p)$. A frequency $p$ with greater probability density means that trajectories spend more time there. Derivation of the stationary distribution $v^*(p)$ under reflecting boundaries is given in Section S4.1 of Supplementary Information. \n\nWhen we ignore the stochastic terms, then Eq.~\\ref{eq:general one dimensional} is an ODE with the same equilibrium points and stabilities as the classic replicator equation, provided $\\delta_2<M$. Whereas if $\\delta_2>M$, then the equilibrium points are the same as the classic replicator equation, but the stabilities are reversed: equilibrium points that are classically unstable become stable, and conversely. And so the value of $\\delta_2$, which determines the payoff-component of offspring variance, can reverse the evolutionary outcome, even from a deterministic perspective.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figure_main\/main_figure4.pdf}\n    \\caption{\\textbf{General evolutionary game dynamics with demographic stochasticity.} \n    We consider three types of representative games, such as prisoner's dilemma (A, D), snowdrift game (B, E), and stag-hunt games (C, F).\n    In the prisoner's dilemma games, when demographic stochasticity is absent or does not meet $\\delta_2>M$, defectors dominate the population (see trajectories sampled in A, left part).\n    While the evolutionary direction can be reversed for $\\delta_2>M$, where cooperation becomes the dominant strategy (see trajectories sampled in A, right part).\n    Shown in (D) is the stationary distribution of cooperators for $\\delta_2=0$, $\\delta_2=25000$, and  $\\delta_2=50000$.\n    Analogously, in the snowdrift game, the demographic stochasticity with $\\delta_2>M$ changes the equilibrium from the coexistence of two strategies (B, left part) to the bi-stability (B, right part), which suggests the transformation of a snowdrift game to a stag-hunt game.\n    We also find that with demographic stochasticity, the evolution in the stag-hunt games proceed ``as if\" the population are playing snowdrift games. \n    Parameters: $B=2$, $D=1$, $s=10^{-3}$, $\\delta_1=2.5$, $\\lambda=10^{-4}$, $x_0=y_0=100$. }\n    \\label{fig5}\n\\end{figure}\n\nMore generally, we can classify three different deterministic scenarios based on the payoff matrix of the two-player, two-action game. For dominance games (Fig.~\\ref{fig5}A), one strategy is always dominant. Here, without loss of generality, we assume defection dominates cooperation ($a<c$ and $b<d$, e.g.,~a prisoner's dilemma). If $\\delta_2<M$, then all trajectories will converge to the full-defector state ($p=0$ stable and $p=1$ unstable). However, if $\\delta_2>M$, cooperation becomes the dominant strategy and all trajectories converge to full-cooperator state ($p=1$ stable and $p=0$ unstable). For coexistence games ($a<c$ and $b>d$, e.g.,~a snowdrift game), the best response is to choose the opposite strategy of the opponent (Fig.~\\ref{fig5}B). If $\\delta_2<M$, there is only one stable equilibrium, $p^*=(d-b)\/(a-b-c+d)$. All trajectories will converge to $p^*$ and therefore cooperators and defectors stably coexist. If $\\delta_2>M$, $p^*$ becomes unstable and $p=0$ and $p=1$ are each stable. Thus, all trajectories converge to either the full-cooperator or the full-defector state, similar to the outcome of a classic coordination game. For coordination games (Fig.~\\ref{fig5}C), the best response is to choose the same strategy as the opponent ($a>c$ and $d>b$, e.g., a stag-hunt game). In this case, $\\delta_2<M$ leads to an unstable internal equilibrium $p^*$ with stable boundaries ($p=0$ and $p=1$). But for $\\delta_2>M$, $p^*$ becomes stable while $p=0$ and $p=1$ are unstable. Most trajectories fluctuate around $p^*$ for a long time, showing similar behavior as a classic coexistence game. In summary, in a population with sufficiently large offspring variance ($\\delta_2>M$), the outcome of each type of game has the dynamical properties classically associated with the opposite type of game in a deterministic setting. In other words, demographic stochasticity effectively transforms the payoff structure of a game in the following way\n\n\\begin{equation}\n    \\begin{pmatrix}\n    a&b\\\\c&d\n    \\end{pmatrix} \\Rightarrow \\begin{pmatrix}\n    -a&-b\\\\-c&-d\n    \\end{pmatrix}.\n\\end{equation}\n\nWe can also characterize general two-player games in term of the stationary frequency distribution of strategies, with reflecting boundaries. This description accounts for more details in the stochastic dynamics, and it reveals a similar, transformative effect of large offspring variance. \nIf $\\delta_2$ is sufficiently large, namely $\\delta_2>M$, then modes of the stationary distribution can be moved from one boundary to the other boundary (dominance games, Fig.~\\ref{fig5}D), from the interior to the two boundaries (coexistence games, Fig.~\\ref{fig5}E), or from the two boundaries to the interior (coordination games, Fig.~\\ref{fig5}F). \nThese results reflect our ODE-based analysis above, and they show that sufficient offspring variance can reverse the evolutionary dynamics in an interacting population. These dramatic effects extend to games with more than two actions, such as rock-paper-scissors (Supplementary Fig.~S4).\n\n\n\n\nThese two analytical perspectives  underscore that large offspring variance can reshape the payoff structure of a game, producing dynamics classically seen in an entirely different game type.  So far, we have focused on the scaling factor $\\delta_2$, which governs how offspring variance grows with payoff, as opposed to $\\delta_1$, which governs the baseline offspring variance. The value of $\\delta_1$ can also profoundly influence evolutionary outcomes, although this cannot be seen from a deterministic perspective alone because $\\delta_1$ has no effect on stabilities of equilibria. Analysis of the stationary distribution shows that a large baseline variance ($\\delta_1B$) can transform any game into a coordination game (see Section S4.1 in Supplementary Information). An example of this result is shown in Fig.~\\ref{fig5}F, where even though $\\delta_2=25,000$ exceeds the carrying capacity, the stationary distribution is not unimodal around intermediate frequency. This is because the effect of  $\\delta_2$ here is offset by the effect of $\\delta_1$.\nThese results show that demographic noise, especially when offspring variance is high, can qualitatively change the evolutionary outcomes compared to  predictions of traditional analysis by replicator equations for fixed or infinite population size \\cite{Hofbauer1998}.\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\nThe question of how cooperation can be maintained is a longstanding and active area of research, spanning multiple disciplines. A large literature has produced compelling explanations for cooperation, but these typically rely on some form of population structure or repeated interactions. Here, we find that even in a well-mixed population with one-shot interactions, natural stochasticity in the total population size alone can favor cooperation that would otherwise be suppressed. For other types of social interactions, as well, demographic stochasticity can reverse the direction of evolutionary trajectories and produce behavioral outcomes that contravene classical expectations.\n\n\nIt is intuitively easier to invade a noisy population than a stable population. And so natural selection near carrying capacity prefers types not only with higher fecundity (greater mean offspring number), but also with lower reproductive noise (smaller offspring variance) \\cite{Parsons2007,Parsons2007a}. The reversal in the direction of selection in a stochastic population reflects this basic trade-off between offspring mean and offspring variance. A larger payoff produces higher fecundity but also greater noise in the reproduction process. Whether it is the mean or the variance in offspring number that dominates the course of evolution is determined by their relative importance, which is governed by $\\delta_2$ in our model.\nClassical models of populations with constant (or infinite) size neglect the effects of offspring variance altogether; but more realistic models, we have seen, permit regimes where offering variance is more important than fecundity.\n\n\n\nAlthough demographic noise has been studied extensively in population models, the underlying mechanism for our results is qualitatively different from those explored in prior studies. Most research on demographic noise has been restricted constant fitness for competing types \\cite{Parsons2007,Parsons2007a,Parsons2010,McKane2005,Butler2009,Hallatschek2007,Stollmeier2018,Wienand2017,Taitelbaum2020,Chotibut2017}, which does not provide a model of social interactions. However, Constable et al.~ analyzed a frequency-dependent fitness model, and they also found that demographic noise can reverse the direction of selection \\cite{Constable2016}. Their model is based on the production and consumption of a public good. One phenotype produces the public good, at a cost that reduces its baseline birth rate, while the other phenotype does not produce the public good. \nThey analyze the case when ``cooperators\" (who produce the public good) have a larger intrinsic carrying capacity than non-producers, and the larger carrying capacity then yields an evolutionary advantage by making producers more robust against invasion. \nThis mechanism is thus a stochastic form of $r$ versus $K$ selection \\cite{pianka1970r}, and it occurs when births and deaths follow Poisson processes. By contrast, in our model, the evolutionary advantage of cooperators arises even though both types have the same baseline birth rate and the same carrying capacity; \n\nand it arises only when the birth process related to payoff is sufficiently over-dispersed. \nThis mechanism is thus fundamentally different from a trade-off between baseline birth rate and carrying capacity of competing types in a Poisson model \\cite{Constable2016,Houchmandzadeh2012,Houchmandzadeh2015}, and it is more closely related to phenomena in population models with heavy-tailed offspring distributions \\cite{schweinsberg2000necessary,Eldon2006,Sargsyan2008,der2012dynamics}.\n\nAside from promoting cooperation in the prisoner's dilemma, demographic stochasticity also transforms outcomes in other forms of social interaction. Stochasticity can convert a snowdrift game into a stage-hunt game, for example, so that the stable co-existence expected in a deterministic or Poisson setting is transformed into bi-stability. Here, again, the underlying mechanism that reverses the evolutionary outcome is over-dispersion in the offspring contribution related to payoff, even when both types have the same baseline birth rate and carrying capacity.\n\nAll of our analyses have assumed a fast-growing population ($\\alpha \\gg s$), which rapidly reaches carrying capacity before any change in the relative frequencies of competing types. The dynamics of competition may be more complicated in a stochastic, slow-growing population, because their analysis cannot be reduced to a one-dimensional slow manifold. In this regime, fixation will take place before reaching carrying capacity. We can nonetheless derive approximations for the fixation probability in this regime as well (Section S4.2 in Supplementary Information), and, in the case of the donation game, we find that cooperation will be favored by selection provided $\\delta_2$ exceeds the initial population size, $\\delta_2>n_0$. This condition is typically easier to satisfy than Eq.~\\ref{eq:condition}, and it is confirmed by both numerical simulations and Monte Carlo simulations of the compound Poisson process (Supplementary Fig.~S5 and Fig.~S6). After cooperators or defectors fix, in this regime of a slow-growing population, the population will tend to grow logistically to its carrying capacity; but in this case the carrying capacity is larger for cooperators (Supplementary Fig.~S7), which provides an additional evolutionary advantage and greater chance of long-term persistence (Supplementary Fig.~S8).\n\n\n\nOur results highlight the strong impact of stochasticity on evolutionary outcomes in populations. The demographic stochasticity we have studied arises from intrinsic properties of birth and death processes, which have size of order $O(\\sqrt{n})$. As the population size grows towards infinity this form of stochasticity has little influence on evolutionary dynamics, which is consistent with the recent finding that migration in finite, group-structured populations can favor cooperators provided the population size is not too large \\cite{Braga2022}. Aside from intrinsic stochasticity during reproduction, real populations may also be subject to external noise, arising from exogenous variation in environmental conditions. Unlike demographic noise, exogenous noise  can be substantial even in population of arbitrary large size. Prior studies on environmental fluctuations, including fluctuations in selection intensity \\cite{Assaf2013}, carrying capacity \\cite{Wienand2017,Taitelbaum2020}, and payoff structure \\cite{Stollmeier2018}, have analyzed their effects by imposing an external noise term onto an otherwise classical, deterministic and continuous system of equations.  The effects of exogenous noise on discrete stochastic systems remain less explored, and they are likely to differ qualitatively from stochastic perturbations of continuous systems  \\cite{Durrett1994}. Coupling intrinsic demographic noise with external environmental noise may produce even more complicated effects, which remains a topic for future research.\n\nThe impact of stochasticity on strategic outcomes likely extends beyond the two-player\/two-action games we focused on, to include many aspects of non-human and human social behavior. Even if behavioral spread is caused by biased imitation, there is nonetheless variance in number of individuals who imitate a type, as well as physical variation in population sizes of interacting social groups as individuals move between social settings. Empirical data has documented burstiness, a form of over-dispersion, in  social interactions \\cite{Stehle2010,Goh2008}. Likewise, in the context of behavior during an epidemic, there is evidence of super-spreading individuals that cause over-dispersion in infectiousness \\cite{Tkachenko2021,Kirkegaard2021}, which may influence frequency-dependent competition among co-circulating variants. Extending our model and analysis to these settings remains an open topic for future research.\n\n\n\n\n\\clearpage\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}