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+{"text":"\\section{Introduction}\nHomotopy associative algebra, or $A_{\\infty}$-algebra is a certain generalization of a differential graded associative algebra. For the first time this concept appeared in the 1960s in the works of Jim Stasheff \\cite{stasheff}, \\cite{stashbook} in the context of Algebraic Topology. The theory continued developing both in the 1970s and the 1980s, but it wasn't until the 1990s that people realized that the same structures appeared in String Physics and Geometry, e.g. in the context of String Field Theory \\cite{zwiebach} and Mirror Symmetry \\cite{konts}. In this article, we are studying several examples of such algebraic objects related to the basic gauge (Super) Field Theories, following previous results obtained in  \\cite{lzgauge}, \\cite{azjpa}, \\cite{azcomm},  and related articles devoted to $L_{\\infty}$-algebras, e.g. \\cite{sftz}, \\cite{azjhep}, \\cite{azijmpa}, see also  \\cite{movshev} and recent paper \\cite{hohm}. \n\nFirst, let us give an idea of what $A_{\\infty}$-algebra is and present one of the reasons why it is a natural object from a mathematical point of view. Suppose, $\\mathcal{F} $ is a differential graded algebra (DGA), i.e. $\\mathcal{F} $ is a chain complex that has an associative bilinear operation satisfying the Leibniz rule with a differential. Assume, there is another complex $\\mathfrak{K}$, homotopically equivalent to $\\mathcal{F} $. Therefore, we can deduce that the cohomology of $\\mathfrak{K}$ is isomorophic to $\\mathcal{F} $, and both have the structure of an associative algebra. The natural question is the following: what happens on the level of chain complexes? It appears that we have a transfer of the algebraic structure from $\\mathcal{F} $ to $\\mathfrak{K}$ but in general the transferred product will not be associative. The structure induced from the bilinear operation and the homotopical equivalence is what we call an $A_{\\infty}$-algebra \\cite{gug}, \\cite{merk},\\cite{konts}. The induced bilinear operation $\\mu_2$ on $\\mathcal{F} $ will satisfy an associativity condition up to homotopy that is described by a trilinear operation $\\mu_3$. Moreover, \n(see e.g.~\\cite{merk},\\cite{konts},\\cite{markltransf},\\cite{hu},\\cite{keller}) one can continue and construct operations $\\mu_n$, satisfying ``higher\" bilinear associativity relations. Thus, if we are given the operations $\\mu_n$, $n=(1, 2,3, ...)$ satisfying such bilinear relations, we say that there is the structure of $A_{\\infty}$-algebra on $\\mathfrak{K}$. In fact, all these relations can be summarized in one: $\\partial^2=0$ (see Appendix A for details). \n\nAn important feature of the $A_{\\infty}$-algebras is that one can define a generalization of the generalized Maurer-Cartan (GMC) equation: \n\\begin{eqnarray}\nQ\\Phi+\\mu_2(\\Phi,\\Phi)+\\mu_3(\\Phi, \\Phi,\\Phi)+...=0,\n\\end{eqnarray}\nwhere $\\Phi$ has degree $1$. This equation possesses the following infinitesimal symmetry transformation:\n\\begin{eqnarray}\n&&\\Phi\\to Q\\Lambda+\\mu_2(\\Phi,\\Lambda)-\\mu_2(\\Lambda,\\Phi)+\\nonumber\\\\\n&&\\mu_3(\\Lambda,\\Phi,\\Phi)-\\mu_3(\\Phi,\\Lambda,\\Phi)+\\mu_3(\\Phi,\\Phi,\\Lambda)...,\n\\end{eqnarray}\nwhere $\\Lambda$ is infinitesimal and has degree 0.  \n\nIn physics, particularly in String Field Theory, $A_{\\infty}$-algebras appear in the context of the Batalin-Vilkovisky (BV) formalism (see e.g.~\\cite{kajiura}). Namely, for a given action functional satisfying the BV master equation, one can construct the nilpotent noncommutative odd vector field $\\mathbf{Q}$ encoding all the operations of the $A_{\\infty}$-algebra (see Appendix A).\n\nIn this article, we will be studying homotopy algebras related to (super)field theories, as well as their transfers. \n\nIn Section 2, as an invitation to the subject, we examine the $N=2$  D=3 Chern-Simons theory in the superfield formulation (this theory appears to be physically relevant, see e.g. \\cite{jafferis},\\cite{klebanov}). Using the corresponding BV action, we \nconstruct the superfield counterpart of the de Rham Complex in three dimensions that turns out to be a homotopy algebra instead of the usual DGA.\n\nIn Section 3, we consider the Yang-Mills (YM) theory in its usual second and  first order formulations. We describe the corresponding $A_{\\infty}$-algebras, for which GMC coincides with YM equations \nof motion, and show how they are related to the Batalin-Vilkovisky (BV) formalism. \nIn this case, the first order formulation gives us just a differential graded algebra, while the second order formulation leads to an $A_{\\infty}$-algebra where all the operations starting from the quadrilinear one vanish.\n\nIn the end of Section 3, we are showing that these two $A_{\\infty}$-algebras are related by the transfer formula. This is one of the simplest non-artificial examples of a transfer formula. \nThis way, we observe that some of the basic field theory reformulations, like the transfer from the second order action to the first order action, \ncan be explained purely on the level of homotopical algebra (see also \\cite{sftz}). \n\nIn Section 4, we generalize these results to the supersymmetric case. We use the superfield formulation of \n$N=1$ supersymmetric (SUSY) YM theory and study the homotopical algebras corresponding to the second and first order formulations. In this case, $A_{\\infty}$-algebras are ``richer\", i.e. we have an infinite chain of polylinear operations in both first and second order formulations. As in the previous example, they are related by the transfer formula.\nIn the end, we discuss some further developments and future tasks, in particular the meaning of the representation of real superfield via complex ones from the homotopical algebra point of view. \\\\\n\n\\noindent {\\bf Acknowledgements.} We are grateful to K. Costello, M. Markl, M. Movshev and J. Stasheff for illuminating discussions. We would like to express our gratitude to the wonderful environment of  Simons Summer Workshops where this work was partially done. A.M.Z. is supported by AMS Simons travel grant.\n\n\\section{ Homotopic de Rham complex in $N=2$ 3D superspace.}\n\\noindent{\\bf 2.1. De Rham complex in 3D and BV Chern-Simons.} Consider the DGA related to the de Rham complex in three dimensions:\n\\begin{eqnarray}\n0\\to\\Omega^0\\xrightarrow{d}\\Omega^1\\xrightarrow{d}\\Omega^2\\xrightarrow{d}\\Omega^3\\to 0,\n\\end{eqnarray}\nwhere $\\Omega^i$ is the space of differential forms of degree $i$. Moreover, this complex possesses a natural \npairing given by the integral of the wedge product of forms and trace, giving a cyclic structure of the de Rham DGA. The same DGA structure can be constructed for the forms with values in $U({\\mathfrak{g}})$ where $\\mathfrak{g}$ is some semisimple Lie algebra. \nThen the formal action functional, associated with this cyclic DGA (see Appendix A), corresponding to the element $\\Phi=c+\\mathbf{A} +\\mathbf{A} ^*+c^*$, where \n$c\\in \\Omega^{0}[1]\\otimes{\\mathfrak{g}}$, $\\mathbf{A} \\in \\Omega^1\\otimes{\\mathfrak{g}}$, $\\mathbf{A} ^*\\in \\Omega^2[-1]\\otimes{\\mathfrak{g}}$, $c^*\\in\\Omega^3[-2]\\otimes{\\mathfrak{g}}$ has the following form:\n\\begin{eqnarray}\nS[\\Phi]=S_{CS}[\\mathbf{A} ]+\\int d^3x{\\rm Tr}(d_{\\mathbf{A} }c\\wedge \\mathbf{A} ^*+[c,c]c^*),\n\\end{eqnarray}\nwhere $S_{CS}[\\mathbf{A} ]=\\int d^3x (\\mathbf{A}  \\wedge d\\mathbf{A} +\\frac{1}{3}[\\mathbf{A} ,\\mathbf{A} ]\\wedge\\mathbf{A} )$ is the usual 3D Chern-Simons action. \\\\\n\n\\noindent {\\bf 2.2. SUSY de Rham complex and SUSY Chern-Simons.} Let us consider N=2 3D Euclidean superspace. \nThis is a superspace where in addition to three even coordinates there are 4 odd coordinates that are the components of Weyl spinors $\\theta^{\\alpha}$, $\\bar\\theta^{\\alpha}$. As in the N=1 4D case, one can consruct  the superderivatives $D_{\\alpha}$, $\\bar D_{\\alpha}$. The difference is that one can make new Lorentz scalars out of them, like $D_{\\alpha}\\bar D^{\\alpha}$ (see Appendix B.3 for details). \nThe relations between superderivatives allow to construct the following complex:\n\\begin{eqnarray}\\label{susydr}\n0\\xrightarrow{ }\\Theta\\xrightarrow{id}\\Sigma\n\\xrightarrow{\\bar D^{\\alpha}D_{\\alpha}}\\t \\Sigma\\xrightarrow{{\\bar D}^2}\\t \\Theta\\to 0.\n\\end{eqnarray} \nHere $\\Theta\\cong\\t \\Theta$ is the space of chiral scalar fields (if $\\Lambda \\in \\Theta$, then $\\bar D_{\\alpha}\\Lambda=0$), and $\\Sigma\\cong\\t\\Sigma$ is the space of complex scalar fields. \nAfterwards, we will denote this complex $(SdR^{\\cdot}, d_{S})$. \nIt is not hard to show that the 3D de Rham complex can be embedded in \\rf{susydr}. Moreover,  one can define a nondegenerate pairing on \\rf{susydr} similar to $\\langle\\cdot, \\cdot \\rangle_\\mathcal{F} $:\n\\begin{eqnarray}\n&&\\langle\\cdot, \\cdot \\rangle: SdR^{i}\\otimes SdR^{3-i}\\to \\mathbb{C},\\nonumber\\\\\n&&\\langle{\\Lambda,\\t\\Lambda}\\rangle=\\int d^3 xd^2\\theta \\Lambda\\t \\Lambda,\\quad \\langle V,\\t V\\rangle=\\int d^3 x d^4\\theta V\\t V,\\nonumber\n\\end{eqnarray}\nwhere $\\Lambda\\in SdR^{0}$, $V\\in SdR^{2}$, $\\t V\\in SdR^{2}$, $\\t \\Lambda\\in SdR^{3}$. This pairing satisfies  \nthe  familiar property \\rf{cycpairing}. Therefore, we can hope for the existence of a cyclic $A_{\\infty}$-algebra related to the complex \\rf{susydr}. \nIn order to construct it, we can consider the following action giving the superfield formulation to the N=2 Chern-Simons theory \\cite{zupnik}:\n\\begin{eqnarray}\\label{zup}\nS_{susyCS}=\\int d^3 xd^4\\theta\\int_0^1dt(V\\bar D^{\\alpha}(e^{-tV}D_{\\alpha}e^{tV})),\n\\end{eqnarray}\nwhere $V\\in \\Sigma\\otimes \\mathfrak{g}$. This action has the symmetry that is the same for all the supersymmetric theories we are considering here, namely  \n\\begin{eqnarray}\ne^V\\to e^{\\Lambda}e^Ve^{\\bar \\Lambda},\n\\end{eqnarray}\nwhere $\\Lambda, \\bar \\Lambda$  are respectively chiral and atichiral scalar fields with values in Lie algebra $\\mathfrak{g}$. The corresponding infinitesimal version of the symmetry is:\n\\begin{eqnarray}\n&&V\\to V+\\delta_{\\Lambda, \\bar \\Lambda}V=V+\\frac{1}{2}L_{V}(\\Lambda-\\bar \\Lambda+\\coth(\\frac{1}{2}L_V)(\\Lambda+\\bar \\Lambda)),\n\\end{eqnarray} \nwhere $L_V\\cdot=[V,\\cdot]$. Let us restrict the symmetry to the chiral transformations: $e^V\\to e^{\\Lambda}e^V$. Then, the BV modification of the action \\rf{zup} is (cf. \\rf{bvsymact}):\n\\begin{eqnarray}\nS^{BV}_{susyCS}=S_{susyCS}+\\int d^3 x d^4 \\theta (\\delta_C (V) V^*)+\n\\int d^3x d^2\\theta ([C,C]C^*),\n\\end{eqnarray}\nwhere $C\\in SdR^{0}[+1]$, $V\\in SdR^{2}$, $V^*\\in SdR^{2}[-1]$, $C^*\\in SdR^{3}[-2]$. This action will generate the odd vector field that according to the results of Appendix B generates the $A_{\\infty}$-algebra. It is obvious that the corresponding chain complex, where the $A_{\\infty}$ operations act, is the one from \\rf{susydr}. \nTherefore, we have the following proposition. \\\\\n\n\\noindent {\\bf Proposition 2.1.} {\\it Complex $(SdR^{\\cdot}, d_{S})$ possesses a nontrivial $A_{\\infty}$ structure (with all the operations nonvanishing) provided by the action functional $S^{BV}_{susyCS}$}.\\\\\n\n\\noindent Let us express the bilinear operation explicitly:\n\n\\begin{center}\n$\\nu_2(f_1,f_2)$=\\\\\n\\vspace{5mm}\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n\\backslashbox{$ f_2$}{$f_1$}&          $\\Lambda_1$ & $V_1$ & $\\t V_1$ & $\\t\\Lambda_1$ \\\\\n\\hline\n$\\Lambda_2$ &                  $\\Lambda_1\\Lambda_2$    &$\\frac{1}{2}V_1\\Lambda_2$ & \n$\\frac{1}{2}\\t V_1\\Lambda_2$ &$\\t\\Lambda_1\\Lambda_2$\\\\\n\\hline\n$V_2$     &  $\\frac{1}{2}\\Lambda_1V_2 $ & $(V_1,V_2)_h$   &  $-\\frac{1}{2}\\bar D^2(\\t V_1V_2)$  &0\\\\\n                    \n\\hline\n$\\t V_2$ & $\\frac{1}{2}\\Lambda_1\\t V_2$ &  $-\\frac{1}{2}\\bar D^2(V_1\\t V_2)$ &    0&0\\\\\n\\hline\n$\\t\\Lambda_2$                     & $\\Lambda_1\\t\\Lambda_2$ &   0 &    0&0\\\\\n\\hline\n\\end{tabular}\\\\\n\\end{center}\n\\noindent where $\\Lambda_1, \\Lambda_2\\in SdR^0$, $V_1, V_2\\in SdR^1$, $\\t V_1, \\t V_2\\in SdR^2$, $\\t \\Lambda_1, \\t \\Lambda_2\\in SdR^3$, and \n\\begin{eqnarray}\n(V_1,V_2)_h=-\\frac{1}{2}D_{\\alpha}V_1\\bar D^{\\alpha}V_2-\\frac{1}{2}\\bar D^{\\alpha}V_1D_{\\alpha}V_2.\n\\end{eqnarray}\n{\\bf Corollary 2.1.} {\\it The operation $\\nu_2$ is homotopy associative on $(SdR^{\\cdot}, d_{S})$}. \\\\\n\n\\noindent The equations of motion for $S_{susyCS}$ are \\cite{zupnik}:\n\\begin{eqnarray}\nf(L_V)\\bar D^{\\alpha}(e^{-V}D_{\\alpha}e^V)=0,\n\\end{eqnarray}\nwhere $f(x)=\\frac{e^{x}-1}{x}$. \nThis equation is the GMC equation for the resulting $A_{\\infty}$-algebra, which is the \nsuperfield generalization of the Maurer-Cartan equation in 3D: $d\\mathbf{A} +\\mathbf{A} \\wedge\\mathbf{A} =0$. \\\\\n\n\\noindent{\\bf Remark 2.1.} Also, we note that in principle, one can extend the complex \\rf{susydr} by means of the space of antichiral scalars (compare to the SUSY Yang-Mills case in subsection 3.4.). The resulting complex will be \n\\begin{eqnarray}\n\\xymatrixcolsep{40pt}\n\\xymatrixrowsep{3pt}\n\\xymatrix{\n0\\ar[r]&\\Theta \\ar[r]^{id} & \\Sigma \\ar[r]^{\\bar D^{\\alpha}D_{\\alpha}}  &\\tilde\\Sigma \\ar[r]^{\\bar D^2}\\ar[ddr]^{D^2} & \\tilde\\Phi\\ar[r]&0\\\\\n &\\oplus&&&\\oplus\n &&          &  &&\\\\\n &\\bar{\\Theta}\\ar[uur]^{-id}&   &&\\tilde{\\bar{\\Theta}}}\\nonumber\n\\end{eqnarray}\n\nThe resulting $A_{\\infty}$-algebra, generated by the action\n\\begin{eqnarray}\n&&S^{fullBV}_{susyCS}=S_{susyCS}+\\int d^3 x d^4 \\theta (\\delta_{C,\\bar C} (V) V^*)+\\nonumber\\\\\n&&\\int d^3x d^2\\theta ([C,C]C^* +[\\bar C,\\bar C]\\bar C^*).\n\\end{eqnarray}\n will describe the full symmetry of the action \\rf{zup}. \n\n\n\\section{Homotopy algebras of Yang-Mills theory.}\n\\noindent {\\bf 3.1. The $A_{\\infty}$-algebra of the Maxwell complex \\cite{lzgauge}.}\nWe consider Maxwell complex\n\\begin{eqnarray}\n0\\to\\mathcal{F}^{0}\\xrightarrow{\\mathcal{Q}}\\mathcal{F}^{1}\\xrightarrow{\n\\mathcal{Q}}\n\\mathcal{F}^{2}\\xrightarrow{\\mathcal{Q}}\\mathcal{F}^{3}\\to 0,\n\\end{eqnarray}\nwhere the spaces $\\mathcal{F}^i$ and the action of $\\mathcal{Q} $ are as follows:\n\\begin{eqnarray}\\label{maxwell}\n0\\xrightarrow{ }\\Omega^{0}(M)\\xrightarrow{\\mathrm{d}}\\Omega^{1}(M)\n\\xrightarrow{\\ud*\\ud}\\Omega^{3}(M)\\xrightarrow{\\mathrm{d}}\\Omega^{4}(M)\\to 0,\n\\end{eqnarray}\nwhere $M$ stands for any four dimensional (pseudo)Riemannian manifold (chain complex \\rf{maxwell} was studied also in \\cite{gover}, \\cite{waldron} in a different context). \nThis complex has a structure of homotopy commutative associative algebra \\cite{lzgauge}. Let us introduce the corresponding multilinear operations:\n\\begin{eqnarray}\n&&(\\cdot, \\cdot)_h: \\mathcal{F}^i\\otimes \\mathcal{F}^j\\to \\mathcal{F}^{i+j},\\nonumber\\\\\n&&(\\cdot, \\cdot, \\cdot)_h: \\mathcal{F}^i\\otimes \\mathcal{F}^j\\otimes \\mathcal{F}^k\\to \n\\mathcal{F}^{i+j+k-1}.\n\\end{eqnarray}\n\nThe bilinear operation is defined by means of the following table:\\\\\n\n$(f_1,f_2)_h$=\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n\\backslashbox{$ f_2$}{$f_1$}&\\makebox{$v$} & \\makebox{$\\mathbf{A} $} & \\makebox{$\\mathbf{V} $} & \\makebox{$a$} \\\\\n\\hline\n$w$ & $vw$& $\\mathbf{A}  w$& $\\mathbf{V}  w$&$a w$\\\\\n\\hline\n$\\mathbf{B} $ &$v\\mathbf{B} $  & $(\\mathbf{A} ,\\mathbf{B} )$   &   $\\mathbf{B} \\wedge\\mathbf{V} $&0\\\\\n\\hline\n$\\mathbf{W} $ & $v\\mathbf{W} $ &  $ \\mathbf{A} \\wedge\\mathbf{W} $ &    0&0\\\\\n\\hline\n$b$ & $vb$ &   0 &    0&0\\\\\n\\hline\n\\end{tabular}\\\\\n\n\\noindent where $f_1$ takes values in the set $\\{v, \\mathbf{A} , \\mathbf{V} , a\\}$ from the first row, and $f_2$ takes values in the set $\\{w, \\mathbf{B} , \\mathbf{W} , b\\}$ from the first column. Other elements in the table represent the value of bilinear operation \n$(f_1,f_2)_h$ for appropriately chosen $f_1$ and $f_2$. In the table above $v,w\\in\\mathcal{F}^0$; \n$\\mathbf{A} ,\\mathbf{B} \\in \\mathcal{F}^1$; $\\mathbf{V} ,\\mathbf{W} \\in\\mathcal{F}^2$; $a,b\\in\\mathcal{F}^3$. \nThe bilinear operation $(\\mathbf{A} ,\\mathbf{B} )$ is defined as follows:\n\\begin{eqnarray}\n&&(\\mathbf{A} ,\\mathbf{B} )=\\mathbf{A} \\wedge(*\\mathrm{d}\\mathbf{B} )-(*\\mathrm{d}\\mathbf{A} )\\wedge\\mathbf{B} +\\mathrm{d} *(\\mathbf{A} \\wedge \\mathbf{B} ).\n\\end{eqnarray}\nThe operation $(\\ \\cdot, \\ \\cdot, \\ \\cdot)_h$ is defined to be nonzero only when all the arguments belong to $\\mathcal{F} ^1$. For $\\mathbf{A} $, \n$\\mathbf{B} $, $\\mathbf{C} $$\\in \\mathcal{F} ^1$ we have: \n\\begin{eqnarray}\n(\\mathbf{A} ,\\mathbf{B}  ,\\mathbf{C}  )_h=\\mathbf{A} \\wedge*(\\mathbf{B} \\wedge\\mathbf{C} )-(*(\\mathbf{A} \\wedge\\mathbf{B} ))\\wedge\\mathbf{C} .\n\\end{eqnarray}\nBy the direct calculations (see \\cite{lzgauge}), one can show that these operations satisfy the following relations providing the structure of homotopy commutative $A_{\\infty}$-algebra. \\\\\n\n\\noindent{\\bf Proposition 3.1.} {\\it Let $a_1,a_2, a_3, a_4,b, c$ $\\in$ $\\mathcal{F} $. Then the following relations hold:\n\\begin{eqnarray}\n&&\\mathcal{Q} (a_1,a_2)_h=(\\mathcal{Q}  a_1,a_2)_h+(-1)^{n_{a_1}}(a_1,\\mathcal{Q}  a_2)_h,\\nonumber\\\\\n&&(a_1,a_2)_h=(-1)^{n_{a_1}n_{a_2}}(a_2,a_1)_h,\\nonumber\\\\\n&&\\mathcal{Q} (a_1,a_2, a_3)_h+(\\mathcal{Q}  a_1,a_2, a_3)_h+(-1)^{n_{a_1}}(a_1,\\mathcal{Q}  a_2, a_3)_h+\\nonumber\\\\\n&&(-1)^{n_{a_1}+n_{a_2}}( a_1, a_2, \\mathcal{Q}  a_3)_h=((a_1,a_2)_h,a_3)_h-(a_1,(a_2,a_3)_h)_h\n,\\nonumber\\\\\n&&(-1)^{n_{a_1}}(a_1,(a_2,a_3,a_4)_h)_h+((a_1,a_2,a_3)_h,a_4)_h=\\nonumber\\\\\n&&((a_1,a_2)_h,a_3,a_4)_h-(a_1,(a_2,a_3)_h, a_4)_h+(a_1,a_2,(a_3,a_4)_h)_h,\n\\nonumber\\\\\n&&((a_1,a_2, a_3)_h,b,c)_h=0.\n\\end{eqnarray} }\nIf we tensor complex $(\\mathcal{F} ^{\\bf\\cdot},\\mathcal{Q} )$ with some universal enveloping algebra for some reductive Lie algebra $\\mathfrak{g}$, one obtains that inherited operations $(\\ \\cdot,\\ \\cdot)_h$ and $(\\ \\cdot,\\ \\cdot, \\ \\cdot)_h$ satisfy the relations of  $A_{\\infty}$-algebra on the resulting complex $(\\mathcal{F} _{\\mathfrak{g}}^{\\bf\\cdot},\\mathcal{Q} )$. \n\nIf the manifold $M$ is compact, or the fields are with the compact support, one can show that the Maxwell complex possesses a pairing \n\\begin{eqnarray}\n\\langle f_1,f_2\\rangle=\\int_M Tr(f_1\\wedge f_2),\n\\end{eqnarray}\n which makes the $A_{\\infty}$-algebra introduced above to be cyclic. \nNamely, one can define\n\\begin{equation}\n\\{\\cdot,..., \\cdot\\}_h: \\mathcal{F} _{\\mathfrak{g}}\\otimes ...\\otimes \\mathcal{F} _{\\mathfrak{g}}\\to \\mathbb{C} \n\\end{equation}\nin the following way:\n\\begin{eqnarray}\n&&\\{f_1,f_2\\}_h=\\langle \\mathcal{Q}  f_1, f_2\\rangle, \\quad \\{f_1,f_2,f_3\\}_h=\\langle (f_1, f_2)_h,f_3\\rangle,\\nonumber\\\\\n&&\\{f_1,f_2,f_3, f_4\\}_h=\\langle (f_1, f_2,f_3)_h,f_4\\rangle.\n\\end{eqnarray}\nFor more details about the cyclic structures see Appendix A. One of the most important applications of the cyclic structure is that one can write the following action functional in the form that is the ``homotopy'' generalization of the Chern-Simons action functional \n\\begin{eqnarray}\nS_{HCS}[f]=\\frac{1}{2}\\{f,f\\}+\\frac{1}{3}\\{f,f,f\\}+\\frac{1}{4}\\{f,f,f,f\\}\n\\end{eqnarray}\nsuch that $f\\in\\mathcal{F} _{\\mathfrak{g}}^1$ and the variation of this functional with respect to $f$ leads to the generalized Maurer-Cartan equation for $f$.\nThe Maurer-Cartan equation and its symmetries in the case of the $A_{\\infty}$-algebra considered above\n\\begin{eqnarray}\n\\mathcal{Q} \\mathbf{A} +(\\mathbf{A} ,\\mathbf{A} )_h+(\\mathbf{A} ,\\mathbf{A} ,\\mathbf{A} )_h=0\\nonumber\\\\\n\\mathbf{A} \\to \\mathbf{A} +\\epsilon (\\mathcal{Q}  u +(u,\\mathbf{A} )_h-(\\mathbf{A} ,u)_h),\n\\end{eqnarray}\nwhich lead to Yang-Mills equations and the gauge symmetry if $f=\\mathbf{A} $, where $\\mathbf{A} \\in\\mathcal{F} _{\\mathfrak{g}}^1$ and $u\\in\\mathcal{F} ^0_\\mathfrak{g}$.\n\nIn the case when $f=c+\\mathbf{A} +\\mathbf{A} ^*+c^*$, where $c\\in\\mathcal{F} _{\\mathfrak{g}}^0[1]$, $\\mathbf{A} \\in \\mathcal{F} _{\\mathfrak{g}}^1$, $\\mathbf{A} ^*\\in \\mathcal{F} _{\\mathfrak{g}}^2[-1]$,  $c^*\\in\\mathcal{F} ^3[-2]$, such that the grading of the element from $\\mathcal{F} ^i_{\\mathfrak{g}}[j]$ is $i+j$,\nthe action above leads to the BV Yang-Mills action:\n\\begin{eqnarray}\nS_{BVYM}=\\int_M{\\rm Tr}({\\bf F}\\wedge *{\\bf F}+\\mathbf{A} ^*\\wedge \\mathrm{d}_{\\mathbf{A} }c+[c,c]c^*).\n\\end{eqnarray}\nIt is well-known that this action satisfies the so-called BV master equation that, according to the general principle, leads to the $A_{\\infty}$-algebra on the complex $(\\mathcal{F} ^{\\cdot},\\mathcal{Q} )$ (see Appendix A). \n\n\n\n\n \nIn the next section, we will construct the associative algebra related to the first order Yang-Mills theory. \\\\\n \n\\noindent {\\bf 3.2. The first order Maxwell complex and the related associative algebra \\cite{costello}.} \nLet us consider the following complex: \n\\begin{eqnarray}\n0\\xrightarrow{ }\\mathfrak{K}^0\\xrightarrow{\\t\\mathcal{Q}  }\\mathfrak{K}^1\\xrightarrow{ \\t\\mathcal{Q} }\\mathfrak{K}^2\\xrightarrow{ \\t\\mathcal{Q} }\\mathfrak{K}^3\\xrightarrow{ }0\n\\end{eqnarray}\nsuch that $\\mathfrak{K}^0=\\Omega^{0}(M)$, $\\mathfrak{K}^{1}=\\Omega^{1}(M)\\oplus\\Omega_+^{2}(M)$, $\\mathfrak{K}^{2}=\\Omega^{3}(M)\\oplus\\Omega^{2}_+(M)$, \n$\\mathfrak{K}^{3}=\\Omega^{4}(M)$ and the differential $\\t \\mathcal{Q} $ acts as follows:\n\\[\n\\xymatrixcolsep{30pt}\n\\xymatrixrowsep{3pt}\n\\xymatrix{\n0\\ar[r]&\\Omega^0(M) \\ar[r]^\\mathrm{d} & \\Omega^1(M) \\ar[ddddr]  &\\Omega^3(M) \\ar[r]^\\mathrm{d} & \\Omega^4(M)\\ar[r]&0\\\\\n &\\quad &     & _{-\\mathrm{d}}\\quad    &&\\\\\n && \\bigoplus & \\bigoplus     &&\\\\\n &&           & _{\\mathrm{d}_{+}}\\quad    &&\\\\\n && \\Omega^2_+(M) \\ar[r]_{-Id} \\ar[uuuur]  & \\Omega^2_+(M)&\n}\n\\]\nwhere $\\mathrm{d}_{+}=\\mathrm{d}+*\\mathrm{d}$ and $\\Omega^2_+(M)$ is the space of self-dual 2-forms on the manifold $M$.  \nOne can define a bilinear operation on the resulting complex:\n\\begin{center}\n$\\mu (f_1,f_2)$=\n\\end{center}\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n\\backslashbox{ $f_2$}{$f_1$}&          $v_1$ & $(\\mathbf{A} _1,{\\bf F}_1)$ & $(\\mathbf{W} _1,{\\bf G}_1)$ & $a_1$ \\\\\n\\hline\n$v_2$ &                  $v_1v_2$    &$(v_2\\mathbf{A} _1,v_2 {\\bf F}_1)$ &$(v_2\\mathbf{W} _1,v_2{\\bf G}_1)$ &$v_2a_1$\\\\\n\\hline\n$(\\mathbf{A} _2,{\\bf F}_2)$     &  $(v_1\\mathbf{A} _2,v_1{\\bf F}_2)$ & $(\\mathbf{A} _2\\wedge {\\bf F}_1-$   &   $-\\mathbf{A} _2\\wedge \\mathbf{W} _1-$&0\\\\\n                               &                               &    $\\mathbf{A} _1\\wedge {\\bf F}_2,2P_{+}(\\mathbf{A} _1\\wedge\\mathbf{A} _2)$   & ${\\bf F}_2\\wedge {\\bf G}_1$&\\\\                    \n\\hline\n$(\\mathbf{W} _2,{\\bf G}_2)$ & $v_1\\mathbf{W} _2$ &  $-\\mathbf{A} _1\\wedge \\mathbf{W} _2-$ &    0&0\\\\\n&&                        ${\\bf F}_1\\wedge {\\bf G}_2$&&\\\\\n\\hline\n$a_2$                     & $v_1a_2$ &   0 &    0&0\\\\\n\\hline\n\\end{tabular}\\\\\n\\vspace{3mm}\n\n\\noindent where $P_{+}=\\frac{1+*}{2}$ is the projection operator on $\\Omega_+^2(M)$. \nHere $f_1,f_2$ take values in the set of variables with prime and  double prime correspondingly. \nOther elements in the table correspond to the appropriate values of $\\mu(f_1,f_2)$. In the table above \n$v_1,v_2\\in \\mathfrak{K}^0$; $(\\mathbf{A} _1,{\\bf F}_2), (\\mathbf{A} _2,{\\bf F}_2)\\in\\mathfrak{K}^1=\\Omega^1(M)\\oplus \\Omega^2_+(M)$;  $(\\mathbf{W} _1,{\\bf G}_1),(\\mathbf{W} _2,{\\bf G}_2)\\in\\mathfrak{K}^2=\\Omega^{3}(M)\\oplus \\Omega_+^{2}(M)$;  $a_1,a_2\\in\\mathfrak{K}^3$. It is not hard to see that this operation gives to the complex above the structure of differential graded abelian algebra. If we tensor it with some Lie algebra $\\mathfrak{g}$, \nwe will find that it possesses the cyclic structure, where the corresponding pairing on $(\\mathfrak{K},\\t Q)$ is defined by the same formula as in the previous section. \nOne can obtain that it is related to the following action:\n\\begin{eqnarray}\nS=\\int_M{\\rm Tr}\\Big({\\bf F}\\wedge{\\bf F}+{\\bf F}\\wedge (\\mathrm{d} \\mathbf{A} +\\mathbf{A} \\wedge\\mathbf{A} )\\Big),\n\\end{eqnarray}\nwhere $F\\in\\Omega^{+}(M)$, which is equivalent to the usual YM theory. \nHere we note that we could choose the space of anti self-dual 2-forms $\\Omega^2_-(M)$ instead of $\\Omega^2_+(M)$ in the complex above, and the resulting complex will also have the structure of differential graded abelian algebra.\n\\\\\n\n\\noindent {\\bf 3.3. Transfer of the $A_{\\infty}$ structure.} \nSuppose, $\\mathcal{F} $ is a differential graded algebra (DGA), i.e. it is a chain complex with a differential $Q'$, and $m$ is an associative bilinear operation on $\\mathfrak{K}$. Suppose we have a chain complex $\\mathfrak{K}$ with a differential $Q'$, which is homotopically equivalent to $\\mathcal{F} $, i.e. there are chain maps $f: \\mathcal{F} \\to \\mathfrak{K}$ and $g:\\mathfrak{K}\\to \\mathcal{F} $, such that  $fg-id=Q'H+HQ'$, where $H$ is of degree $-1$. Then $\\mathcal{F} $ and $\\mathfrak{K}$ are quasiisomorhic, i.e. we have isomorphism on the level of cohomology: $H_{Q'}^*(\\mathfrak{K})\\cong H_{Q}^*(\\mathcal{F} )$ and $\\mu_2(\\cdot,\\cdot)=g\\circ m(f(\\cdot), f(\\cdot))$ is an associative bilinear operation on $H_{Q'}^*(\\mathfrak{K})$. Therefore, we have a $transfer$ of the associative multiplication on the level of cohomology. \nAs we explained in the introduction, there is a transfer on the level of complexes, but not of the associative algebra. The structure induced from the operation $m$ and the homotopical equivalence is the $A_{\\infty}$-algebra. An elementary calculation shows that \n\\begin{eqnarray}\n&&\\mu_2(\\mu_2(a,b),c)-\\mu_2(a,\\mu_2(b,c))=\\\\\n&&Q\\mu_3(a,b,c)+\\mu_3(Qa,b,c)+(-1)^{|a|}\\mu_3(a,Qb,c)+(-1)^{|a|+|b|}\\mu_3(a,b,Qc),\\nonumber\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\\label{tran}\n&&\\mu_3(a,b,c)=\\\\\n&&g\\circ m(H\\circ m(f(a),f(b)),f(c))-(-1)^{|a|}g\\circ m(f(a),H\\circ m(f(a),f(b))\\nonumber\n\\end{eqnarray}\nis a trilinear operation of degree $-1$. One can find that $\\mu_3$ also satisfies bilinear ``higher associativity\" relations with $\\mu_2$ and $\\mu_1\\equiv Q'$ and leads to next operation $\\mu_4$. One can continue the process and in general obtain the infinite amount of operations satisfying certain billinear relations. The general construction of such $\\mu_n$ can be described by \ncertain sum over the expressions parametrized by tree graphs with vertices corresponding to operation $m$ and edges corresponding to homotopy $H$ \\cite{merk}, \\cite{konts}. \nIn fact, there is the more general statement: the differential graded algebra structure can be replaced by $A_{\\infty}$ structure, and it again will be transferred to the $A_{\\infty}$-algebra (see Appendix A and e.g. \\cite{markltransf}). \n\nNow we will show that one can transfer the $A_{\\infty}$ structure from the first order complex to the Maxwell one, according to \\cite{markltransf}. Firstly, we will show that the Maxwell complex and the first order complex are quasiisomorphic.  \nNamely let us construct the maps $g:(\\mathfrak{K}^{\\cdot}, \\t \\mathcal{Q} )\\to  (\\mathcal{F} ^{\\cdot}, \\mathcal{Q} )$, $f:\n(\\mathcal{F} ^{\\cdot}, Q)\\to  (\\mathfrak{K}^{\\cdot}, \\t \\mathcal{Q} )$ such that their composition is homotopic to identity. \nThe explicit expression for $f$ and $g$ are:\n\\begin{eqnarray}\n&& f(u)=u,\\quad f(\\mathbf{A} )=(\\mathbf{A} ,2P_{+}\\mathrm{d} \\mathbf{A} ), \\quad f(\\mathbf{V} )=(\\mathbf{V} ,0),\\quad f(v)=v,\\nonumber\\\\\n&& g(u)=u, \\quad g((\\mathbf{A} ,{\\bf F}))=\\mathbf{A} ,\\quad g((\\mathbf{V} , \\mathbf{G}))=d\\mathbf{G}+\\mathbf{V} ,\\quad g(v)=v.\n\\end{eqnarray}\nHere $u\\in\\Omega^{0}$, $\\mathbf{A} \\in\\Omega^{1}$, ${\\bf F},\\mathbf{G}\\in\\Omega_{+}^2$, \n$\\mathbf{V} \\in \\Omega^3$, $v\\in\\Omega^4$.\nTherefore, \n\\begin{eqnarray}\ng\\circ f=id,\\quad  f\\circ g=id +\\t \\mathcal{Q}  H +H\\t \\mathcal{Q} \n\\end{eqnarray}\nHere $H$ is the homotopy on the complex $(\\mathfrak{K}^{\\cdot},\\t \\mathcal{Q} )$, i.e. the map \n$H: \\mathfrak{K}^{i}\\to\\mathfrak{K}^{i-1}$, and it is nonzero only on $\\mathfrak{K}^2$. The explicit formula is:\n\\begin{eqnarray}\n H((\\mathbf{V} ,\\mathbf{G}))=(0,\\mathbf{G})\n\\end{eqnarray}\nHence,\n\\begin{eqnarray}\n&& f\\circ g((\\mathbf{A} ,\\mathbf{F}))=(\\mathbf{A} ,2P_{+}d\\mathbf{A} ), \\quad f\\circ g((\\mathbf{V} ,\\mathbf{G}))=(d\\mathbf{G}+\\mathbf{V} ,0),\\nonumber\\\\\n&& \\t \\mathcal{Q}  H((\\mathbf{V} ,\\mathbf{G}))=(d\\mathbf{G},-\\mathbf{G}), \\quad H\\t \\mathcal{Q} ((\\mathbf{A} ,\\mathbf{F}))=(0,2P_{+}d\\mathbf{A} -\\mathbf{F}).\n\\end{eqnarray}\nTherefore we have a Proposition.\\\\\n\n\\noindent{\\bf Proposition 3.2.} {\\it \ni)The complexes $(\\mathcal{F} ^{\\cdot},\\mathcal{Q} )$, $(\\mathfrak{K}^{\\cdot}, \\t \\mathcal{Q} )$ are \nquasiisomorphic and homotopically equivalent provided by the maps $f,g$.\\\\\nii)Under the homotopy equivalence, the structure of DGA (from subsection 2.2) on $(\\mathfrak{K}^{\\cdot}, \\t \\mathcal{Q} )$ is transferred to the described above $A_{\\infty}$ structure (from subsection 2.1)on $(\\mathcal{F} ^{\\cdot},\\mathcal{Q} )$ .}\\\\\n\n\\noindent{\\bf Proof.} We have proved $(i)$ above. Let us prove $(ii)$. Namely, consider for example how the \n$A_{\\infty}$ structure is transferred from $(\\mathfrak{K}^{\\cdot},\\t \\mathcal{Q} )$ to $(\\mathcal{F} ^{\\cdot},\\mathcal{Q} )$. \nThe bilinear and trilinear operations that are transferred have the following form \\rf{tran}:\n\\begin{eqnarray}\n&&(a_1,a_2)'_h=g\\circ\\mu(f(a_1),f (a_2)), \\nonumber\\\\\n&&(a_1,a_2,a_3)'_h=g\\circ\\mu(f(a_1),H\\circ \\mu(f(a_2),f(a_3))-\\nonumber\\\\\n&&(-1)^{|a_1|}\\mu\\circ (H\\circ\\mu(f(a_1),f(a_2)), f(a_3)), \n\\end{eqnarray}\nwhere $a_i\\in (\\mathcal{F} ^{\\cdot},\\mathcal{Q} )$ $(i=1,2,3)$. The simple calculation shows that $(a_1,a_2)'_h$, $(a_1,a_2,a_3)'_h$ \ncoincide with the defined earlier $(a_1,a_2)_h$, $(a_1,a_2,a_3)_h$. $\\blacksquare$\\\\\n\n\\noindent  The example we have considered in this subsection shows that some of the field theory reformulations can be described in terms of homological algebra only. In the next section we will see the same picture in the case of supersymmetric four-dimensional Yang-Mills theory and its first order reformulation.\n\n\\section{$A_{\\infty}$-algebras of superforms in 4D.}\n\n\\noindent{\\bf 4.1. SUSY Maxwell complex for complex superfields.} We  claim that the Maxwell complex \n\n\\begin{equation}\\label{max}\n0\\xrightarrow{ }\\Omega^{0}(M)\\xrightarrow{\\mathrm{d}}\\Omega^{1}(M)\n\\xrightarrow{\\ud*\\ud}\\Omega^{3}(M)\\xrightarrow{\\mathrm{d}}\\Omega^{4}(M)\\to 0\n\\end{equation}\nand the complex of the first order theory:\n\\begin{eqnarray}\\label{fo}\n\\xymatrixcolsep{30pt}\n\\xymatrixrowsep{3pt}\n\\xymatrix{\n0\\ar[r]&\\Omega^0(M) \\ar[r]^\\mathrm{d} & \\Omega^1(M) \\ar[ddddr]  &\\Omega^3(M) \\ar[r]^\\mathrm{d} & \\Omega^4(M)\\ar[r]&0\\\\\n &\\quad &     & _{-\\mathrm{d}}\\quad    &&\\\\\n && \\bigoplus & \\bigoplus     &&\\\\\n &&           & _{\\mathrm{d}_{+}}\\quad    &&\\\\\n && \\Omega^2_+(M) \\ar[r]_{-Id} \\ar[uuuur]  & \\Omega^2_+(M)&\n}\n\\end{eqnarray}\nwhere $M=\\mathbb{R}^{1,3}$, can be naturally embedded into certain complex in N=1 superspace. This is a superspace where in addition to four even spacetime coordinates, there are two odd Weyl spinor coordinates \n$\\theta^{\\alpha}, \\theta^{\\dot\\alpha}$.  \nWe keep the notation close to \\cite{susybook}, see also Appendix B. Denote the space of complex scalar superfields $V(x,\\theta,\\bar \\theta)$ as $\\Sigma$ and the space of chiral spinor superfields  $W_{\\alpha}(x,\\theta)$ (corresponding to field strength) as $\\Theta$. Finally, we denote the space of chiral superfields corresponding to gauge transformations $\\Lambda(x,\\theta)$ as $\\Phi$. The following complex:\n\\begin{eqnarray}\\label{msusy}\n0\\xrightarrow{ }\\Xi\\xrightarrow{id}\\Sigma\n\\xrightarrow{D^{\\alpha}{\\bar D}^2D_{\\alpha}}\\t \\Sigma\\xrightarrow{{\\bar D}^2}\\t \\Xi\\to 0.\n\\end{eqnarray}\nwhere $\\Xi\\cong\\t \\Xi$ and $\\Sigma\\cong \\t\\Sigma$ is the supersymmetric generalization of the Maxwell complex, it is clear that the Maxwell complex \\rf{max} naturally embeds into the complex \\rf{msusy}. \nMoreover, one can consider the complex\n\\begin{eqnarray}\\label{sfo}\n\\xymatrixcolsep{30pt}\n\\xymatrixrowsep{3pt}\n\\xymatrix{\n0\\ar[r]&\\Xi \\ar[r]^{id} & \\Sigma \\ar[ddddr]  &\\tilde\\Sigma \\ar[r]^{\\bar D^2} & \\tilde\\Xi\\ar[r]&0\\\\\n &\\quad &     & _{\\widehat{div}}\\quad    &&\\\\\n && \\bigoplus & \\bigoplus     &&\\\\\n &&           & \\quad _{\\bar D^2D_{\\alpha}}  &&\\\\\n && \\Theta \\ar[r]_{Id} \\ar[uuuur]  & \\tilde\\Theta&\n}\n\\end{eqnarray}\nwhere  \n$\\widehat{div}{W}=D^{\\alpha}W_{\\alpha}$, $W$ is a chiral spinor field from \n$\\Sigma$ and $\\tilde\\Theta\\cong \\Theta$. One can show that \\rf{fo} can be embedded in \\rf{sfo}. \nAs we noted in the section 2.2, one could consider the first order Maxwell complex with antiselfdual 2-forms. This complex will be embedded into antichiral version of the \\rf{sfo} (i.e. the spaces $\\Xi$ and $\\Theta$ would be replaced by their antichiral counterparts). \n\nAfterwards, we will denote the supersymmetric Maxwell complex as  $(\\mathcal{F}_{susy} ^{\\cdot}, \\mathfrak{D})$ and the  \nsupersymmetric complex for the first order theory as $( \\mathcal{K}_{susy} ^{\\cdot}, {\\mathbb{D}})$. \nAs one may suspect, these complexes are homotopy equivalent as in the previous case. \nOne can construct the supersymmetric version of maps $f,g$ from the previous subsection, i.e. maps $g^s:(\\mathcal{K}_{susy} ^{\\cdot}, {\\mathbb{D}})\\to  (\\mathcal{F}_{susy} ^{\\cdot},  \\mathfrak{D})$, $f^s: \n(\\mathcal{F}_{susy} ^{\\cdot},  \\mathfrak{D})\\to  (\\mathcal{K}_{susy} ^{\\cdot}, {\\mathbb{D}})$:\n\\begin{eqnarray}\\label{fsgs}\nf^s(\\Lambda)=\\Lambda, \\quad f^s(V)=(-V,\\bar D^2D_{\\alpha}V), \\quad f^s(\\t V)=(\\t V, 0),\\quad f^s(\\t \\Lambda)=\\t \\Lambda,\\\\\ng^s(\\Lambda)=\\Lambda, \\quad g^s((V,W))=-V, \\quad g^s((\\t V, \\t W_{\\alpha})=\\t V-D^{\\alpha}\\t W_{\\alpha},\\quad g^s(\\t \\Lambda)=\\t \\Lambda.\\nonumber\n\\end{eqnarray}\nMoreover, $g^s\\circ f^s=id$ and $f^s\\circ g^s=id +{\\mathbb{D}} H+H{\\mathbb{D}}$ where the homotopy operator is nonzero only on $\\mathcal{K}_{susy} ^2$ and is defined by the formula:\n\\begin{equation}\nH((V,W))=((0,-W)).\n\\end{equation}\nThe precise statement is the following.\\\\\n\n\\noindent{\\bf Proposition 4.1.} {\\it The complexes $(\\mathcal{K}_{susy} ^{\\cdot}, {\\mathbb{D}})$, $(\\mathcal{F}_{susy} ^{\\cdot},  \\mathfrak{D})$ are quasiisomorphic and homotopically equivalent where the corresponding maps $g^s:(\\mathcal{K}_{susy} ^{\\cdot}, {\\mathbb{D}})\\to  (\\mathcal{F}_{susy} ^{\\cdot},  \\mathfrak{D})$, $f^s:(\\mathcal{F}_{susy} ^{\\cdot}, Q)\\to  (\\mathcal{K}_{susy} ^{\\cdot}, {\\mathbb{D}})$ are given by \\rf{fsgs}.}\\\\\n\nAnother issue about two complexes above is that one can define graded antisymmetric bilinear forms on \n$\\mathcal{F} _{susy}$, $\\mathfrak{K}_{susy}$ such that \n\\begin{equation}\n\\langle\\cdot, \\cdot \\rangle_{\\mathfrak{K}}:  \\mathfrak{K}^i_{susy}\\otimes  \\mathfrak{K}^{3-i}_{susy}\\to \\mathbb{C},\\quad\n\\langle\\cdot, \\cdot \\rangle_{\\mathcal{F} }:  \\mathcal{F} ^i_{susy}\\otimes  \\mathcal{F} ^{3-i}_{susy}\\to \\mathbb{C}.\n\\end{equation}\nOn $\\mathfrak{K}_{susy}$ they are defined by the following formulas:\n\\begin{eqnarray}\n&&\\langle{\\Lambda,\\t\\Lambda}\\rangle_{\\mathfrak{K}}=\\int d^4 xd^2\\theta \\Lambda\\t \\Lambda,\\quad \\langle V,\\t V\\rangle{\\mathfrak{K}}=\\int d^4 x d^4\\theta V\\t V,\\nonumber\\\\\n&& \\langle W,\\t W\\rangle_{\\mathfrak{K}}=\\frac{1}{2}\\int d^4 xd^2\\theta W_{\\alpha}\\t W^{\\alpha}.\n\\end{eqnarray}\n\n\\noindent {\\bf Proposition 4.2.} {\\it The graded symmetric bilinear forms $\\langle\\cdot, \\cdot \\rangle_{\\mathcal{F} }$, $\\langle\\cdot, \\cdot \\rangle_{\\mathfrak{K}}$ obey the relation \n\\begin{eqnarray}\\label{cycpairing}\n\\langle a_1, Qa_2\\rangle+(-1)^{a_1+a_2+a_1a_2}\\langle a_2,Qa_1\\rangle=0, \n\\end{eqnarray}\nwhen $a_1,a_2\\in \\mathcal{F} ^r$ or $a_1,a_2\\in \\mathfrak{K}^r$ and $Q$ stands for $ \\mathfrak{D}$ or ${\\mathbb{D}}$.}\\\\\n\n\\noindent The bilinear form on $\\mathcal{F} _{susy}$ is defined by the restriction of the form on $\\mathfrak{K}_{susy}$. Therefore, from now on we suppress the index notation for the bilinear forms.\n\nOne of the immediate consequences of the Proposition above is that if one can construct the $A_{\\infty}$-algebra on the complex $(\\mathcal{K}_{susy} ^{\\cdot}, {\\mathbb{D}})$, then the $A_{\\infty}$ structure on \n$(\\mathcal{F}_{susy} ^{\\cdot},  \\mathfrak{D})$ can be obtained by the transfer procedure as in the previous section. \\\\\n\n\\noindent {\\bf 4.2. Reminder of N=1 SUSY YM.} Let us consider the action of the N=1 SUSY YM theory in 4 dimensions:\n\\begin{eqnarray}\\label{asusy}\nS_{SYM}[V]=\\frac{1}{2}\\int d^4xd^4 \\theta Tr (e^{-V}D_{\\alpha}e^V{\\bar D}^2(e^{-V}D_{\\alpha}e^V)),\n\\end{eqnarray}\nwhere $V$ is the real superfield, taking values in some Lie algebra $\\mathfrak{g}$, i.e. $V\\in\\Sigma\\otimes{\\mathfrak{g}}$. It is invariant under the following transformations:\n\\begin{eqnarray}\n\\label{transf}\ne^V\\to e^{\\bar\\Lambda}e^{V}e^{\\Lambda},\n\\end{eqnarray}\nwhere $\\Lambda$ is a chiral scalar superfield. \nIt is useful to define the supercurvature, i.e. the spinor chiral superfield $W\\in \\Theta\\otimes{\\mathfrak{g}}$\n\\begin{eqnarray}\nW_{\\alpha}=-\\bar D^2(e^{-V}D_{\\alpha}e^V).\n\\end{eqnarray} \nUsing the global transformation formula \\rf{transf}, one can find the infinitesimal transformations of $V$ and $W$:  \n\\begin{eqnarray}\n&&V\\to V+\\delta_{\\Lambda, \\bar \\Lambda}V=V+\\frac{1}{2}L_{V}(\\Lambda-\\bar \\Lambda+\\coth(\\frac{1}{2}L_V)(\\Lambda+\\bar \\Lambda))\\nonumber\\\\\n&&W_{\\alpha}\\to W_{\\alpha}+[W_{\\alpha},\\Lambda]\n\\end{eqnarray} \nwhere $\\Lambda, \\bar{\\Lambda}$ are infinitesimal and $L_V\\cdot=[V,\\cdot]$.\nVarying the action $S_{SYM}$ with respect to $V$, one finds the following expression:\n\\begin{eqnarray}\\label{eqm}\nf(L_V)[\\nabla_{\\alpha},W^{\\alpha}]=0, \n\\end{eqnarray}\nwhere $\\nabla_{\\alpha}=e^{-V}D_{\\alpha}e^V$, and $f(x)=\\frac{e^{x}-1}{x}$. \nOne can see that this equation is equivalent to the \"physically covariant\" equation $[\\nabla_{\\alpha},W^{\\alpha}]=0$ because operator $f(L_V)$ is invertible. \nHere we note (this is important for the next subsection) that the equation \\rf{eqm} is the equation of motion for the functional \\rf{asusy} even when $V$ is not real.  The reality condition gives the constraint:\n\\begin{eqnarray}\n[\\nabla_{\\alpha},W^{\\alpha}]=[\\nabla_{\\dot{\\alpha}},W^{\\dot{\\alpha}}].\n\\end{eqnarray}\nBelow we are showing that the equation \\rf{eqm} is covariant from the homological point of view, i.e. we \nare observing that it arises as GMC equation for the certain cyclic $A_{\\infty}$-algebra. \\\\\n\n\\noindent {\\bf 4.3. Chiral $A_{\\infty}$-algebra.} \nOne can also rewrite the action \\rf{asusy} in the first order form the way we we did in the case of usual Yang-Mills: \n\\begin{eqnarray}\n&&S_{fo}[V,W]=\\frac{1}{2}\\int d^4x d^2 \\theta \\frac{1}{2}Tr(W_{\\alpha}W^{\\alpha}+\\bar{D}^2 (e^{-V}D_{\\alpha}e^V)W^{\\alpha}).\n\\end{eqnarray} \nWe see that both actions $S_{SYM}$ and $S_{fo}$ in the case of abelian Lie algebra \ncan be written as $\\frac{1}{2}\\langle \\psi,Q\\psi\\rangle$, where $\\psi$ is an element of degree \n$1$ and $Q$ stands either for $ \\mathfrak{D}$ or ${\\mathbb{D}}$.\nLet us forget about $\\bar \\Lambda$-symmetry in this section and consider only the one, generated by chiral scalar superfields $\\Lambda$.  \nThen the BV generalization of the last action looks as follows:\n\\begin{eqnarray}\n&& S^{BV}_{fo}=S_{fo}+\\nonumber\\\\\n&&\\int d^4xd^4\\theta Tr(\\delta_{C}(V)V^*)+ \\int d^4x d^2\\theta Tr([W^{\\alpha},C]W^*_{\\alpha}+\\frac{1}{2}[C,C]C^*).\n\\end{eqnarray} \nHere, as usual, ghosts and antifields are $c^{+}\\in\\Phi[1], V^*\\in\\t \\Sigma[-1]$ and is a real superfield, $W^*\\in\\t \\Sigma[-1]$, $c\\in\\t \\Theta[-2]$. In the actions above $\\delta_{c}(V)=\\frac{1}{2}L_{V}(c+\\coth(\\frac{1}{2}L_V)(c))$. \nIt is feasible to check that these actions satisfy the Master equation. \nTherefore, an odd vector field ${\\bf Q}$ \nthat is defined by means of the formula ${\\bf Q}\\cdot=(S_{fo}^{BV},\\cdot)_{BV}$ acts on the fields \nin the following way:\n\\begin{eqnarray}\\label{bfq}\n&&{\\bf Q}c=\\frac{1}{2}[C,C],\\nonumber\\\\\n&&{\\bf Q}V=\\delta_{C}(V),\\nonumber\\\\\n&&{\\bf Q}V^*=f(L_V)[e^{-V}D_{\\alpha}e^V,W^{\\alpha}]+\\frac{\\delta}{\\delta V}\\delta_{C}(V)V^*,\\nonumber\\\\\n&&{\\bf Q}W_{\\alpha}=[W_{\\alpha},C],\\nonumber\\\\\n&&{\\bf Q}W_{\\alpha}^*=[W_{\\alpha}^*,C],\\nonumber\\\\\n&&{\\bf Q}{C^*}=[C,C^*]-\\frac{1}{2}L_V(V^*-\\coth(\\frac{1}{2}L_V)(V^*)) +[W_{\\alpha},{W^{\\alpha}}^*].\n\\end{eqnarray}\nThen we have the following Proposition.\\\\\n\n\\noindent{\\bf Proposition 4.3.} {\\it The odd vector field ${\\bf Q}$ satisfies the nilpotency condition ${\\bf Q}^2=0$ and therefore determines the $A_{\\infty}$ structure on the complex \\rf{fo}.}\\\\\n\n\\noindent Therefore, the action $S^{BV}_{fo}$  has the form of the homotopy Chern-Simons theory, \ni.e.:\n\\begin{eqnarray}\n\\frac{1}{2}\\langle\\psi,{\\mathbb{D}}\\psi\\rangle+\\sum_{n\\ge 2}\\frac{1}{n+1}\\langle\\mu_n(\\psi,...,\\psi),\\psi\\rangle,\n\\end{eqnarray}\nwhere $\\mu_n$ stand for bilinear and higher operations in the $A_{\\infty}$-algebra. \nOne of the specific features of the corresponding $A_{\\infty}$-algebra is that the \ncorresponding $L_{\\infty}$-algebra vanishes in the case of abelian Lie algebra \n$\\mathfrak{g}$. One can see that directly from \\rf{bfq} or from the fact that the BV action in this case becomes bilinear in fields. Therefore, for example, the bilinear operation for this  \n$A_{\\infty}$-algebra in the case of abelian $\\mathfrak{g}$ is commutative, like in the usual Yang-Mills case.  \nLet us write the expression for this bilinear operation explicitly.\n\n\\begin{center}\n$\\mu_2(f_1,f_2)$=\n\\end{center}\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n\\backslashbox{$ f_2$}{$f_1$}&          $\\Lambda_1$ & $(V_1,W_1)$ & $(\\t V_1,\\t W_1)$ & $\\t\\Lambda_1$ \\\\\n\\hline\n$\\Lambda_2$ &                  $\\Lambda_1\\Lambda_2$    &$(\\frac{1}{2}V_1\\Lambda_2,W_1\\Lambda_2)$ & \n$(\\frac{1}{2}\\t V_1\\Lambda_2,\\t W_1\\Lambda_2)$ &$\\t\\Lambda_1\\Lambda_2$\\\\\n\\hline\n$(V_2,W_2)$     &  $(\\frac{1}{2}\\Lambda_1V_2,\\Lambda_1W_2)$ & $((V_1,W_1), (V_2,W_2))_h$   &   $\\t W_{1,\\alpha}W_2^{\\alpha}$ &0\\\\\n                               &                               &       & $-\\frac{1}{2}\\bar D^2(\\t V_1V_2)$&\\\\                    \n\\hline\n$(\\t V_2,\\t W_2)$ & $(\\frac{1}{2}\\Lambda_1\\t V_2,\\Lambda_1\\t W_2)$ &  $-W_1^{\\alpha}\\t W_{2,\\alpha}$ &    0&0\\\\\n&&                        $-\\frac{1}{2}\\bar D^2(V_1\\t V_2)$&&\\\\\n\\hline\n$\\t\\Lambda_2$                     & $\\Lambda_1\\t\\Lambda_2$ &   0 &    0&0\\\\\n\\hline\n\\end{tabular}\\\\\n\\vspace{3mm}\n\nHere $f_1,f_2$ take values in the set of variables with indices $1$ and $2$ correspondingly. In the table above $\\Lambda_i\\in \\mathcal{K}_{susy} ^0$, $(V_i,W_i)\\in \\mathcal{K}_{susy} ^1 (i=1,2)$, $(\\t V_i,\\t W_i)\\in \\mathcal{K}_{susy} ^2 (i=1,2)$, \n$\\t \\Lambda_i\\in \\mathcal{K}_{susy} ^3$, and \n\\begin{eqnarray}\n&&((V_1,W_1), (V_2,W_2))_h=\\\\\n&&(D_{\\alpha}V_1W_2^{\\alpha}+W_1^{\\alpha}D_{\\alpha}V_2+\\frac{1}{2}(V_1D_{\\alpha}W^{\\alpha}_2-D_{\\alpha}W^{\\alpha}_1V_2),-\\frac{1}{2}\\bar D^2(V_1D_{\\alpha}V_2-D_{\\alpha}V_1V_2).\\nonumber\n\\end{eqnarray}\n{\\bf Corollary 4.1.} {\\it The operation $\\mu$ on the complex \\rf{sfo} is homotopy associative on \n$(\\mathfrak{K}^{\\cdot}, \\tilde{\\mathfrak{D}})$}.\\\\\n\n\\noindent In Appendix B, we explicitly prove this proposition.\n\nKnowing all the operations of $A_{\\infty}$-algebra based on complex \\rf{sfo}, one can construct ones on the complex \\rf{msusy}. The resulting bilinear operation is:\n\\begin{center}\n$m_2(f_1,f_2)$=\n\\end{center}\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n\\backslashbox{$ f_2$}{$f_1$}&          $\\Lambda_1$ & $V_1$ & $\\t V_1$ & $\\t\\Lambda_1$ \\\\\n\\hline\n$\\Lambda_2$ &                  $\\Lambda_1\\Lambda_2$    &$\\frac{1}{2}V_1\\Lambda_2$ & \n$\\frac{1}{2}\\t V_1\\Lambda_2$ &$\\t\\Lambda_1\\Lambda_2$\\\\\n\\hline\n$V_2$     &  $\\frac{1}{2}\\Lambda_1V_2 $ & $(V_1,V_2)_h$   &  $-\\frac{1}{2}\\bar D^2(\\t V_1V_2)$  &0\\\\\n                    \n\\hline\n$\\t V_2$ & $\\frac{1}{2}\\Lambda_1\\t V_2$ &  $-\\frac{1}{2}\\bar D^2(V_1\\t V_2)$ &    0&0\\\\\n\\hline\n$\\t\\Lambda_2$                     & $\\Lambda_1\\t\\Lambda_2$ &   0 &    0&0\\\\\n\\hline\n\\end{tabular}\\\\\n\n\\noindent where \n\\begin{eqnarray}\n&& (V_1,V_2)_h=\\frac{1}{2}D^{\\alpha}\\bar D^2(V_1D_{\\alpha}V_2-V_2D_{\\alpha}V_1)-D_{\\alpha}V_1\\bar D^2D^{\\alpha}V_2-\\nonumber\\\\\n&&D_{\\alpha}V_2\\bar D^2D^{\\alpha}V_1D_{\\alpha}V_2+\\frac{1}{2}(V_2D_{\\alpha}\\bar D^2D^{\\alpha}V_1-V_1D_{\\alpha}\\bar D^2D^{\\alpha}V_2).\n\\end{eqnarray}\n\n\\noindent Therefore, we have another proposition.\\\\\n\n\\noindent {\\bf Proposition 4.4.} {\\it Operation $m_2$ is homotopy associative on the supersymmetric generalization of the Maxwell complex \\rf{msusy}.}\\\\\n\n\\noindent As one could expect, the operation $m_2$ comes from the BV functional\n \\begin{eqnarray}\\label{bvsymact}\n&&S^{BV}_{SYM}[V]=\\frac{1}{2}\\int d^4xd^4 \\theta Tr(e^{-V}D_{\\alpha}e^V{\\bar D}^2(e^{-V}D_{\\alpha}e^V))+\\nonumber\\\\\n&&\\int d^4xd^4\\theta Tr(\\delta_{C}(V)V^*) +\\int d^4xd^2\\theta\\frac{1}{2}Tr([C,C]C^*).\n\\end{eqnarray}\n\nAgain, as in the case of nonsupersymmetric Yang-Mills, we see that the $A_{\\infty}$-algebras of the second and the first order theories are related by means of transfer formula. \\\\\n\n\\noindent {\\bf 4.4. Full symmetry and real representation.} So far we neglected the full symmetry \\rf{transf} of the action \\rf{asusy}. Namely, \nwe considered only the chiral part of the symmetry. In order to obtain full symmetry on the homological level there are two options. The first one is to consider the following extension of the complex \\rf{msusy}:\n\\begin{eqnarray}\n\\xymatrixcolsep{40pt}\n\\xymatrixrowsep{3pt}\n\\xymatrix{\n0\\ar[r]&\\Theta \\ar[r]^{id} & \\Sigma \\ar[r]^{D_{\\alpha}\\bar D^2D^{\\alpha}}  &\\tilde\\Sigma \\ar[r]^{\\bar D^2}\\ar[ddr]^{D^2} & \\tilde\\Theta\\ar[r]&0\\\\\n &\\oplus&&&\\oplus\n &&          &  &&\\\\\n &\\bar{\\Theta}\\ar[uur]^{-id}&   &&\\tilde{\\bar{\\Theta}}}\\nonumber\n\\end{eqnarray}\nHere $\\bar{\\Theta}\\cong \\bar{\\tilde \\Theta}$ is a space of antichiral superfields. From now on, we denote it as $(\\mathcal{F} _{full}^{\\cdot}, \\mathcal{D})$. \n To reproduce this complex, one should consider the full BV action:\n\\begin{eqnarray}\n&&S^{BV, full}_{SYM}[V]=\\nonumber\\\\\n&&\\frac{1}{2}\\int d^4x d^4 \\theta Tr(e^{-V}D_{\\alpha}e^V{\\bar D}^2(e^{-V}D_{\\alpha}e^V))+\\int d^4xd^4\\theta Tr(\\delta_{C,\\bar C}(V)V^*)+\\nonumber\\\\\n&&\\int d^4x d^2\\theta\\frac{1}{2}Tr([C,C]C^{*})+\\int d^4x d^2\\bar \\theta\\frac{1}{2}Tr([\\bar C,\\bar C]\\bar C^*).\n\\end{eqnarray}\nHere $\\delta_{C,\\bar C}(V)=\\frac{1}{2}L_V(C-\\bar C+coth(\\frac{1}{2}L_V)(C+\\bar C))$. The corresponding \nbilinear operation should be modified in this way:\n\\begin{center}\n$m^{full}_2(f_1,f_2)$=\n\\end{center}\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n\\backslashbox{$ f_2$}{$f_1$}&          $(\\Lambda_1, \\bar\\Lambda_1)$ & $V_1$ & $\\t V_1$ & $(\\t\\Lambda_1, \n\\bar{\\t{\\Lambda}}_1)$ \\\\\n\\hline\n$(\\Lambda_2, \\bar \\Lambda_2)$ &                  $(\\Lambda_1\\Lambda_2, \\bar \\Lambda_1\\bar\\Lambda_2)$    &$\\frac{1}{2}V_1(\\Lambda_2+\\bar\\Lambda_2)$ & \n$\\frac{1}{2}\\t V_1(\\Lambda_2+\\bar\\Lambda_2)$ &$(\\t\\Lambda_1\\Lambda_2,\\bar{\\t{\\Lambda}}_1\\bar{\\t{\\Lambda}}_2)$\\\\\n\\hline\n$V_2$     &  $\\frac{1}{2}(\\Lambda_1+\\bar\\Lambda_1)V_2 $ & $(V_1,V_2)_h$   &  $(-\\frac{1}{2}\\bar D^2(\\t V_1V_2), \\frac{1}{2}D^2(\\t V_1V_2))$  &0\\\\\n                    \n\\hline\n$\\t V_2$ & $\\frac{1}{2}(\\Lambda_1+\\bar\\Lambda_1)\\t V_2$ &  $-\\frac{1}{2}\\bar D^2(V_1\\t V_2)$ &    0&0\\\\\n\\hline\n$(\\t\\Lambda_2, {\\bar{\\t\\Lambda}}_2)$                     & $(\\Lambda_1\\t\\Lambda_2, \\bar\\Lambda_1{\\bar{\\t\\Lambda}}_2)$ &   0 &    0&0\\\\\n\\hline\n\\end{tabular}\\\\\n\n\\noindent Using the transfer formula, one can find the modification for the bilinear operation of the first order model complex. \n\nSometimes it is useful to define the following representation of the field $V$:\n\\begin{eqnarray}\\label{chiral}\ne^V=e^{\\bar \\Omega}e^{\\Omega},\n\\end{eqnarray}\nsuch that under \\rf{transf}, the transformation of $\\Omega$ and $\\bar{\\Omega}$ is:\n\\begin{eqnarray}\ne^{\\Omega}\\to e^Ke^{\\Omega}e^{\\Lambda}, \\quad e^{\\bar \\Omega}\\to e^{\\bar \\Lambda}e^{\\bar \\Omega}e^{-K},\n\\end{eqnarray}\nwhere the new $K$-symmetry appears. \nTherefore, another approach to get full symmetry into the picture is to represent $V$ in terms of $\\Omega$ and $\\bar \\Omega$ and treat them as separate varibles in the BV action. The resulting $A_{\\infty}$-algebra is based on the following complex:\n\\begin{eqnarray}\n\\xymatrixcolsep{40pt}\n\\xymatrixrowsep{3pt}\n\\xymatrix{\n&\\Theta \\ar[r]^{id} & \\Sigma_c\\ar[ddddr] \\ar[r]^{D_{\\alpha}\\bar D^2D^{\\alpha}}  &\\tilde\\Sigma_c \\ar[ddr]^{id}\\ar[r]^{\\bar D^2} & \\tilde\\Theta\\\\\n &\\oplus&& &\\oplus&&          &  &&\\\\\n 0\\ar[r]&\\Upsilon\\ar[uur]^{id} \\ar[ddr]^{-id}&\\oplus&\\oplus&\\t \\Upsilon\\ar[r]&0\\\\\n &\\oplus&&&\\oplus\n &&          &  &&\\\\\n &\\bar{\\Theta}\\ar[r]^{-id}&\\bar\\Sigma_c\\ar[uuuur]  \\ar[r]^{D_{\\alpha}\\bar D^2D^{\\alpha}} &\\bar{\\tilde\\Sigma}_c\\ar[uur]^{-id}\\ar[r]^{D^2}&\\tilde{\\bar{\\Theta}}}\\nonumber\n\\end{eqnarray}\nHere all maps in the middle of the diagram are given by the action of the operator $D_{\\alpha}\\bar D^2D^{\\alpha}$, and the spaces $\\Sigma_r, \\tilde\\Sigma_r, \\bar\\Sigma_r, \\bar{\\tilde\\Sigma}_r,\\Upsilon$ are isomorphic to the space $\\Sigma$ (see above). \nWe will denote the resulting complex as $(\\mathcal{F} ^{\\cdot}_{real}, \\mathcal{D})$. The immediately arising interesting question is: how the $A_{\\infty}$ structure on this complex is related to $A_{\\infty}$-algebra on \n$(\\mathcal{F} _{full}^{\\cdot}, \\mathcal{D})$? Below  we will show that they are homotopically equivalent. \nLet us explicitly construct the chain maps $f:\\mathcal{F} _{full}^{\\cdot}\\to \\mathcal{F} _{real}^{\\cdot}$ and $g:\\mathcal{F} _{real}^{\\cdot}\\to \\mathcal{F} _{full}^{\\cdot}$:\n\\begin{eqnarray}\n&&f((\\Lambda, \\bar \\Lambda))=(\\Lambda, -\\frac{1}{2}(\\Lambda+\\bar \\Lambda), \\bar \\Lambda),\\quad \nf(V)=(\\frac{1}{2}V, \\frac{1}{2}V), \\nonumber\\\\\n&& f(\\t V)=(\\t V, \\t V), \\quad f((\\Lambda, \\bar \\Lambda))=(\\Lambda, 0, \\bar \\Lambda)\\nonumber\\\\\n&&\\nonumber\\\\\n&&g((\\Lambda, K, \\bar \\Lambda))=(\\Lambda, \\bar \\Lambda), \\quad g((\\Omega, \\bar \\Omega))=\\Omega+\\bar \\Omega\\nonumber\\\\\n&&g((\\t \\Omega, \\t{\\bar\\Omega}))=\\frac{1}{2}(\\t \\Omega+ \\t {\\bar\\Omega}), \\quad g((\\Lambda, K, \\bar \\Lambda))=\n(\\Lambda-\\frac{1}{2}{\\bar D}^2K, \\bar \\Lambda-\\frac{1}{2}{D}^2K)\n\\end{eqnarray}\nIt is feasible to check that $gf=id$ and $fg=id+[Q,H]$ where $H$ is an operator of degree $-1$ on the complex \n$\\mathcal{F} _{chiral}^{\\cdot}$ which is nonzero on $\\mathcal{F} ^1_{real}$ and $\\mathcal{F} ^3_{real}$, such that \n\\begin{eqnarray}\n&&H((\\Omega, \\bar \\Omega))=(0, \\frac{1}{2}(\\bar \\Omega-\\Omega),0),\\nonumber\\\\\n&&H((\\t \\Lambda, \\t K, \\t {\\bar \\Lambda}))=(-\\frac{1}{2}K,\\frac{1}{2}K) \n\\end{eqnarray}\nTherefore, one can formulate the following Proposition.\\\\\n\n\\noindent {\\bf Proposition 4.5.} {\\it Complex $(\\mathcal{F} _{full}^{\\cdot}, \\mathcal{D})$ is homotopically equivalent to \n$\\mathcal{F} _{real}^{\\cdot}$.}\\\\\n\nThis allows us to transfer the homotopy algebra structure from $\\mathcal{F} _{real}^{\\cdot}$ to $\\mathcal{F} _{full}^{\\cdot}$. Here we also note the following. If we look at  \"Baker-Campbell-Hausdorff\" change of variables \n$e^V=e^{\\bar\\Omega}e^{\\Omega}$ in the corresponding equation of motion (Maurer-Cartan equation), we find out that it becomes part of the transfer of the $A_{\\infty}$-algebra. \n\n\n\\section*{Appendix A: $A_{\\infty}$-algebras and the BV formalism.}\nIn this appendix we summarize all necessary information about $A_{\\infty}$-algebras. For more details see \ne.g. \\cite{stashbook}, \\cite{kajiura}, \\cite{markltransf}. \\\\\n\n\\noindent {\\bf A1. $A_{\\infty}$-algebras.} \nThe $A_{\\infty}$-algebra is a generalization of an associative algebra with a differential. Namely, consider a graded vector space $V=\\oplus_k V_k$ with a differential $Q$. Consider the multilinear operations $\\mu_r: V^{\\otimes r}\\to V$ of the degree $2-r$, such that $\\mu_1=Q$. \\\\\n\n\\noindent {\\bf Definition.}{\\it The space V is an $A_{\\infty}$-algebra if $\\mu_n$ satisfy the following bilinear identity:\n\\begin{eqnarray}\\label{arel}\n\\sum^{n-1}_{i=1}(-1)^{i}M_i\\circ M_{n-i+1}=0\n\\end{eqnarray}\non $V^{\\otimes n}$. \nHere $M_s$ acts on $V^{\\otimes m}$ ($m\\ge s$)  as the sum of all possible operators of the form \n${\\bf 1}^{\\otimes^l}\\otimes\\mu_s\\otimes{\\bf 1}^{\\otimes^{m-s-l}}$ taken with appropriate signs. In other words, \n$M_s:V^{\\otimes^m}\\to V^{\\otimes^{m-s+1}}$ and \n\\begin{eqnarray}\nM_s=\\sum^{m-s}_{l=0}(-1)^{l(s+1)}{\\bf 1}^{\\otimes^l}\\otimes\\mu_s\\otimes{\\bf 1}^{\\otimes^{m-s-l}}.\n\\end{eqnarray}}\nLet's write several relations which are satisfied by $Q$, $\\mu_1$, $\\mu_2$, $\\mu_3$:\n\\begin{eqnarray}\n&&Q^2=0,\\\\\n&&Q\\mu_2(a_1,a_2)=\\mu_2(Q a_1,a_2)+(-1)^{n_{a_1}}\\mu_2(a_1,Q a_2),\\nonumber\\\\\n&&Q\\mu_3(a_1,a_2, a_3)+\\mu_3(Q a_1,a_2, a_3)+(-1)^{n_{a_1}}\\mu_3(a_1,Q a_2, a_3)_h+\\nonumber\\\\\n&&(-1)^{n_{a_1}+n_{a_2}}\\mu_3( a_1, a_2, Q a_3)=\\mu_2(\\mu_2(a_1,a_2),a_3)-\\mu_2(a_1,\\mu_2(a_2,a_3)).\\nonumber\n\\end{eqnarray}\nIn such a way we see that if $\\mu_n=0$, $n\\ge 3$ , then we have just graded differential associative algebra. \nIt appears that the relations \\rf{arel} can be encoded into one equation $\\partial^2=0$. To see this, one applies the desuspension operation (the operation which shifts the grading $s^{-1}: V_{k}\\to V_{k-1}$) to $\\mu_n$, i.e. one can define operations of degree $1$: $\\nu_n=s\\nu_n {s^{-1}}^{\\otimes^n}$. \nMore explicitly, \n\\begin{eqnarray}\n\\nu_n(s^{-1} a_1,...,s^{-1} a_n)=(-1)^{s(a)}s^{-1}\\mu_n(a_1,...,a_n),\n\\end{eqnarray}\nsuch that $s(a)=(1-n)n_{a_1}+(2-n)n_{a_2}+...+a_{n-1}$. \nThe relations between $\\nu_n$ operations can be summarized in the following simple equations:\n\\begin{eqnarray}\\label{M2}\n\\sum^n_{i=1}N_i\\circ N_{n+1-i}=0.\n\\end{eqnarray}\non $V^{\\otimes n}$. Here each $N_s$ acts on $V^{\\otimes^m}$ ($m\\ge s$) as the sum of all operators ${\\bf 1}^{\\otimes^l}\\otimes\\nu_s\\otimes{\\bf 1}^{\\otimes^k}$, such that $l+s+k=m$.  Combining them into one operator $\\partial=\\sum_n\\nu_n$, acting on a space $\\oplus_kV^{\\otimes^k}$ the relations \\rf{arel} can be combined into one equation $\\partial^2=0$. \n\nAnother way to represent the relations \\rf{arel} as the nilpotency condition of some operator is by using the differential operators on a noncommutative manifold. Let the set $\\{e_i\\}$ be the homogeneous elements,  \nwhich form a basis of $V$. Introduce noncommutative supercoordinates $x^i$ such that $X=x^ie_i$ has degree degree 1 on $V$. One can write a noncommutative vector field, such that  \n\\begin{eqnarray}\\label{Q}\n{\\bf Q}x^i=\\nu_{n;j_1,...,j_n}x^{j_1}...x^{j_n},\n\\end{eqnarray}\nwhere $\\nu_{n;j_1,...,j_n}=\\nu_n(e_{j_1},...,e_{j_n})$. Then the relations \\rf{M2} can be formulated as \n${\\bf Q}^2=0$. \\\\\n\n\\noindent{\\bf A2. Cyclic structures, BV formalism and the generalized Maurer-Cartan equation.} \nFirst we define what cyclic structure is.\\\\\n\n\\noindent{\\bf Definition.} {\\it The $A_{\\infty}$-algebra on a space V is called $cyclic$ if there exists a nondegenenerate  pairing $\\langle\\cdot, \\cdot \\rangle$, such that it is graded symmetric\n\\begin{eqnarray}\n\\langle a, b \\rangle=-(-1)^{(n_a+1)(n_b+1)}\\langle b,a  \\rangle\n\\end{eqnarray}\nand satisfies the following conditions for $\\mu_n$:\n\\begin{eqnarray}\n&&\\langle a_1,\\mu_{n-1}(a_2,...,a_n)\\rangle=\\nonumber\\\\\n&&(-1)^{(n-1)(a_1+a_2+1)+a_1(a_2+...+a_n)}\\langle a_2,\\mu_{n-1}(a_3,...,a_n,a_1)\\rangle.\n\\end{eqnarray}}\nIt makes sense to define $\\psi_n(a_1,...,a_n)=\\langle a_1, \\mu_{n-1}(a_2, ..., a_n)\\rangle$. Consider again the element of degree 1 $X=x^ie_i$, where $x^i$ are noncommutative \nsupercoordinates and define the formal $action$ $functional$:\n\\begin{eqnarray}\\label{hcs}\nS[X]=\\sum^{\\infty}_{n=2}\\frac{1}{n}\\Psi_n(X,...,X).\n\\end{eqnarray}\nLet $\\omega_{ij}=\\langle e_i, e_j\\rangle$. Then one can define a BV bracket:\n\\begin{eqnarray}\n(\\alpha,\\beta)_{BV}=\\frac{{\\overleftarrow{\\partial}}\\alpha}{\\partial x^i}\\omega^{ij}\\frac{\\overrightarrow{\\partial}\\beta}{\\partial x^j}\n\\end{eqnarray}\non the space of polynomials of $x^i$, such that $S$ satisfies classical Master equation:\n\\begin{eqnarray}\n(S,S)_{BV}=0\n\\end{eqnarray}\nUsing this condition, one can find that $S$ defines an odd vector field such that\n\\begin{eqnarray}\n(S,x^i)_{BV}=\\nu_{n;j_1,...,j_n}x^{j_1}...x^{j_n},\n\\end{eqnarray}\nwhich coincides with odd vector field ${\\bf Q}$ was defined in \\rf{Q}. Moreover, for a given action \n\\begin{eqnarray}\nS=v_{i_1i_2}x^{i_1}x^{i_2}+\\sum_{n\\ge3}v_{i_1...i_n}x^{i_1}....x^{i_n},\n\\end{eqnarray}\nwhich is cyclic in $x^i$ and satisfies Master equation, we obtain the cyclic $A_{\\infty}$-algebra (see e.g. \\cite{kajiura},\\cite{zwiebach}).\n\nVarying the action \\rf{hcs} with respect to $X$, one finds the equation of motion which are known as generalized Maurer-Cartan equation:\n\\begin{eqnarray}\nQX+\\sum_{n\\ge2}\\mu_n(X,...,X)=0.\n\\end{eqnarray}\nThis equation is known to have the following symmetry\n\\begin{eqnarray}\nX\\to Q\\alpha+\\sum_{n\\ge2,k}(-1)^{n-k}\\mu_n(X,...,\\alpha,...,X),\n\\end{eqnarray}\nwhere $\\alpha$ is an element of degree 0 and $k$ means the position of $\\alpha$ in $\\mu_n$. \\\\\n\n\\noindent {\\bf A3. Transfer of the $A_{\\infty}$ structure.} Suppose you have two complexes which are quasiisomorphic and moreover homotopically equivalent. Moreover, suppose that on one of them there exists the structure of $A_{\\infty}$-algebra. Then there exists an $A_{\\infty}$-algebra on another complex. The explicit formulas are given in \\cite{markltransf} following the results of \\cite{konts} and \\cite{merk}.  \n\nIn fact, in \\cite{markltransf} there is even a more general statement. \nLet us formulate it in a precise way. Consider two complexes $(\\mathcal{F} ,Q)$ and $(\\mathfrak{K}, Q')$ such that there are maps \n$f:(\\mathcal{F} ,Q)\\to (\\mathfrak{K}, Q')$, $g:(\\mathfrak{K},Q')\\to (\\mathcal{F} ,Q)$ such that $fg$ is chain homotopic to identity. In other words, there exists a map $H: (\\mathfrak{K},Q')\\to (\\mathfrak{K}, Q')$ of degree ${-1}$, such that $fg=id+Q' H+H Q'$. Then, given an $A_{\\infty}$-algebra structure $\\hat{\\mu}_{\\{n\\}}$ on $\\mathfrak{K}$, such that $\\hat{\\mu}_1=Q'$, one can construct an $A_{\\infty}$-algebra on $(\\mathcal{F} ,Q)$ by means \nof the formula \n\\begin{eqnarray}\n\\mu_n=g\\circ p_n \\circ f^{\\otimes^n},\n\\end{eqnarray}\nwhere $p_n$ is obtained by means of consequtive recurrent procedure of applying the homotopy operators to $\\t \\mu_n$. The explicit formula is:\n\\begin{eqnarray}\np_n=\\sum_B(-1)^{\\theta(r_1,...,r_k)}\\hat{\\mu}_k(H\\circ p_{r_1},...,  H\\circ p_{r_k}),\n\\end{eqnarray}\nwhere $B=\\{k, r_1, . . . , r_k | 2 \\le k \\le n, r_1, ... , r_k \\ge 1, r_1 + ... + r_k = n\\}$, $\\theta(r_1,...,r_k)=\\sum_{1\\le\\alpha\\le\\beta\\le_r}r_{\\alpha}(r_{\\beta}+1)$ and $p_2\\equiv \\hat{\\mu}_2$, $p_1\\circ H\\equiv id$. \n\n\n\\section*{Appendix B: Explicit calculations for $A_{\\infty}$ superalgebras of superforms.}\n{\\bf B1.} {\\bf Notations for $N=1$ 4D superspace.} We keep the notations from \\cite{susybook}. \nThe world-volume metric in the 4D space with the coordinates $x^{\\mu}$  is $\\eta_{\\mu\\nu}=\\rm{diag}(-1,+1,+1,+1)$. \nThe $N=1$ four-dimensional space has two anticommutative Weyl spinor coordinates $\\theta^{\\alpha}, \\theta^{\\dot{\\alpha}}$. The spinor indices are raised and lowered by means of $C_{\\alpha\\beta}, C_{\\dot{\\alpha}\\dot{\\beta}}$, such that $C_{21}=-C_{12}=i$. One can define superderivatives: \n\\begin{eqnarray}\nD_{\\alpha}=\\partial_{\\alpha}+\\frac{i}{2}\\theta^{\\dot{\\beta}}\\partial_{\\alpha\\dot{\\beta}}, \\quad \\bar D_{\\dot{\\alpha}}=\\bar \\partial_{\\alpha}+\\frac{i}{2}{\\theta}^{\\beta}\\partial_{\\beta\\dot{\\alpha}},\n\\end{eqnarray}\nwhere $\\partial_{\\alpha\\dot{\\beta}}=\\sigma_{\\alpha\\dot{\\beta}}^{\\mu}\\partial_{\\mu}$. \nTherefore, one can introduce the (anti)chiral scalar superfields, which are defined by the nilpotency of certain antiderivatives $D_{\\dot{\\alpha}}\\Lambda(x,\\theta)=0$ ($D_{\\alpha}\\bar \\Lambda(x,\\theta)=0$)\nThe relations \nbetween superderivatives are:\n\\begin{eqnarray}\n\\{D_{\\alpha},D_{\\dot{\\beta}}\\}=i\\partial_{\\alpha\\dot{\\beta}}, \\quad \n\\{D_{\\alpha},D_{\\dot{\\beta}}\\}=0, \\quad \\{D_{\\dot{\\alpha}},\\bar D_{\\dot{\\beta}}\\}=0.\n\\end{eqnarray}\nFor calculations it is also useful to introduce operators $D^2=C^{\\alpha\\beta}D_{\\alpha}D_{\\beta}$, $\\bar D^2=C^{\\dot{\\alpha}\\dot{\\beta}}D_{\\dot{\\alpha}}D_{\\dot{\\beta}}$.\\\\\n\n\\noindent {\\bf B2.} {\\bf Explicit calculations for the $A_{\\infty}$-algebra of $N=1$ SUSY Yang-Mills theory.}  \nIn this subsection we show by explicit calculation that the bilinear operation we defined on the complex \\rf{fo} is homotopy associative and the differential acts on it as a derivation.\n\nFor simplicity we denote $\\mu_n(\\cdot, ..., \\cdot)$ as $(\\cdot, ..., \\cdot)_h$. \nWe remind that  we defined the graded symmetric bilinear operation $(\\cdot,\\cdot)_h$, i.e. \n$(a_1,a_2)_h=(-1)^{|a_1||a_2|}(a_2,a_1)_h$ on the complex \\rf{sfo} in the following way:\n\\begin{eqnarray}\n&&(\\Lambda_1,\\Lambda_2)_h=\\Lambda_1\\Lambda_2,\\nonumber\\\\\n&&(\\Lambda,V)_h=\\frac{1}{2}\\Lambda V,\\nonumber\\\\\n&&(\\Lambda,W_{\\alpha})_h=\\Lambda W_{\\alpha},\\nonumber\\\\\n&&(\\Lambda,\\tilde W_{\\alpha})_h=\\Lambda\\tilde W_{\\alpha},\\nonumber\\\\\n&&(\\Lambda,\\tilde V)_h=\\frac{1}{2}\\Lambda\\tilde V,\\nonumber\\\\\n&&(\\Lambda,\\tilde\\Lambda)_h=\\Lambda\\tilde\\Lambda,\\nonumber\\\\\n&&(V_1,V_2)_{h,{\\alpha}}=-\\frac{1}{2}\\bar D^2(V_1D_{\\alpha}V_2-D_{\\alpha}V_1V_2),\\nonumber\\\\\n&&(V,W)_h=D_{\\alpha}VW^{\\alpha}+\\frac{1}{2}D^{\\alpha}W_{\\alpha}V,\\nonumber\\\\\n&&(\\tilde W,W)_h=\\tilde W^{\\alpha}W_{\\alpha},\\nonumber\\\\\n&&(\\tilde V, V)=-\\frac{1}{2}{\\bar{D}}^2(\\t VV),\\nonumber\\\\\n&&(W_1,W_2)_h=0,\\quad (V, \\tilde W)_h=0,\n\\end{eqnarray}\nsuch that $\\Lambda_1,\\Lambda_2\\in \\Phi$; $V,V_1,V_2\\in \\Sigma$; $W,W_1,W_2\\in\\Theta$, \n$\\tilde W\\in \\tilde \\Theta$; $\\tilde V\\in \\tilde \\Sigma$; $\\tilde\\Lambda\\in \\tilde\\Xi$. \n\nNow we start checking that this operation satisfies all necessary relations of \nhomotopy associative algebra. The first relation is the Leibniz rule, i.e. \n\\begin{eqnarray}\n{\\mathbb{D}} (a_1,a_2)_h=({\\mathbb{D}} a_1,a_2)_h+(-1)^{|a_1|}(a_1,{\\mathbb{D}} a_2)_h.\n\\end{eqnarray}\nLet us check it step by step:\n\\begin{eqnarray}\n&&{\\mathbb{D}} (\\Lambda,V)_h=\\frac{1}{2}\\bar D^2D_{\\alpha}(\\Lambda V)=\\nonumber\\\\\n&&\\frac{1}{2}\\bar D^2(D_{\\alpha}\\Lambda V-\\Lambda \nD_{\\alpha}V)+\\Lambda{\\bar D}^2 D_{\\alpha}V=\\nonumber\\\\\n&&({\\mathbb{D}}\\Lambda,V)_h+(\\Lambda,{\\mathbb{D}} V)_h,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&&{\\mathbb{D}} (\\Lambda,W)_h=D^{\\alpha}(\\Lambda W_{\\alpha})+\\Lambda W_{\\alpha}=\\nonumber\\\\\n&& D_{\\alpha}\\Lambda W^{\\alpha}+\\frac{1}{2}\\Lambda D^{\\alpha}W_{\\alpha}+\\Lambda W_{\\alpha}+\n\\frac{1}{2}\\Lambda D^{\\alpha}W_{\\alpha}=\\nonumber\\\\\n&&({\\mathbb{D}}\\Lambda,W)_h+(\\Lambda,{\\mathbb{D}} W)_h,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n{\\mathbb{D}} (V,W)_h=\\bar D^2(D_{\\alpha}VW^{\\alpha}+\\frac{1}{2}VD^{\\alpha}W_{\\alpha})=\\nonumber\\\\\n\\bar D^2 D^{\\alpha}VW_{\\alpha}+\\frac{1}{2}\\bar D^2VD^{\\alpha}W_{\\alpha}=\\nonumber\\\\\n({\\mathbb{D}} V,W)_h-(V,{\\mathbb{D}} W)_h,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n{\\mathbb{D}} (\\Lambda,\\tilde V)_h=\\frac{1}{2}\\bar D^2(\\Lambda \\tilde V)=-\\frac{1}{2}\\Lambda \\bar D^2\\tilde V+\n\\Lambda \\bar D^2\\tilde V=\n({\\mathbb{D}}\\Lambda,\\tilde V)_h+(\\Lambda,{\\mathbb{D}}\\tilde V)_h.\n\\end{eqnarray}\n\nOur next task is to verify that $(\\cdot,\\cdot)_h$ satisfies homotopy associativity relation:\n\\begin{eqnarray}\n&& (a_1,(a_2,a_3)_h)_h-((a_1,a_2)_h,a_3)_h+{\\mathbb{D}} (a_1,a_2,a_3)_h+ ({\\mathbb{D}} a_1,a_2,a_3)_h+\\nonumber\\\\\n&&(-1)^{|a_1|}(a_1,{\\mathbb{D}} a_2,a_3)_h+(-1)^{|a_1|+|a_2|}(a_1,a_2,{\\mathbb{D}} a_3)_h=0,\n\\end{eqnarray}\n\nwhere $(\\cdot,\\cdot,\\cdot)_h$ is the trilinear operation we will derive below.\n\nLet us proceed as above, checking associativity step by step:\n\n\\begin{eqnarray}\n&&(\\Lambda,(V_1,V_2)_h)_h-((\\Lambda,V_1)_h,V_2)_h=\\nonumber\\\\\n&&-\\frac{1}{2}\\bar D^2(\\Lambda V_1D_{\\alpha}V_2-\\Lambda D_{\\alpha}V_1 V_2)+\n\\frac{1}{4}\\bar D^2(\\Lambda V_1D_{\\alpha}V_2-V_1D_{\\alpha}(\\Lambda V_2))=\\nonumber\\\\\n&&-\\frac{1}{4}\\bar D^2(\\Lambda(V_1D_{\\alpha}V_2-\nD_{\\alpha}V_1 V_2)+D_{\\alpha}\\Lambda V_1V_2)=\\nonumber\\\\\n&&-{\\mathbb{D}} (\\Lambda,V_1,V_2)_h-({\\mathbb{D}}\\Lambda,V_1,V_2)_h-(\\Lambda,{\\mathbb{D}} V_1,V_2)_h+(\\Lambda,V_1,{\\mathbb{D}} V_2)_h,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&&(V_1,(\\Lambda,V_2)_h)_h-((V_1,\\Lambda)_h,V_2)_h=\\nonumber\\\\\n&&-\\frac{1}{4}\\bar D^2 (V_1D_{\\alpha}(\\Lambda V_2)-D_{\\alpha}V\\Lambda V_2)+\\frac{1}{4}\\bar D^2 (V_1\\Lambda D_{\\alpha}V_2-D_{\\alpha}(V_1\\Lambda V_2)=\\nonumber\\\\\n&&-\\frac{1}{2}\\bar D^2(V_1D_{\\alpha}\\Lambda V_2)=\\nonumber\\\\\n&&-{\\mathbb{D}} (V_1,\\Lambda,V_2)_h-({\\mathbb{D}} V_1,\\Lambda,V_2)_h+(V_1,{\\mathbb{D}}\\Lambda,V_2)_h+(V_1,\\Lambda,{\\mathbb{D}} V_2)_h,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&&(\\Lambda_1,(\\Lambda_2,V)_h)_h-((\\Lambda_1,\\Lambda_2)_h,V)_h=\\frac{1}{4}(\\Lambda_1\\Lambda_2 V)-\\frac{1}{2}(\\Lambda_1\\Lambda_2V)=\\nonumber\\\\\n&&-\\frac{1}{4}\\Lambda_1\\Lambda_2 V=\\nonumber\\\\\n&&-{\\mathbb{D}} (\\Lambda_1,\\Lambda_2,V)_h-({\\mathbb{D}}\\Lambda_1,\\Lambda_2,V)_h-(\\Lambda_1,{\\mathbb{D}}\\Lambda_2,V)_h-(\\Lambda_1,\\Lambda_2,{\\mathbb{D}} V)_h.\n\\end{eqnarray}\nSince we have \n\\begin{eqnarray}\n((\\Lambda_1,V)_h,\\Lambda_2)_h=(\\Lambda_1,(V,\\Lambda_2)_h)_h,\n\\end{eqnarray}\nwe obtain \n\\begin{eqnarray}\n&&(V_1,V_2,V_3)_h=\\frac{1}{6}\\bar D^2(V_1V_2D_{\\alpha}V_3-2V_1(D_{\\alpha}V_2)V_3+(D_{\\alpha}V_1)V_2V_3)\\nonumber\\\\\n&&(\\Lambda,V_1,V_2)_h=(V_1,V_2,\\Lambda)_h=\\frac{1}{12}\\Lambda V_1 V_2\\nonumber\\\\\n&&(V_1,\\Lambda,V_2)_h=\\frac{1}{6}\\Lambda V_1V_2\n\\end{eqnarray}\n\nOne can check that these expressions fit the formulas above.\nThen we need to study the associativity relation involving \n$W$-terms, i.e.\n\\begin{eqnarray}\n&&(\\Lambda, (V,W)_h)_h-((\\Lambda,V)_h,W)_h=\\nonumber\\\\\n&&\\frac{1}{2}\\Lambda D_{\\alpha}VW^{\\alpha}+\\frac{1}{4}D^{\\alpha}W_{\\alpha}V\\Lambda-\\frac{1}{2}\nD_{\\alpha}(\\Lambda V)W^{\\alpha}-\\frac{1}{4}\\Lambda VD^{\\alpha}W_{\\alpha}=\\nonumber\\\\\n&&-\\frac{1}{2}D_{\\alpha}\\Lambda VW^{\\alpha}=\\nonumber\\\\\n&&-{\\mathbb{D}}(\\Lambda,V,W)_h-({\\mathbb{D}}\\Lambda,V,W)_h-(\\Lambda,{\\mathbb{D}} V,W)_h+(\\Lambda,V,{\\mathbb{D}} W)_h,\n\\end{eqnarray}\nwhere $(V_1,V_2, W)_h= \\frac{1}{2}D_{\\alpha}V_1V_2W^{\\alpha}+\\frac{1}{6}V_1V_2D_{\\alpha}W^{\\alpha}$ and $\n(\\Lambda, V, \\t V)=\\frac{1}{6}V_1V_2\\t V$. Another terms involving $W$ are\n\\begin{eqnarray}\n&&(V_1,(V_2,W)_h)_h-((V_1,V_2)_h,W)_h=\\nonumber\\\\\n&&-\\frac{1}{2}\\bar D^2(\\frac{1}{2}V_1(D_{\\alpha}V_2W^{\\alpha}+\\frac{1}{2}D^{\\alpha}W_{\\alpha}V_2)+\\frac{1}{2}\\bar D^2(V_1D_{\\alpha}V_2-D_{\\alpha}V_1V_2)W^{\\alpha})=\\nonumber\\\\\n&&-\\bar D^2(\\frac{1}{2}D_{\\alpha}V_1V_2W^{\\alpha}+\\frac{1}{4}V_1D^{\\alpha}W_{\\alpha}V_2)\\nonumber\\\\\n&&-{\\mathbb{D}} (V_1,V_2,W)_h-({\\mathbb{D}} V_1,V_2,W)_h+(V_1,{\\mathbb{D}} V_2,W)_h-(V_1,V_2,{\\mathbb{D}} W)_h,\n\\end{eqnarray}\n \\begin{eqnarray}\n&&(V_1, (W,V_2)_h)_h-((V_1, W)_h,V_2)_h)=\\frac{1}{2}{\\bar D^2}(V_1(D^{\\alpha}V_2W_{\\alpha}+\\frac{1}{2}D^{\\alpha}W_{\\alpha}V_2)+\\nonumber\\\\\n&&V_2(D^{\\alpha}V_1W_{\\alpha}+\\frac{1}{2}D^{\\alpha}W_{\\alpha}V_1)=\\nonumber\\\\\n&&-{\\mathbb{D}} (V_1,W,V_2)_h-({\\mathbb{D}} V_1,W,V_2)_h+(V_1,{\\mathbb{D}} W,V_2)_h-(V_1,W,{\\mathbb{D}} V_2)_h.\n\\end{eqnarray}\nHere \n\\begin{eqnarray}\n&& (V_1,V_2,W)_h=(W,V_1,V_2)_h\n=\\frac{1}{2}D_{\\alpha}V_1V_2W^{\\alpha}+\\frac{1}{6}V_1V_2D_{\\alpha}W^{\\alpha},\\nonumber\\\\\n&& (V_1,W,V_2)_h=-\\frac{1}{2}(V_1D_{\\alpha}V_2+D_{\\alpha}V_1V_2)W^{\\alpha}-\\frac{1}{3}V_1D^{\\alpha}W_{\\alpha}V_2,\\nonumber\\\\\n&& (V_1,V_2,\\t V)_h=(\\t V, V_1,V_2)=\\frac{1}{12}\\bar D^2(V_1V_2\\t V),\\nonumber\\\\\n&& (V_1,\\t V, V_2)=\\frac{1}{6}\\bar D^2 (V_1\\t V V_2).\n\\end{eqnarray}\n\n\nTherefore from the calculations above one can obtain that algebraically up to the third order these equations look as follows:\n\\begin{eqnarray}\n&&{\\mathbb{D}} U+(U,U)_h+(U,U,U)_h+...=0,\\nonumber\\\\\n&&U\\to U+{\\mathbb{D}}\\Lambda+(U,\\Lambda)_h+\\nonumber\\\\\n&&(U,U,\\Lambda)_h-(U,\\Lambda,U)_h+(\\Lambda,U,U)_h+...,\n\\end{eqnarray}\nwhere $U$ is the element of the grading 1 from the complex \\rf{fo} corresponding to the pair $(V,W)$.\\\\\n\n\\noindent{\\bf B.3. Notations for N=2 3D superspace.} We work with three-dimensional space with coordinates $x^{\\mu}$ and euclidean metric $\\eta_{\\mu\\nu}=\\rm{diag}(+1,+1,+1)$. The Dirac matrices are \n$(\\gamma^{\\mu})^{\\beta}_{\\alpha}=i(\\sigma_1,\\sigma_2,\\sigma_3)$. We can raise and lower the corresponding spinor indices via $C^{\\alpha\\beta}$, i.e. $\\xi^{\\alpha}=C^{\\alpha\\beta}\\xi_{\\beta}$ and $\\xi_{\\alpha}=C_{\\alpha\\beta}\\xi^{\\beta}$, such that $C^{12}=-C_{12}=i$. \n\nThe $N=2$ 3D superspace has two 2-component anticommuting coordinates $\\theta^{\\alpha}_1$ and $\\theta^{\\alpha}_2$. Therefore (this is similar to $N=1$ superspace) one can define complex coordinates $\\theta^{\\alpha}=\\theta^{\\alpha}_1-i\\theta^{\\alpha}_2$ and   $\\bar\\theta^{\\alpha}=\\theta^{\\alpha}_1+i\\theta^{\\alpha}_2$.  This allows to  define  superderivatives (cf. $N=1$ D=4 case):\n\\begin{eqnarray}\nD_{\\alpha}=\\partial_{\\alpha}+\\frac{i}{2}\\bar{\\theta}^{\\beta}\\partial_{\\alpha\\beta}, \\quad \\bar D_{\\alpha}=\\bar \\partial_{\\alpha}+\\frac{i}{2}{\\theta}^{\\beta}\\partial_{\\alpha\\beta}\n\\end{eqnarray}\nwhere $\\partial_{\\alpha\\beta}=\\gamma^{\\mu}_{\\alpha\\beta}\\partial_{\\mu}$. One defines the (anti)chiral scalar fields via the \nfamiliar equation: $\\bar D_{\\alpha}\\Lambda(x,\\theta)=0$, $D_{\\alpha}\\bar \\Lambda(x,\\theta)=0$.    \nFor the calculations we will need the following commutation relations between superderivatives:\n\\begin{eqnarray}\n\\{D_{\\alpha},\\bar D_{\\beta}\\}=i\\gamma^{\\mu}_{\\alpha\\beta}\\partial_{\\mu}=i\\partial_{\\alpha\\beta}, \\quad \n\\{D_{\\alpha},D_{\\beta}\\}=0, \\quad \\{\\bar D_{\\alpha},\\bar D_{\\beta}\\}=0.\n\\end{eqnarray}\nAs well as in $N=1$ D=4 case, it is useful to introduce operators $D^2=C^{\\alpha\\beta}D_{\\alpha}D_{\\beta}$, $\\bar D^2=C^{\\alpha\\beta}\\bar D_{\\alpha}\\bar D_{\\beta}$.\\\\\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLattice QCD predicted that at sufficiently high temperatures\nand\/or densities the quarks and gluons confined inside\nhadrons get deconfined into a medium of quarks and gluons coined\nas quark-gluon Plasma. In the last few decades a large number\nof experiment has been involved in identifying this new state \nof matter in ultrarelativistic heavy-ion collisions (URHICs) at\nRHIC and LHC.  However, for the noncentral events in URHICs, \na strong \nmagnetic field is generated at the very early \nstages of the collisions due to very high relative velocities \nof the spectator quarks with respect to the fireball. \nDepending on the centralities of the collisions, the \nstrength of the magnetic fields may vary from $m_{\\pi}^2$ \n($\\sim 10^{18}$ Gauss) at RHIC to 10 $m_{\\pi}^2$ at \nLHC~\\cite{Skokov:IJMPA24'2009,Voronyuk:PRC83'2011}. \nMotivated by this, in the recent past many \ntheoretical works have started emerging to explore \nthe effects of this strong magnetic field on the various  \nQCD phenomena~\\cite{Fukushima:PRD78'2008,\nBraguta:PRD89'2014,Kharzeev:PRL106'2011,Gusynin:PRL73'1994}.\nEarlier the nascent strong magnetic field was thought to decay \nvery fast with time, resulting the magnetic field of weaker \nstrength. However, it was \nlater found that the realistic \nestimates of electrical conductivity of the medium \nmay elongate the life-time of the magnetic field\n~\\cite{Tuchin:AHEP2013'2013,Mclerran:NPA929'2014,\nRath:PRD100'2019}. It thus becomes imperative to investigate\nthe effects of both strong and weak magnetic field on the \nsignature of the novel matter produced in URHICs.\n\nThe heavy quarkonia is one of the \nprobe to study the properties of nuclear matter under\nextreme condition of temperature and magnetic field, because \nthe heavy quark pairs are formed in URHICs on a very short time-scale \n$\\sim 1\/2m_Q$ (where $m_Q$ is the mass of \nthe charm or bottom quark), which is similar to the \ntime-scale at which the magnetic field is generated. \nTherefore the study of the effects of magnetic field on the\nproperties of heavy quarkonia is worth of investigation.\nWe have recently studied the properties of quarkonia \nin strong magnetic field. However, as we know the quarkonia, \nthe physical resonances of $Q \\bar Q$ states, are formed in the \nplasma frame at a time, $t_F$ (=$\\gamma \\tau_F$), \nwhich is order of $1-2fm$ depending on the resonances and \ntheir momenta. By the time elapsed, the magnetic field may \nbecome weak, so in our present study, we aim to understand \ntheoretically the properties of heavy quarkonia and their \ndissociation in the presence of weak magnetic field\n($T^2>|q_fB|$, $T^2>m_f^2$, where $|q_f|$ ($m_f$) is the \nabsolute electric charge (mass) of \nthe $f$-th quark flavour). As we \nknow that, in order to study the dissociation of quarkonia \nthe perturbative computation of heavy quarkonium potential \nis needed. \n\nOur understanding of heavy quarkonium has taken a major step forward in computing effective field theories (EFT) from the underlying theory - QCD, such as non-relativistic QCD (NRQCD)\\cite{Bodwin:PRD51'1995} and potential NRQCD~\\cite{Brambilla:NPB566'2000}, which\nare synthesized successively by separating the intrinsic scales of heavy quark bound states (e.g. mass, velocity, binding energy) as well as the thermal medium-related scales (e.g. $T$, $gT$, $g^2 T$) in the weak-coupling system, in overall comparison with $\\Lambda_{QCD}$. However, in the relativistic collisions that are created at URHICs, the separation of scales in an EFT is not always apparent, meaning it is often difficult to construct a potential model. An alternative approach is a  first-principle lattice QCD simulation in which one studies spectral functions derived from Euclidean meson \ncorrelation~\\cite{Alberico:PRD77'2008}. The construction of spectral functions, however, is problematic\nbecause the temporal range at large temperatures decreases.\nFor this reason studies of quarkonia using finite temperature potential models are useful as a complement to lattice studies.\nThe perturbative computations of the potential at \nhigh temperatures show that the potential of \n$Q \\bar Q$ is complex~\\cite{Laine:JHEP03'2007 }, \nwhere the real part is screened due to the existence \nof deconfined color charges~\\cite{Matsui:PLB178'1986 } \nand the imaginary part ~\\cite{Beraudo:NPA806'2008 } \nassigns the thermal width to the resonance. Therefore \nthe physics of quarkonium dissociation in a medium \nhas been refined in the last two decades, where the \nresonances were initially thought to be dissociated \nwhen the screening is strong enough, {\\em i.e.} the \nreal-part of the potential is too weak to keep \nthe $Q\\bar Q$ pair together. Nowadays, the dissociation \nis thought to be  primarily because of the widening of \nthe resonance width arising either from the \ninelastic parton scattering mechanism mediated \nby the spacelike gluons, known as \nLandau damping~\\cite{Laine:JHEP03'2007 } or \nfrom the gluo-dissociation process during which\ncolor singlet state undergoes into a color octet \nstate by a hard thermal gluon\n~\\cite{Brambilla:JHEP1305'2013}. The latter processes \ntake precedence when the medium temperature\nis lower than the binding energy of the particular \nresonance. This dissociates the quarkonium even at \nlower temperatures where the probability of color \nscreening is negligible.\nRecently one of us estimated the imaginary-part of the potential \nperturbatively, where the inclusion of a \nconfining string term makes the (magnitude) imaginary component \nsmaller~\\cite{Lata:PRD89'2014,Lata:PRD88'2013}, compared to\nthe medium modification of the perturbative term alone\n\\cite{Adiran:PRD79'2009}. Gauge-gravity duality also indicates \nthat in strong coupling limit the potential also develops an \nimaginary component beyond a critical separation of \n$Q \\bar Q$ pair~\\cite{Binoy:PRD92'2015,Binoy:PRD91'2015}.\nMoreover lattice studies have also shown that the potential \nmay have a sizable imaginary part~\\cite{Rothkopf:PRL'2012}.\nThere are, however, other processes which may cause the \ndepopulation of the resonance states either through the \ntransition from ground state to the excited states during \nthe non adiabatic evolution of quarkonia~\\cite{Bagchi:MPLA30'2015} \nor through the swelling or shrinking of states due to the Brownian \nmotion of $Q \\bar Q$ states in the parton \nplasma~\\cite{Binoy:NPA708'2002}. Very recently the change in the \nproperties of heavy quarkonia immersed in a weakly-coupled\nthermal QCD medium has been described by HTL permittivity\n~\\cite{Lafferty:arxiv:1906.00035}. They used the generalized \nGauss law in conjunction to linear response theory to obtain \nthe real and imaginary parts of the heavy quark potential, \nwhere a logarithmic divergence in imaginary part is found \ndue to string contribution at large $r$.\nThey have circumvented by regularizing weak infrared\ndiverging ($1\/p$) term in the resummed gluon propagator \nby choosing the regulation scale in terms of Debye mass.\nThere is another recent work~\\cite{Guo:PRD100'2019}, where \na nonperturbative term induced by the dimension two gluon \ncondensate besides the usual HTL resummed contribution is \nincluded in the resummed gluon propagator to obtain \nthe string contribution in the potential, in \naddition to the Karsch Mehr Satz (KMS) potential~\\cite{KMS}.\n\nThe abovementioned studies are \nattributed for a thermal medium in the absence of a \nmagnetic field. However, as mentioned earlier that \na magnetic field is also \ngenerated in the heavy ion collisions, thus the influence of \na homogeneous and constant external magnetic field \non the heavy meson spectroscopy has been investigated \nquantum mechanically subjected to a three-dimensional harmonic \npotential and Cornell potential plus spin-spin \ninteraction term~\\cite{Alford:PRD88'2013,Bonati:PRD92'2015}.\nFurther, the effect of a constant uniform magnetic field \non the static quarkonium potential at zero and finite \ntemperature~\\cite{Bonati:PRD94'2016} and on the screening \nmasses~\\cite{Bonati:PRD95'2017} have been investigated. \nThe momentum diffusion coefficients of heavy quarks \nin a strong magnetic field along the directions parallel \nand perpendicular to the magnetic field at the leading order \nin QCD coupling constant has been \nstudied~\\cite{Fukushima:PRD93'2016}. Recently \nwe have explored the effects of strong magnetic \nfield on the properties of the heavy-quarkonium in \nfinite temperature by computing the real part of \nthe $Q \\bar Q$ potential~\\cite{Mujeeb:EPJC77'2017} \nin the framework of perturbative thermal QCD and \nstudied the dissociation of heavy quarkonia due to \nthe color screening. Successively, we made an attempt \nto study the dissociation of heavy quarkonia \ndue to Landau damping in presence\nof strong magnetic field by calculating the real\nand imaginary parts of the heavy quark potential in\npresence of strong magnetic field~\\cite{Mujeeb:NPA995'2020}.  \nThe complex heavy quark potential in presence of strong \nmagnetic field has also been obtained in~\\cite{Balbeer:PRD97'2018}. Very recently we have \nalso investigated the strong magnetic field-induced \nanisotropic interaction in heavy quark bound \nstates~\\cite{Salman:2004.08868}. The effects \nof strong magnetic field on the wakes in the induced charge density and in \nthe potential due to the passage of highly energetic \npartons through a thermal QCD medium has also been investigated~\\cite{Mujeeb:1901.03497}. \nRecently, the dispersion \nspectra of a gluon in hot QCD medium \nin presence of strong as well as weak magnetic field limit \nis studied~\\cite{karmakar:EPJC79'2019}. The effect of the \nstrong magnetic field on the collisional energy loss of heavy quark moving in a magnetized thermal partonic medium has \nbeen studied~\\cite{Balbeer:arxiv2002.04922}. Also the anisotropic momentum diffusion and the drag coefficients of heavy quarks have been computed in a strongly magnetized quark-gluon plasma beyond the static limit within the framework of Langevindynamics~\\cite{Balbeer:arxiv2004.11092}.\n\nIn the present study, we aim to obtain the complex heavy quark \nanti-quark potential in an environment of temperature and weak \nmagnetic field. For that purpose, we first start with the\nevaluation of gluon self energy in the similar environment\nusing the imaginary-time formalism.\nAs the quark-loop is only affected with the magnetic field \nthus, the quark-loop in the said environment \nis now dictated by both the scales namely the \nmagnetic field as well as  the \ntemperature, whereas for the gluon-loop, the temperature \nis the only available scale in the medium as the \ngluon-loop is not affected with the magnetic field. \nFurthermore, we have revisited the general structure of \ngluon self energy tensor in presence of weak magnetic field in \nthermal medium and obtained the relevant structure functions. \nHence the real and imaginary parts of the resummed gluon \npropagator have been obtained, which give the real and imaginary \nparts of the dielectric permittivity. The real and imaginary \nparts of the dielectric permittivity will inturn give the real \nand imaginary parts of the complex heavy quark potential. \nThe real part of the potential is \nused in the Schr\\\"{o}dinger equation to obtain the \nbinding energy of heavy quarkonia whereas the imaginary \npart is used to calculate the thermal width. Finally, \nwe have obtained the dissociation temperatures \nof heavy quarkonia and studied how the dissociation\ntemperatures get affected in presence of magnetic field. \n\nThus, our work proceeds as follows. In section 2, we will\ncalculate the gluon self energy in a weak magnetic field \nwherein, we will discuss the general structure of gluon \nself energy and resummed gluon propagator at finite \ntemperature in presence of weak magnetic field and will\ncalculate the relevant form factors in subsection 2.1 \nand subsection 2.2, respectively. Thus, the real and \nimaginary parts of the resummed gluon propagator will \ngive the real and imaginary parts of the dielectric \npermittivity in subsection 3.1, which gives the \nreal and imaginary parts of complex heavy quark potential \nin subsection 3.2. We will use the real and imaginary \nparts of the potential to obtain the binding energy and \nthermal width in subsection 4.1 and 4.2, respectively, \nwhich will then give the dissociation temperatures \nof heavy quarkonia in subsection 4.3. Finally,\nwe will conclude our findings in section 5.    \n\n\n\\section{Gluon self energy in a weak magnetic field}\nIn this section we will evaluate the gluon self energy in \na weak magnetic field. As we know that for the evaluation\nof gluon self energy, we need to evaluate both the quark\nloop and gluon loop contributions in presence of \nweak magnetic field. Because of weak magnetic field,\nonly the quark loop will get affected whereas the gluon \nloop remain as such. Now, we will first start with the \nquark-loop contribution to gluon self energy\n\\begin{eqnarray}\ni\\Pi^{\\mu\\nu}_{ab}(Q)&=&-\\int\\frac{d^4K}{(2\\pi)^4}Tr\n\\left[ i g t_b \\gamma^\\nu i S(K) i g t_a \\gamma^\\mu i S(P)\n\\right],\n\\nonumber\\\\\n&=&\\sum_f\\frac{g^2\\delta_{ab}}{2}\\int\\frac{d^4K}{(2\\pi)^4}Tr\n\\left[ \\gamma^\\nu i S(K) \\gamma^\\mu i S(P)\\right],\n\\label{self_energy}\n\\end{eqnarray}\nwhere $P=(K-Q)$ and $Tr(t_a t_b)=\\frac{\\delta_{ab}}{2}$. \nThe $S(k)$ is the quark propagator in a weak magnetic field\nwhich can be written upto order of $O(q_fB)^2$ as\n\\cite{ayala:1805.07344} \n\\begin{eqnarray}\niS(K)=i\\frac{(\\slashed{K}+m_f)}{K^2-m^2_f}-q_fB\n\\frac{\\gamma_1\\gamma_2(\\slashed{K}_\\parallel+m_f)}{(K^2-m_f^2)^2}\n-2i(q_f B)^2\\frac{[K_\\perp^2(\\slashed{K}_\\parallel+m_f)\n+\\slashed{K}_\\perp(m_f^2-K_\\parallel^2)]}\n{(K^2-m_f^2)^4},\n\\label{weak_propagator}\n\\end{eqnarray}\nwhere $m_f$ and $q_f$ are the mass and charge of the \n$f^{th}$ flavor quark. According to the following \nchoice of metric tensors,\n\\begin{eqnarray*}\ng^{\\mu\\nu}_\\parallel&=& {\\rm diag} (1,0,0-1),\\\\\n~g^{\\mu\\nu}_\\perp&=&{\\rm diag} (0,-1,-1,0),\n\\end{eqnarray*}\nthe four-momentum suitable in a magnetic field \ndirected along the $z$ axis, \n$n^\\mu =(0,0,0,-1)$, is given by\n\\begin{eqnarray}\nK^{\\mu}_\\parallel&=&(k_0,0,0,k_z),\\label{momentum_parallel}\\\\\nK^{\\mu}_\\perp&=&(0,k_x,k_y,0),\\label{momentum_perpendicular}\\\\\nK_{\\parallel}^2&=&k_{0}^2-k_{z}^2,\\\\ \nK_{\\perp}^2&=&k_{x}^2+k_{y}^2.\n\\end{eqnarray}\nThe above Eq.\\eqref{weak_propagator} can be recast in the \nfollowing form \n\\begin{eqnarray}\niS(K)=S_0(K)+S_1(K)+S_2(K), \n\\label{weak_propagator1}\n\\end{eqnarray}\nwhere \n$S_0(K)$ is the contribution of the order $O[(q_fB)^0]$,\n$S_1(K)$ is the contribution of the order $O[(q_fB)^1]$\nand $S_2(K)$ is the contribution of the order $O[(q_fB)^2]$.\nUsing Eq.\\eqref{weak_propagator1}, the Eq.\\eqref{self_energy} \ncan be written as\n\\begin{eqnarray}\n\\Pi^{\\mu\\nu}(Q)=-\\sum_f\\frac{ig^2}{2}\\int\\frac{d^4K}\n{(2\\pi)^4}Tr\\left[\\gamma^\\nu \\lbrace S_0(K)+S_1(K)+S_2(K)\n\\rbrace \\gamma^\\mu \\lbrace S_0(P)+S_1(P)+S_2(P)\\rbrace \n\\right].\n\\label{self_energy1}\n\\end{eqnarray}\nAfter simplifying, the above gluon self energy given by \nEq.\\eqref{self_energy1} can be expressed as follows\n\\begin{eqnarray}\n\\Pi^{\\mu\\nu}(Q)=\\Pi^{\\mu\\nu}_{(0,0)}(Q)+\\Pi^{\\mu\\nu}_{(1,1)}(Q)\n+2\\Pi^{\\mu\\nu}_{(2,0)}(Q)+O[(q_fB)^3],\n\\label{self_energy2}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\Pi^{\\mu\\nu}_{(0,0)}(Q)&=&-\\sum_f\\frac{ig^2}{2}\\int\\frac{d^4K}\n{(2\\pi)^4}Tr[\\gamma^\\nu S_0(K)\\gamma^\\mu S_0(P)],\\label{pi_00}\\\\\n\\Pi^{\\mu\\nu}_{(1,1)}(Q)&=&-\\sum_f\\frac{ig^2}{2}\\int\\frac{d^4K}\n{(2\\pi)^4}Tr\\lbrace \\gamma^\\nu S_1(K) \\gamma^\\mu S_1(P)\n\\rbrace,\\label{pi_11}\\\\\n\\Pi^{\\mu\\nu}_{(2,0)}(Q)&=&-\\sum_f\\frac{ig^2}{2}\\int\\frac{d^4K}\n{(2\\pi)^4}Tr\\left[\\gamma^\\nu S_2(K) \\gamma^\\mu S_0(P)\\right].\n\\label{pi_20}\n\\end{eqnarray}\nThe term $\\Pi^{\\mu\\nu}_{(0,0)}$ is of the order $O[(q_fB)^0]$, where \n$\\Pi^{\\mu\\nu}_{(1,1)}$ and $\\Pi^{\\mu\\nu}_{(2,0)}$ both are of the order \n$O[(q_fB)^2]$. The term which is of the order $O[(q_fB)^1]$ \nvanishes. Substituting the values of $S_0$, $S_1$ and $S_2$ \nin Eq.\\eqref{pi_00}, \nEq.\\eqref{pi_11} and Eq.\\eqref{pi_20} by comparing Eq.\\eqref{weak_propagator} \nwith Eq.\\eqref{weak_propagator1},  we get \n\n\\begin{eqnarray}\n\\Pi^{\\mu\\nu}_{(0,0)}(Q)&=&\\sum_f\\frac{ig^2}{2}\\int\\frac{d^4K}{(2\\pi)^4}\n\\frac{Tr[\\gamma^\\nu(\\slashed{K}+m_f)\\gamma^\\mu(\\slashed{P}+m_f)]}\n{(K^2-m^2_f)(P^2-m_f^2)},\\nonumber\\\\\n&=&\\sum_f i2g^2\\int\\frac{d^4K}{(2\\pi)^4}\\frac{\\left[P^\\mu K^\\nu+\nK^\\mu P^\\nu-g^{\\mu\\nu}(K\\cdot P-m_f^2)\\right]}\n{(K^2-m^2_f)(P^2-m_f^2)},\\\\\n\\Pi^{\\mu\\nu}_{(1,1)}(Q)&=&-\\sum_f\\frac{ig^2(q_fB)^2}{2}\\int\\frac{d^4K}\n{(2\\pi)^4}\\frac{Tr[\\gamma^\\nu\\gamma_1\\gamma_2\n(\\slashed{K}_\\parallel+m_f)\\gamma^\\mu\\gamma_1\\gamma_2\n(\\slashed{P}_\\parallel+m_f)]}\n{(K^2-m^2_f)^2(P^2-m_f^2)^2},\\nonumber\\\\\n&=&\\sum_f 2ig^2(q_fB)^2\\int\\frac{d^4K}{(2\\pi)^4}\n\\frac{\\left[P_\\parallel^\\mu K_\\parallel^\\nu +K_\\parallel^\\mu \nP_\\parallel^\\nu +(g_\\parallel^{\\mu\\nu}-g_\\perp^{\\mu\\nu})\n(m_f^2-K_\\parallel\\cdot P_\\parallel)\\right]}\n{(K^2-m^2_f)^2(P^2-m_f^2)^2},\\\\\n\\Pi^{\\mu\\nu}_{(2,0)}(Q)&=&-\\sum_f\\frac{2ig^2(q_fB)^2}{2}\\int\\frac{d^4K}{(2\\pi)^4}\n\\frac{Tr\\left[\\gamma^\\nu\\lbrace K_\\perp^2\n(\\slashed{K}_\\parallel+m_f)+\\slashed{K}_\\perp\n(m_f^2-K_{\\parallel}^2)\\rbrace\\gamma^\\mu(\\slashed{P}+m_f)\\right]}\n{(K^2-m_f^2)^4(P^2-m_f^2)},\n\\nonumber\\\\\n&=&-\\sum_f 4ig^2(q_fB)^2\\int\\frac{d^4K}{(2\\pi)^4}\n\\left[\\frac{M^{\\mu\\nu}}{(K^2-m_f^2)^4(P^2-m_f^2)}\\right],\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nM^{\\mu\\nu}&=&K_\\perp^2\\left[P^\\mu K_\\parallel^\\nu+\nK_\\parallel^\\mu P^\\nu-g^{\\mu\\nu}(K_\\parallel\\cdot P-m_f^2)\\right]\n+(m_f^2-K_\\parallel^2)\\left[P^\\mu K_\\perp^\\nu +K_\\perp^\\mu \nP^\\nu-g^{\\mu\\nu}(K_\\perp\\cdot P)\\right].~~~\n\\end{eqnarray}\nHere the strong coupling $g$ runs with the magnetic field and \ntemperature both, which is recently obtained in~\\cite{ayala:PRD98'2018}\n\\begin{eqnarray}\n\\alpha_s(\\Lambda^2,eB)=\\frac{g^2}{4\\pi}=\\frac{\\alpha_s(\\Lambda^2)}{1+\nb_1\\alpha_s(\\Lambda^2)\\ln\\left(\\frac{\\Lambda^2}\n{\\Lambda^2+eB}\\right)},\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\n\\alpha_s(\\Lambda^2)=\\frac{1}{\nb_1\\ln\\left(\\frac{\\Lambda^2}\n{\\Lambda_{\\overline{MS}}^2}\\right)},\n\\end{eqnarray}\nwhere $\\Lambda$ is set at $2\\pi T$, $b_1=\\frac{11N_c-2N_f}{12\\pi}$ and\n$\\Lambda_{\\overline{MS}}=0.176GeV$.\n\nBefore evaluating further, we will first discuss the structure\nof gluon self energy in thermal medium in presence of weak\nmagnetic field in the next subsection.\n \n\\subsection{Structure of gluon self energy and resummed gluon propagator \nfor thermal medium in the presence of weak magnetic field}\nIn this subsection, we will briefly discuss the general structure \nof gluon self energy tensor and resummed gluon propagator for thermal medium \nin the presence of weak\nmagnetic field. The general structure of gluon self energy in a \nthermal medium defined by the heat bath in local rest frame, \n$u^\\mu=(1,0,0,0)$ and in the presence of magnetic field directed \nalong the $z$-direction, $n_\\mu=(0,0,0,-1)$ is recently obtained \nas follows~\\cite{karmakar:EPJC79'2019} \n\\begin{eqnarray}\n\\Pi^{\\mu\\nu}(Q)=b(Q)B^{\\mu\\nu}(Q)+c(Q)R^{\\mu\\nu}(Q)+d(Q)M^{\\mu\\nu}(Q)\n+a(Q)N^{\\mu\\nu}(Q),\n\\label{self_decomposition}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nB^{\\mu\\nu}(Q)&=&\\frac{{\\bar{u}}^\\mu{\\bar{u}}^\\nu}{{\\bar{u}}^2},\\\\\nR^{\\mu\\nu}(Q)&=&g_{\\perp}^{\\mu\\nu}-\\frac{Q_{\\perp}^{\\mu}Q_{\\perp}^{\\nu}}\n{Q_{\\perp}^2},\\\\\nM^{\\mu\\nu}(Q)&=&\\frac{{\\bar{n}}^\\mu{\\bar{n}}^\\nu}{{\\bar{n}}^2},\\\\\nN^{\\mu\\nu}(Q)&=&\\frac{{\\bar{u}}^\\mu{\\bar{n}}^\\nu+{\\bar{u}}^\\nu{\\bar{n}}^\\mu}\n{\\sqrt{{\\bar{u}}^2}\\sqrt{{\\bar{n}}^2}},\n\\end{eqnarray}\nthe four vectors ${\\bar{u}}^\\mu$ and ${\\bar{n}}^\\mu$ used in the \nconstruction of above tensors are defined as follows\n\\begin{eqnarray}\n\\bar{u}^\\mu &=&\\left(g^{\\mu\\nu}-\\frac{Q^\\mu Q^\\nu}{Q^2}\\right)u_\\nu,\\\\\n\\bar{n}^\\mu &=&\\left(\\tilde{g}^{\\mu\\nu}-\\frac{\\tilde{Q}^\\mu\\tilde{Q}^\\nu}\n{\\tilde{Q}^2}\\right)n_\\nu,\n\\end{eqnarray}\nwhere ${\\tilde{g}}^{\\mu\\nu}=g^{\\mu\\nu}-u^\\mu u^\\nu$ and \n$\\tilde{Q}^\\mu=Q^\\mu-(Q.u)u^\\mu$. Using the properties of \nprojection tensors, the form factors appear in \n\\eqref{self_decomposition} can be obtained as\n\\begin{eqnarray}\nb(Q)&=&B^{\\mu\\nu}(Q)\\Pi_{\\mu\\nu}(Q),\n\\label{form_b}\\\\\nc(Q)&=&R^{\\mu\\nu}(Q)\\Pi_{\\mu\\nu}(Q),\n\\label{form_c}\\\\\nd(Q)&=&M^{\\mu\\nu}(Q)\\Pi_{\\mu\\nu}(Q),\n\\label{form_d}\\\\\na(Q)&=&\\frac{1}{2}N^{\\mu\\nu}(Q)\\Pi_{\\mu\\nu}(Q)\n\\label{form_a}.\n\\end{eqnarray}\nNow we can obtained the resummed gluon propagator in thermal medium\nin presence of weak magnetic field. The general form of the resummed \ngluon propagator in Landau gauge can be written \nas~\\cite{karmakar:EPJC79'2019} \n\\begin{eqnarray}\nD^{\\mu\\nu}(Q)=\\frac{(Q^2-d)B^{\\mu\\nu}}{(Q^2-b)(Q^2-d)-a^2}\n+\\frac{R^{\\mu\\nu}}{Q^2-c}+\\frac{(Q^2-b)M^{\\mu\\nu}}{(Q^2-b)(Q^2-d)-a^2}\n+\\frac{aN^{\\mu\\nu}}{(Q^2-b)(Q^2-d)-a^2}.\n\\end{eqnarray}\nThe point to be noted here is that, we required only the ``00''-component \nof resummed gluon propagator for deriving the heavy quark potential. \nHence the ``00''-component of the propagator can be obtained as \n\\begin{eqnarray}\nD^{00}(Q)=\\frac{(Q^2-d)\\bar{u}^2}{(Q^2-b)(Q^2-d)-a^2},\n\\label{propagator_00}\n\\end{eqnarray}\nwhere $R^{00}=M^{00}=N^{00}=0$. Now we will obtained the form factors\nappear in the above propagator \\eqref{propagator_00}. We will first\nstart with the form factor $a$, which can be obtained using Eq.\\eqref{form_a}\nwith Eq.\\eqref{self_energy2} as \n\\begin{eqnarray}\na(Q)=a_0(Q)+a_2(Q),\n\\end{eqnarray}\nwhere $a_0$ is of the order of $O(q_fB)^0$ and \n$a_2$ is of the order of $O(q_fB)^2$. An important point to be\nnoted here is that the zero magnetic field contribution of form \nfactor $a$ vanishes, that is $a_0=0$, whereas $a_2$ gives the \ncontribution of order $O(q_fB)^2$. However the contribution \nof form factor $a$ in the denominator of the \npropagator \\eqref{propagator_00} appear as $a^2$, which becomes \nof the order of $O(q_fB)^4$. Since in the current theoretical \ncalculation we are considering contribution upto $O(q_fB)^2$, \nso we can neglect the contribution appear from the form factor \n$a$. Thus, the ``00''-component of resummed gluon \npropagator upto $O(q_fB)^2$ can be written as\n\\begin{eqnarray}\nD^{00}(Q)=\\frac{\\bar{u}^2}{(Q^2-b)},\n\\label{propagator_final}\n\\end{eqnarray}\nso we end up with only one form factor $b$, which we will \nevaluate in the next subsection.\n    \n\\subsection{Real and imaginary parts of the form factor $b(Q)$}\nIn this subsection, we will calculate the real and imaginary\nparts of the form factor \n$b$. Using Eq.\\eqref{form_b}, the form factor $b$ \ncan be evaluated as follows \n\\begin{eqnarray}\nb(Q)&=&B_{\\mu\\nu}(Q)\\Pi^{\\mu\\nu}(Q),\\nonumber\\\\\nb(Q)&=&\\frac{{\\bar{u}}_\\mu{\\bar{u}}_\\nu}{{\\bar{u}}^2}\\Pi^{\\mu\\nu}(Q),\\nonumber\\\\\n&=&\\left[\\frac{u_\\mu u_\\nu}{{\\bar{u}}^2}-\\frac{(Q.u)u_\\nu Q_\\mu}{\\bar{u}^2Q^2}-\\frac{(Q.u)u_\\mu Q_\\nu}{\\bar{u}^2Q^2}+\\frac{(Q.u)^2Q_\\nu Q_\\mu}{\\bar{u}^2Q^4}\\right]\\Pi^{\\mu\\nu}(Q),\\nonumber\\\\\n&=&\\frac{u_\\mu u_\\nu}{{\\bar{u}}^2}\\Pi^{\\mu\\nu}(Q),\n\\label{correct_form}\n\\end{eqnarray}\nwhere we have used transversality condition $Q_\\mu\\Pi^{\\mu\\nu}(Q)=Q_\\nu\\Pi^{\\mu\\nu}(Q)=0$, to arrive at Eq.\n\\eqref{correct_form}. Thus using Eq.\\eqref{self_energy2}, the form factor b can be \nwritten upto $O[(q_fB)^2]$  as\n\\begin{eqnarray}\nb(Q)=b_0(Q)+b_2(Q),\n\\label{formfactor_b}\n\\end{eqnarray}\nwhere the form factors $b_0$ and $b_2$ are defined as follows\n\\begin{eqnarray}\nb_0(Q)&=&\\frac{u_\\mu u_\\nu}{\\bar{u}^2}\\Pi^{\\mu\\nu}_{(0,0)}(Q),\n\\label{formfactor_b0}\\\\\nb_2(Q)&=&\\frac{u_\\mu u_\\nu}{\\bar{u}^2}[\\Pi^{\\mu\\nu}_{(1,1)}(Q)+\n2\\Pi^{\\mu\\nu}_{(2,0)}(Q)].\n\\label{formfactor_b2}\n\\end{eqnarray}\n\\textbf{\\underline{Form factor $b_0(Q)$ (order of $O[(q_fB)^0]$)}}:\\\\\n\\\\\nHere we will solve the form factor $b_0$. Using \nEq.\\eqref{formfactor_b0}, the form factor can be\nwritten as\n\\begin{eqnarray}\nb_0(Q)&=&\\frac{u_\\mu u_\\nu}{\\bar{u}^2}\\Pi^{\\mu\\nu}_{(0,0)}(Q),\n\\nonumber\\\\\n&=&\\sum_f \\frac{i2g^2}{\\bar{u}^2}\\int\\frac{d^4K}{(2\\pi)^4}\\frac\n{\\left[2k_0^2-K^2+m_f^2\\right]}\n{(K^2-m^2_f)(P^2-m_f^2)}.\n\\end{eqnarray}\nNow we will solve the form factor $b_0$ using the imaginary-time \nformalism, the detailed calculation for which has been shown \nin appendix~\\ref{b_0}. \nThus, the real and imaginary parts of the \nform factor $b_0$ in the static limit \nare given as follows\n\\begin{eqnarray}\n{\\rm Re}~b_0(q_0=0)&=&g^2T^2\\frac{N_f}{6},\\\\\n\\left[\\frac{{\\rm Im}~b_0(q_0,q)}{q_0}\\right]_\n{q_0=0}&=&\\frac{g^2T^2N_f}{6}\\frac{\\pi}{2q}.\n\\end{eqnarray} \nNow we will evaluate the gluonic contribution. The \ntemporal component of gluon self energy due to the \ngluon-loop contribution can be calculated as\n~\\cite{Weldon:PRD26'1982,Pisarski:PRL63'1989},\n\\begin{eqnarray}\n\\Pi^{00}(q_0,q)=-g^2 T^2 \\frac{N_c}{3}\\left(\\frac{q_0}\n{2q}\\ln\\frac\n{q_0+q+i\\epsilon}{q_0-q+i\\epsilon}-1\\right)~,\n\\end{eqnarray}\nwhich gives the real and imaginary parts of form \nfactor $b_0$ due to the gluonic contribution in the\nstatic limit\n\\begin{eqnarray}\n{\\rm Re}~b_0(q_0=0)&=&g^2T^2\\left(\\frac{N_c}{3}\\right),\\\\\n\\left[\\frac{{\\rm Im}~b_0(q_0,q)}{q_0}\\right]_{q_0=0}\n&=&g^2T^2\\left(\\frac{N_c}{3}\\right)\\frac{\\pi}{2q}.\n\\label{img_b0}\n\\end{eqnarray}\nNow we add the quark and gluon-loop contributions together to\nobtain the real and imaginary parts of form factor $b_0$ in the \nstatic limit as follows \n\\begin{eqnarray}\n{\\rm Re}~b_0(q_0=0)&=&g^2T^2\\left(\\frac{N_c}{3}+\\frac{N_f}{6}\\right),\n\\label{real_b0}\\\\\n\\left[\\frac{{\\rm Im}~b_0(q_0,q)}{q_0}\\right]_{q_0=0}\n&=&g^2T^2\\left(\\frac{N_c}{3}+\\frac{N_f}{6}\\right)\\frac{\\pi}{2q}.\n\\label{img_b0}\n\\end{eqnarray}\nThus we can see that the form factor $b_0$ is independent of the\nmagnetic field as it is $O[(q_fB)^0]$ and depends only\non the temperature of the medium. This form factor $b_0$  \ncoincides with the HTL form factor $\\Pi_L$ in absence of \nthe magnetic field~\\cite{Weldon:PRD26'1982,Pisarski:PRL63'1989}.\n\n\\textbf{\\underline{Form factor $b_2(Q)$ (order of $O[(q_fB)^2]$)}}:\\\\\n\\\\\nHere we will discuss the form factor $b_2$, which is of the \norder of $O[(q_fB)^2]$. Using Eq.\\eqref{formfactor_b2}, the \nform factor is given by \n\\begin{eqnarray}\nb_2(Q)&=&\\frac{u_\\mu u_\\nu}{\\bar{u}^2}[\\Pi^{\\mu\\nu}_{(1,1)}(Q)+\n2\\Pi^{\\mu\\nu}_{(2,0)}(Q)],\n\\nonumber\\\\\n&=&\\sum_f \\frac{i2g^2(q_fB)^2}{\\bar{u}^2}\\left[\\int\\frac{d^4K}{(2\\pi)\n^4}\\left\\lbrace\\frac{\\left(2k_0^2-K_\\parallel^2+m_f^2\\right)}\n{(K^2-m^2_f)^2(P^2-m_f^2)^2}\n-\\frac{\\left(8k_0^2K_\\perp^2\\right)}{(K^2-m^2_f)^4(P^2-m_f^2)}\\right\n\\rbrace\\right].~~~\n\\end{eqnarray}\nWe have calculated the real and imaginary parts of the \nform factor $b_2$ in the appendix~\\ref{b_2}, which gives the \nreal and imaginary parts of the form factor $b_2$ in the \nstatic limit as follows\n\\begin{eqnarray}\n{\\rm Re}~b_2(q_0=0)=\\sum_f\\frac{g^2}{12\\pi^2 T^2}(q_fB)^2\n\\sum_{l=1}^{\\infty}(-1)^{l+1}l^2K_0(\\frac{m_fl}{T}).\n\\label{real_b2}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\left[\\frac{{\\rm Im}~b_2(q_0,q)}{q_0}\\right]_\n{q_0=0}&=&\\frac{1}{q}\\left[\\sum_f\\frac{g^2(q_fB)^2}{16\\pi T^2}\n\\sum_{l=1}^\\infty(-1)^{l+1}l^2K_0\\left(\\frac{m_f l}{T}\\right)\n\\right. \\nonumber\\\\ &&\\left. \n-\\sum_f\\frac{g^2(q_fB)^2}{96\\pi T^2}\n\\sum_{l=1}^\\infty(-1)^{l+1}l^2K_2\\left(\\frac{m_f l}{T}\\right)\n\\right. \\nonumber\\\\ &&\\left.\n+\\sum_f\\frac{g^2(q_fB)^2}{768\\pi}\\frac{(8T-7\\pi m_f)}{m_f^2 T}\n\\right],\n\\label{img_b2}\n\\end{eqnarray}\nwhere $K_0$ and $K_2$ are the modified Bessel functions \nof second kind.\n\nAfter obtaining the real and imaginary parts of\nthe form factor $b_0$ and $b_2$, we can write the \nreal and imaginary parts of form factor $b$ using \nEq.\\eqref{formfactor_b} as follows\n\\begin{eqnarray}\n{\\rm Re}~b(q_0=0)&=&\ng^2T^2\\left(\\frac{N_c}{3}+\\frac{N_f}{6}\\right)+\n\\sum_f\\frac{g^2}{12\\pi^2 T^2}(q_fB)^2\n\\sum_{l=1}^{\\infty}(-1)^{l+1}l^2K_0(\\frac{m_fl}{T}),\n\\label{real_b}\\\\\n\\left[\\frac{{\\rm Im}~b(q_0,q)}{q_0}\\right]_\n{q_0=0}&=&g^2T^2\\left(\\frac{N_c}{3}+\\frac{N_f}{6}\\right)\\frac{\\pi}{2q}\n+\\frac{1}{q}\\left[\\sum_f\\frac{g^2(q_fB)^2}{16\\pi T^2}\n\\sum_{l=1}^\\infty(-1)^{l+1}l^2K_0\\left(\\frac{m_f l}{T}\\right)\n\\right. \\nonumber\\\\ &-&\\left. \n\\sum_f\\frac{g^2(q_fB)^2}{96\\pi T^2}\n\\sum_{l=1}^\\infty(-1)^{l+1}l^2K_2\\left(\\frac{m_f l}{T}\\right)\n+\\sum_f\\frac{g^2(q_fB)^2}{768\\pi}\\frac{(8T-7\\pi m_f)}{m_f^2 T}\n\\right],\n~~\\label{img_b}\n\\end{eqnarray}\nwhere Eq.\\eqref{real_b} is the real-part of the form factor \nin the static limit which gives the Debye screening mass in \nthe presence of weak magnetic field as follows\n\\begin{eqnarray}\nM_D^2=g^2T^2\\left(\\frac{N_c}{3}+\\frac{N_f}{6}\\right)+\n\\sum_f\\frac{g^2}{12\\pi^2 T^2}(q_fB)^2\n\\sum_{l=1}^{\\infty}(-1)^{l+1}l^2K_0(\\frac{m_fl}{T}).\n\\label{debyemass}\n\\end{eqnarray}\nThus, it is observed that Debye screening mass of the thermal \nmedium in the presence of weak magnetic field is affected \nby both the temperature and magnetic field. Now in order to see\nhow the Debye mass is changed in the presence of weak magnetic field \nwe have mentioned the leading order result of Debye mass \nfor thermal medium in absence of magnetic \n(termed as ``Pure Thermal'')~\\cite{Shuryak:ZETF'1978}.\n\\begin{eqnarray}\nM^2_D({\\rm Pure~Thermal})=g^2 T^2 \n\\left(\\frac{N_c}{3} +\\frac{N_f}{6}\\right).\n\\end{eqnarray}\nIn the left panel of Fig.\\ref{debye}, we have quantitatively\nstudied the variation of Debye mass with varying strength \nof weak magnetic field for a fixed value of temperature. We \nhave observed that the debye mass is found to increase\nwith the varying strength of magnetic field. On the other hand, \nin the right panel\nof Fig.\\ref{debye}, we have studied the variation with the \ntemperature for a fixed value of magnetic field and \nobserved that the Debye mass is also found to increase \nwith increasing temperature, but \nthe increase of Debye mass with temperature is fast \nas compared to the slow increase with magnetic field. In\naddition to this, we have also made a comparison of Debye\nmass in presence of magnetic field with the one in absence \nof magnetic field and observed that the Debye mass in presence \nof weak magnetic field is found to be slightly higher as \ncompared to the one in pure thermal case. \n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{c c}\n\\includegraphics[width=7.6cm,height=8.4cm]{debye_mag.eps}&~~~~~\n\\includegraphics[width=7.5cm,height=7.6cm]{debye_temp.eps}\\\\\n\\end{tabular}\n\\caption{Variation of Debye mass with magnetic field (left panel)\nand with temperature (right panel).}\n\\label{debye}\n\\end{center}\n\\end{figure}\n\n\n\\section{Medium modified heavy quark potential}\nIn this section we will discuss the medium modification to \nthe potential between a heavy quark $Q$ and its anti-quark \n$\\bar{Q}$ in the presence of weak magnetic field at \nfinite temperature.\nSince the mass of the heavy quark ($m_Q$) is very large, \nso the requirements - $m_Q \\gg T \\gg \\Lambda_{QCD}$ and \n$m_Q \\gg \\sqrt{eB}$ are satisfied for the description of \nthe interactions between a pair of heavy quark and \nanti-quark at finite temperature in a weak \nmagnetic field in terms of quantum mechanical \npotential, that leads to the validity of \ntaking the static heavy quark potential. \nThus we can obtain the \nmedium-modification to the vacuum potential in the \npresence of magnetic field by correcting both its \nshort and long-distance part \nwith a dielectric function $\\epsilon(q)$ \nas \n\\begin{equation}\nV(r;T,B)=\\int\\frac{d^3q}{(2\\pi)^{3\/2}}\n({e^{iq.r}-1})\\frac{V(q)}{\\epsilon(q)},\n\\label{pot_defn}\n\\end{equation}\nwhere the $r$-independent term has subtracted to renormalize \nthe heavy quark free energy, which is the perturbative \nfree energy of quarkonium at infinite separation. The Fourier \ntransform,  $V(q)$  of the perturbative part of the Cornell \npotential ($V(r)=-\\frac{4\\alpha_s}{3r}$) \nis given by\n\\begin{equation}\n{V}(q)=-\\frac{4}{3}\\sqrt{\\frac{2}{\\pi}} \n\\frac{\\alpha_s}{q^2},\n\\label{ft_pot}\n\\end{equation}\nand the dielectric permittivity, $\\epsilon(q)$, \nembodies the effects of confined medium in the presence of \nmagnetic field is to be calculated next. The important \npoint to be noted here is that we have taken the Fourier \ntransform of the perturbative part of the vacuum potential \nonly, the reason for this is that we can not use the same \nscreening scale for both Coulomb and string terms because \nof the non-perturbative \nnature of the string term. To include the non-perturbative \npart of the potential, we will use the method of dimension \ntwo gluon condensate. \n\\subsection{The complex permittivity for a hot QCD medium \nin a weak magnetic field}\nThe complex dielectric permittivity, $\\epsilon (q)$ is \ndefined by the static limit of ``00''-component \nof resummed gluon propagator from the linear\nresponse theory\n\\begin{equation}\n\\frac{1}{\\epsilon (q)}=-\\displaystyle\n{\\lim_{q_0 \\rightarrow 0}}{q}^{2}D^{00}(q_{0}, \nq).\n\\label{dielectric}\n\\end{equation}\nNow we will evaluate the ``00''-component of resummed \ngluon propagator. The real-part of the resummed \ngluon propagator in the static limit \ncan be evaluated by using \nEq.\\eqref{propagator_final} and Eq.\\eqref{real_b} \n\\begin{eqnarray}\n{\\rm Re}~D^{00}(q_0=0)=\\frac{-1}{q^2+M_D^2}.\n\\label{real_resummed}\n\\end{eqnarray}\nThe imaginary part of resummed\ngluon propagator can be written in terms\nof the real and imaginary parts of the \nform factor by using the following \nformula~\\cite{Weldon:PRD42'1990}\n\\begin{eqnarray}\n{\\rm Im}~D^{00}(q_0,q)=\\frac{2T}{q_0}\\frac{{\\rm Im}~b(q_0,q)}\n{(Q^2-{\\rm Re}~b(q_0,q))^2+({\\rm Im}~b(q_0,q))^2},\n\\end{eqnarray}\nwhich can be recast into the following form\n\\begin{eqnarray}\n{\\rm Im}~D^{00}(q_0,q)=2T\\frac{\\left[\\frac{{\\rm Im}~b(q_0,q)}\n{q_0}\\right]}\n{(Q^2-{\\rm Re}~b(q_0,q))^2+(q_0\\left[\\frac{{\\rm Im}~b(q_0,q)}\n{q_0}\\right])^2},\n\\end{eqnarray}\nin the static limit the above equation reduces to \nthe simplified form\n\\begin{eqnarray}\n{\\rm Im}~D^{00}(q_0=0)=2T\\frac{\\left[\\frac{{\\rm Im}~\nb(q_0,q)}{q_0}\\right]_{q_0=0}}{(q^2+M_D^2)^2},\n\\label{resummed}\n\\end{eqnarray}\nwhere we have substituted ${\\rm Re}~b(q_0=0)=M_D^2$. \nUsing Eq.\\eqref{img_b} and the above Eq.\\eqref{resummed}, \nthe imaginary part of ``00''-component of resummed gluon\npropagator can be written as follows \n\\begin{eqnarray}\n{\\rm Im}~D^{00}(q_0=0,q)=\\frac{\\pi T M^2_{(T,B)}}{q(q^2+M_D^2)^2},\n\\label{img_resummed}\n\\end{eqnarray}\nwhere we have defined the quantity $M^2_{(T,B)}$ as follows\n\\begin{eqnarray}\nM_{(T,B)}^2&=&g^2T^2\\left(\\frac{N_c}{3}+\\frac{N_f}{6}\\right)\n+\\left[\\sum_f\\frac{g^2(q_fB)^2}{8\\pi^2 T^2}\n\\sum_{l=1}^\\infty(-1)^{l+1}l^2K_0\\left(\\frac{m_f l}{T}\\right)\n\\right. \\nonumber\\\\ &&-\\left. \n\\sum_f\\frac{g^2(q_fB)^2}{48\\pi^2 T^2}\n\\sum_{l=1}^\\infty(-1)^{l+1}l^2K_2\\left(\\frac{m_f l}{T}\\right)\n+\\sum_f\\frac{g^2(q_fB)^2}{384\\pi^2}\\frac{(8T-7\\pi m_f)}{m_f^2 T}\n\\right].~\n\\end{eqnarray}\nNow we will obtain the real and imaginary parts \nof dielectric permittivity, before evaluating them\nwe will discuss the procedure to handle the \nnonperturbative part of the heavy quark potential.\nThe handling of the nonperturbative part of the potential is recently \nbeen discussed in~\\cite{Guo:PRD100'2019}. The procedure \nis to include\na nonperturbative term in the real and imaginary parts\nof the ``00''-component of resummed gluon propagator along\nwith the usual Hard Thermal Loop (HTL) propagator which \nwe have obtained earlier. The real and imaginary parts of the \nnonperturbative (NP) term by using the dimension two gluon\ncondensate are given as follows\n\\begin{eqnarray}\n{\\rm Re}~D^{00}_{NP}(q_0=0,q)=-\\frac{m_G^2}{(q^2+M_D^2)^2},\\\\\n{\\rm Im}~D^{00}_{NP}(q_0=0,q)=\\frac{2\\pi TM^2_{(T,B)}m_G^2}\n{q(q^2+M_D^2)^3},\n\\end{eqnarray}\nwhere $m_G^2$ is a dimensional constant, which can be related \nto the string tension through the relation \n$\\sigma=\\alpha m_G^2\/2$. Thus, the real and \nimaginary parts of the ``00''-component of the \nresummed gluon propagator that consists of \nboth the HTL and the NP \ncontributions can be written as follows\n\\begin{eqnarray}\n{\\rm Re}~D^{00}(q_0=0,q)&=&-\\frac{1}\n{q^2+M_D^2}-\\frac{m_G^2}\n{(q^2+M_D^2)^2}\n\\label{real_propagator},\\\\\n{\\rm Im}~D^{00}(q_0=0,q)&=&\\frac{\\pi T M^2_{(T,B)}}\n{q(q^2+M_D^2)^2}+\\frac{2\\pi T M^2_{(T,B)}m_G^2}\n{q(q^2+M_D^2)^3}.\n\\label{imaginary_propagator}\n\\end{eqnarray}\nNow substituting Eq.\\eqref{real_propagator} and \nEq.\\eqref{imaginary_propagator} in Eq.\\eqref{dielectric} \ngives the real and imaginary parts of the dielectric \npermittivity, respectively\n\\begin{eqnarray}\n\\frac{1}{{\\rm Re}~\\epsilon (q)}&=&\\frac{q^2}\n{q^2+M_D^2}+\\frac{q^2 m_G^2}\n{(q^2+M_D^2)^2}\n\\label{real_dielectric},\\\\\n\\frac{1}{{\\rm Im}~\\epsilon (q)}&=&-\\frac{q\\pi T M^2_{(T,B)}}\n{(q^2+M_D^2)^2}-\\frac{2q\\pi T M^2_{(T,B)}m_G^2}\n{(q^2+M_D^2)^3}.\n\\label{imaginary_dielectric}\n\\end{eqnarray}\nWe are now going to derive the real and imaginary \nparts of the complex potential from the real and imaginary \nparts of dielectric permittivities, respectively in \nthe next subsection. The important point to be \nnoted here is that the non perturbative terms in the\nreal and imaginary parts of the dielectric permittivity\nwill lead to the string contribution in the \nreal and imaginary parts of the potential.\n\n\\subsection{Real and Imaginary parts of the potential}\nHere we will calculate the real and imaginary parts \nof the heavy quark potential in presence of weak magnetic field.\nThe real-part of the dielectric permittivity \nin Eq.\\eqref{real_dielectric} is substituted into the definition \nof potential in Eq.\\eqref{pot_defn} to obtain the real-part of \n$Q \\bar Q$ potential in the presence of weak magnetic \nfield \n(with $\\hat{r}=rM_{D}$)\n\\begin{eqnarray}\n\\rm{Re} V(r;T,B)&=&-\\frac{4}{3}\\alpha_s\\left(\\frac{e^{-\\hat{r}}}\n{r}+M_D\\right)+\\frac{4}{3}\\frac{\\sigma}{M_D}\n\\left(1-e^{-\\hat{r}}\\right),\n\\label{real_potential}\n\\end{eqnarray}\nwhere the temperature and magnetic field dependence in the\npotential enters through the \nDebye mass. While plotting the real-part of the \npotential we have excluded the non-local terms which are \nhowever, required to reduce the potential in the medium \n$V(r;T,B)$ to the vacuum potential in $(T, B) \\rightarrow 0 $ \nlimit. In Fig.\\ref{realb}, we have plotted the real-part of the \npotential as a function of interquark distance ($r$). In the left\npanel of Fig.\\ref{realb}, we have plotted the real-part of the \npotential for different strengths of weak magnetic field like \n$eB=0.5m_\\pi^2$ and $2m_\\pi^2$ for a fixed value of temperature \n$T=2T_c$. We observed that on increasing \nthe value of magnetic field the real-part become more screened.\nWhereas in the right panel of Fig.\\ref{realb}, the real-part\nis plotted for different strengths of temperature like $T=1.5T_c$\nand $T=2T_c$ and found to be more screened on increasing \nthe value of temperature. Thus, the real-part of the potential\nis found to be more screened on increasing the value of both \ntemperature and magnetic field. This observation of the \nreal-part of the potential can be understood in terms \nof the observation of the Debye mass which is found \nto be increased both with temperature and magnetic field \nas shown earlier in Fig.\\ref{debye}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{c c}\n\\includegraphics[width=7.5cm,height=7.5cm]{rpot_mag.eps}\n&~~~~~\n\\includegraphics[width=7.5cm,height=7.5cm]{rpot_temp.eps}\\\\\n\\end{tabular}\n\\caption{Real-part of the potential for different strengths of \nmagnetic field (left panel) and for different strengths of\ntemperature (right panel).}\n\\label{realb}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=7.5cm,height=7.5cm]{rpot_mag_comp.eps}\n\\caption{Real-part of the potential in the absence and \npresence of weak magnetic field.}\n\\label{real_comp}\n\\end{center}\n\\end{figure}\nWe have made a comparison in \nFig.\\ref{real_comp} to see how the magnetic field \nwill affect the real-part of the potential, for \nthat we have plotted the real-part of the \npotential in presence of magnetic field with the \none for pure thermal case. As we have \nseen in the right panel of Fig.\\ref{debye} that the Debye \nmass in presence of magnetic field is slightly higher\nas compared to the Debye mass in pure thermal medium,\nthat leads to the slightly more screening of the \nreal-part of the potential in presence of weak magnetic \nfield as compared to the same in the pure thermal case.   \n\nWe will now evaluate the imaginary-part of the\npotential in presence of weak magnetic field. The \nimaginary-part of the potential is obtained by substituting \nthe imaginary part of dielectric permittivity from \nEq.\\eqref{imaginary_dielectric} into the \ndefinition of the potential Eq.\\eqref{pot_defn}\n\\begin{eqnarray}\n\\rm{Im} V_C(r;T,B)&=&-\\frac{4}{3}\\frac{\\alpha_s T M^2_{(T,B)}}{M_D^2}\\phi_2(\\hat{r}),\\\\\n\\rm{Im} V_S(r;T,B)&=&-\\frac{4 \\sigma T M^2_{(T,B)}}{M_D^4}\n\\phi_3(\\hat{r}),\n\\label{imaginary_potential}\n\\end{eqnarray}\nwhere the function $\\phi_2(\\hat{r})$ and $\\phi_3(\\hat{r})$\nare given in~\\cite{Guo:PRD100'2019}\n\\begin{eqnarray}\n\\phi_2(\\hat{r})&=&2\\int_0^{\\infty}\\frac{zdz}{(z^2+1)^2}\n\\left[1-\\frac{\\sin z\\hat{r}}{z\\hat{r}}\\right],\\\\\n\\phi_3(\\hat{r})&=&2\\int_0^{\\infty}\\frac{zdz}{(z^2+1)^3}\n\\left[1-\\frac{\\sin z\\hat{r}}{z\\hat{r}}\\right],\n\\end{eqnarray}\nand in the small $\\hat{r}$ limit $(\\hat{r}\\ll 1)$, the above functions \nbecome \n\\begin{eqnarray}\n\\phi_2(\\hat{r})&\\approx &-\\frac{1}{9}{\\hat{r}}^2\\left(3\\ln \\hat{r}-\n4+3\\gamma_E\\right),\\\\\n\\phi_3(\\hat{r})&\\approx&\\frac{{\\hat{r}}^2}{12}+\\frac{{\\hat{r}}^4}{900}\n\\left(15\\ln\\hat{r}-23+15\\gamma_E\\right).\n\\end{eqnarray}\nIt is worth mentioning that we considered the imaginary part of the potential within the\nsmall distance limit ($\\hat{r}=rM_D\\ll 1$), so that it can be viewed as a perturbation. This could be relevant for the bound states of very heavy quarks, where Bohr radii, $r_B$ (=$\\frac{n^2}{g^2 m_Q}$) of quarkonia are smaller than the Debye length, $\\frac{1}{M_D}$. As we know that the former ($r_B$) is related to the scales of nonrelativistic heavy quark bound states in vacuum ($T=0$) \nand the scales associated to the thermal medium. In fact, the above condition ($r_B < \\frac{1}{M_D}$) is translated to the hierarchy for the validity of potential approach ($m_Q> T~ {\\rm or}~ gT$).\n\nSimilar to the real-part of the potential we have plotted the \nimaginary-part of the potential as a function \nof interquark distance ($r$) in Fig.\\ref{imgb}. We have calculated \nthe imaginary-part of the potential for different strengths of \nweak magnetic field like $eB=0.5m_\\pi^2$ and $2m_\\pi^2$ in the left\npanel of Fig.\\ref{imgb}. We found that on increasing \nthe value of magnetic field the magnitude of imaginary-part \ngets increased. On the other hand, in the right panel of \nFig.\\ref{imgb}, the imaginary-part is calculated for different \nstrengths of temperature like $T=1.5T_c$\nand $T=2T_c$, here also the imaginary-part is found to increase \nwith the temperature. Hence the magnitude of the imaginary-part of \nthe potential gets increased with the value of temperature \nand magnetic field both. This observation also attributed to the \nfact that the Debye mass is found to be increased with temperature \nand magnetic field both.\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{c c}\n\\includegraphics[width=7.5cm,height=7.5cm]{ipot_mag.eps}\n&~~~~~\\includegraphics[width=7.5cm,height=7.5cm]{ipot_temp.eps}\\\\\n\\end{tabular}\n\\caption{Imaginary-part of the potential for different strengths of \nmagnetic field (left panel) and for different strengths of\ntemperature (right panel).}\n\\label{imgb}\n\\end{center}\n\\end{figure}\nHere also we have calculated the imaginary-part of the potential \nin presence of magnetic field with the one for pure thermal \ncase in Fig.\\ref{img_comp}, where we observed that the \nimaginary-part of the potential in presence of magnetic field \nis increased slightly as compared to the one in pure thermal \ncase. \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=7.5cm,height=7.5cm]{ipot_mag_comp.eps}\n\\caption{Imaginary-part of the potential in the absence and \npresence of weak magnetic field.}\n\\label{img_comp}\n\\end{center}\n\\end{figure}\n\\newpage\n\n\n\\section{Properties of quarkonia}\nIn this section we first explore the effects of weak magnetic \nfield on the properties of heavy quarkonia. The obtained real\nand imaginary parts of the heavy quark potential will be \nused to evaluate the binding energy and thermal width of \nthe heavy quarkonia, respectively.\n\n\\subsection{Binding energy}\nIn this subsection, we have obtained the binding energy \nof $J\/\\psi$ and \n$\\Upsilon$. In order to calculate the \nbinding energy, the real part of \nthe potential Eq.\\eqref{real_potential} is put into the radial \npart of the Schr\\\"{o}dinger equation, which is then solved \nnumerically to obtain the energy eigenvalues that inturns give \nthe binding energies of quarkonia. To see how the presence of \nweak magnetic \nfield affects the binding of quarkonia, we have plotted the \nbinding energies of $J\/\\psi$ as a function of $T\/T_c$ for \ndifferent strengths of magnetic field in the left panel \nof Fig.\\ref{psi}. We observed that the binding energy is \nfound to decrease with the temperature and magnetic field\nboth, we can attribute this finding in terms of the increasing \nof screening with the temperature and magnetic field that \nwe have observed in the real-part of the potential. The point \nto be noted here is that the difference between the values \nof binding energies plotted for the magnetic field $eB=0.5m_\\pi^2$ \nand $eB=2m_\\pi^2$ is pronounced at higher\ntemperature, this is in accordance with the validity of our \nwork in the weak field limit $(T^2>|q_fB|)$. \n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{c c}\n\\includegraphics[width=7.5cm,height=7cm]{binding_psi.eps}&\n~~~~~\\includegraphics[width=7.5cm,height=7cm]{binding_psi_comp.eps}\n\\end{tabular}\n\\caption{The binding energy of $J\/\\psi$ as a function of \ntemperature.} \n\\label{psi}\n\\end{center}\n\\end{figure}   \n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{c c}\n\\includegraphics[width=7.5cm,height=7cm]{binding_upsilon.eps}&\n~~~~~\\includegraphics[width=7.5cm,height=7cm]{binding_upsilon_comp.eps}\n\\end{tabular}\n\\caption{The binding energy of $\\Upsilon$ as a function of temperature.}\n\\label{upsilon}\n\\end{center}\n\\end{figure} \n  \nIn the right panel of Fig.\\ref{psi}, we have also compared the \nbinding energy of $J\/\\psi$ in presence of weak magnetic field with\nthe pure thermal case. We found that the binding energy in presence \nof magnetic field is smaller as compared to the one in \npure thermal case, this is because the real-part of the potential\nin presence of magnetic becomes more screened as compared to \npure thermal case. The similar observation has also been\nobserved for $\\Upsilon$, except that the value of binding \nenergy for $\\Upsilon$ is higher as compared to the value for \n$J\/\\Psi$. The variation of binding energy for $\\Upsilon$ is \nstudied in the left and right panel of Fig.\\ref{upsilon}.  \n\\newpage\n\n\\subsection{Thermal width}\nWe will now use the imaginary part of the potential obtained in presence \nof weak magnetic field to estimate the broadening of the resonance states \nin a thermal medium. So\nusing the first-order time-independent perturbation theory, the width \n($\\Gamma$) has been evaluated by folding with ($\\Phi(r)$),\n\\begin{eqnarray}\n\\Gamma({\\rm T,B})=-2\\int_0^\\infty \\rm{Im}~V(r;T,B) |\\Phi(r)|^2 d\\tau,\n\\label{gammaT}\n\\end{eqnarray}\nthe wave function $\\Phi(r)$ is taken to be the Coloumbic wave function \nfor the ground state\n\\begin{eqnarray}\n\\Phi(r)=\\frac{1}{\\sqrt{\\pi a_0^3}}e^{-r\/a_0},\n\\end{eqnarray}\nwhere $a_0$ is the Bohr radius of the \nheavy quarkonium system. \nHere we have used the imaginary-part of the potential\nas a perturbation to obtain the thermal width, and for that \npurpose we have obtained the imaginary-part of the potential\nin the small distance limit.   \n \\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{c c}\n\\includegraphics[width=7.5cm,height=7.5cm]{width_psi.eps}&\n~~~~\\includegraphics[width=7.5cm,height=7.5cm]{width_psi_comp.eps}\n\\end{tabular}\n\\caption{Variation of the thermal widths with the temperature for \n$J\/\\psi$.}\n\\label{decay_psi}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{c c}\n\\includegraphics[width=7.5cm,height=7.5cm]{width_upsilon.eps}&\n~~~~\\includegraphics[width=7.5cm,height=7.5cm]{width_upsilon_comp.eps}\n\\end{tabular}\n\\caption{Variation of the thermal widths with the temperature for \n$\\Upsilon$.}\n\\label{decay_upsilon}\n\\end{center}\n\\end{figure}\nWe have obtained the thermal width numerically and observed that it depend on the temperature as well as the weak \nmagnetic field.\nTo explore the effects of the weak magnetic field on the \nthermal width of heavy quarkonia, we have plotted the \nthermal width of $J\/\\psi$ and $\\Upsilon$ as a function of \n$T\/T_c$ for different strengths of magnetic field in \nFig.\\ref{decay_psi} and Fig.\\ref{decay_upsilon}, respectively. \nWe observed that the thermal widths for $J\/\\psi$ and \n$\\Upsilon$ get increased both with the temperature and \nmagnetic field as depicted in the left panels of \nFig.\\ref{decay_psi} and Fig.\\ref{decay_upsilon}. We can \nunderstood this finding in terms of the increase of the \nimaginary-part of the potential, the magnitude of which \ngets enhanced both with temperature and magnetic field. \nWe also made a comparison of thermal width in presence \nof weak magnetic field with its counter part in absence \nof magnetic field in the right panels of \nFig.\\ref{decay_psi} and Fig.\\ref{decay_upsilon}, where we \nfound that the decay widths for $J\/\\Psi$ and $\\Upsilon$ \nget increased in the presence of magnetic field as compared \nto the pure thermal case.\n\n\\subsection{Dissociation of quarkonia}\nIn the previous subsections, we have obtained the binding energies \nand thermal widths of heavy quarkonia, $J\/\\psi$ and $\\Upsilon$. \nNow we will study the quasi-free dissociation of heavy quarkonia \nin a thermal QCD medium and see how the dissociation temperatures \nof quarkonia are affected in the presence of weak magnetic field. \nFor that purpose we use the criterion on the width of the resonance \n($\\Gamma$): $\\Gamma \\ge 2 ~{\\rm{BE}}$ \\cite{Mocsy:PRL99'2007} \n(where ${\\rm{BE}}$ is the binding energy of the heavy quarkonia) \nto estimate the dissociation temperature for $J\/\\psi$ \nand $\\Upsilon$.\n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n&\\multicolumn{2}{|c|}{Dissociation Temperatures $T_d$ in $T_c$} \\\\\n\\hline\nState & $J\/\\psi$ & $\\Upsilon$ \\\\ \n\\hline \nPure Thermal ($eB=0$)& 1.80 & 3.50 \\\\ \n\\hline \n$eB=0.5m_\\pi^2$ & 1.74 & 3.43 \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Dissociation temperatures in absence and \npresence of weak magnetic field.} \n\\label{table_diss}\n\\end{table}\nWe have obtained the dissociation temperatures of \n$J\/\\Psi$ and $\\Upsilon$ in the absence and presence of \nweak magnetic field in Table.\\ref{table_diss}, and \nobserved that the dissociation temperatures become \nslightly lower in the presence of weak magnetic field.\n{\\em For example}, with $eB = 0m_\\pi^2$ the $J\/\\psi$ and \n$\\Upsilon$ are dissociated at $1.80T_c$ and $3.50T_c$, \nrespectively whereas with $eB = 0.5m_\\pi^2$ the $J\/\\psi$ \nand $\\Upsilon$ are dissociated at $1.74T_c$ and $3.43T_c$.\nThis observation leads to the slightly early dissociation \nof heavy quarkonia in the presence of the weak magnetic field.\n\n\\section{Conclusions}\nIn the present theoretical study, we have explored the \neffects of weak magnetic field on the dissociation of \nquarkonia in a thermal QCD by calculating the \ncomplex heavy quark potential perturbatively\nin the aforesaid medium. For that purpose, we first evaluate \nthe gluon self-energy in a similar environment \nusing the imaginary-time formalism.\nFurthermore, we have revisited the general structure of \ngluon self-energy tensor in the presence of weak magnetic field in \nthermal medium and obtained the relevant structure functions, that \nin turn give rise to the real and imaginary parts of the\nresummed gluon propagator, which give the real and imaginary \nparts of the dielectric permittivity. To include the medium \nmodification to the non-perturbative part of the vacuum heavy \nquark potential, we have included a non-perturbative term in \nthe resummed gluon propagator induced by the dimension two \ngluon condensate besides the usual hard thermal\nloop resummed contribution. Thus, the real and imaginary \nparts of the dielectric permittivity will be used to evaluate \nthe real and imaginary parts of the complex heavy quark \npotential. We have studied the effects of weak magnetic field\non the real and imaginary parts of the potential. We have \nfound that the real-part of the potential is found to be more \nscreened on increasing the value of temperature and magnetic field \nboth. In addition to this, we have observed that the real-part\ngets slightly more screened in the presence of weak magnetic \nfield as compared to its counter part in the \nabsence of magnetic field. \nOn the other hand,  \nthe magnitude of the imaginary-part of the potential gets \nincreased with the value of both temperature and magnetic field,\nand its magnitude also gets increased in the presence \nof weak magnetic field as compared to pure thermal case.\nThe real part of the potential is used in the Schr\\\"{o}dinger \nequation to obtain the binding energy of heavy quarkonia, \nwhereas the imaginary part is used to calculate the \nthermal width. We observed that the binding energies of \n$J\/\\Psi$ and $\\Upsilon$ are found to decrease with the \ntemperature and magnetic field both, we can attribute this \nfindings in terms of the increasing of screening of the\nreal-part of the potential. We also observed that the binding energy of $J\/\\Psi$\nand $\\Upsilon$ in the presence of magnetic field are \nsmaller as compared to the one in the pure thermal case. The \nincrease in the magnitude of the imaginary-part of the \npotential will leads to the increase of decay width with \ntemperature and magnetic field both. The thermal width for \n$J\/\\Psi$ and $\\Upsilon$ get increased in presence of magnetic \nfield as compared to pure thermal case. With the observations \nof binding energy and decay width in hands, we have \nfinally studied the dissociation of quarkonia in the presence \nof weak magnetic field. The dissociation temperatures for \n$J\/\\Psi$ and $\\Upsilon$ become slightly lower in the the presence \nof weak magnetic field. {\\em For example}, with $eB = 0 m_\\pi^2$ \nthe $J\/\\psi$ and $\\Upsilon$ \nare dissociated at $1.80T_c$ and $3.50T_c$, respectively whereas \nwith $eB = 0.5m_\\pi^2$ the $J\/\\psi$ and $\\Upsilon$ are \ndissociated at $1.74T_c$ and $3.43T_c$. This observation leads \nto the slightly early dissociation of quarkonia because of the \npresence of a weak magnetic field.\n\n\n\\section*{Acknowledgements}\nOne of the author BKP is thankful to the CSIR (Grant No.03 (1407)\/17\/EMR-II), \nGovernment of India for the financial assistance.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \n\nIn   the late 1990's, M. Gromov in \\cite{Gromov} introduced the notion of mean dimension for a topological dynamical system $(X,\\phi)$\n($X$ is a compact topological space and $\\phi$ is a continuous map on $X$), which is, as well as the topological entropy, an invariant under conjugacy.    In \\cite{lind}, Lindenstrauss and Weiss showed  that the mean dimension is zero if the topological dimension\nof $X$ is finite. They  gave some examples where the mean dimension is positive. For instance, they proved that the mean dimension of   $(([0,1]^m)^{\\mathbb{Z}}, \\sigma)$, where $\\sigma$ is the two-sided full shift map on $([0,1]^m)^{\\mathbb{Z}}$,  which has infinite topological entropy,   is equals to $m$ and that any non-trivial factor of  $( ([0,1]^m)^{\\mathbb{Z}},\\sigma)$\nhas positive mean dimension. \n\n\\medskip\n\n\nGiven a dynamical system  $(X,\\phi)$, an interesting question related to such a system is the following: under what conditions is it possible to imbed  such a system in the shift space $(([0,1]^\\mathbb{N})^{\\mathbb{Z}}, \\sigma)$? That is, what properties the system must have to guarantee the existence of a continuous map $i: X\\to ([0,1]^\\mathbb{N})^{\\mathbb{Z}}$  satisfying $\\sigma \\circ i = i \\circ \\varphi$?  In \\cite{lind} the authors proved that \n a necessary condition for an invertible system $(X,\\phi)$ to be embedded in \n$(([0,1]^m)^{\\mathbb{Z}},\\sigma)$   is   that $\\text{mdim}(X,\\phi)\\leq m$, where  $\\text{mdim}(X,\\phi)$ denotes the mean dimension of the system $(X,\\phi)$.\nIn \\cite{lind3}  it was  proved that if $ (X, \\phi)$ is an invertible system which is an extension of a minimal system, and $K$ is  a convex set with non-empty interior such that $\\text{mdim} (X,\\phi)< \\text{dim} K \/36$, then  $(X, \\phi)$ can be embedded in the shift space $(K^{\\mathbb{Z}},\\sigma)$. In particular, if $\\text{mdim} (X,\\phi)< m\/36$, then  $(X, \\phi)$ can be embedded in $(([0,1]^m)^{\\mathbb{Z}},\\sigma)$. More recently, Gutman and Tsukamoto  \\cite{Gutman} showed that, that if $(X, \\phi)$ is a minimal system with $\\text{mdim}(X, \\phi) <N\/\n2$ then we can embed it in\n$(([0, 1]^{N})^{\\mathbb{Z}},\\sigma)$. In \\cite[Theorem 1.3]{lind4}, Lindenstrauss and Tsukamoto \n constructed a minimal system with mean dimension equal to $N\/2$ which cannot be\nembedded into $(([0, 1]^{N})^{\\mathbb{Z}},\\sigma)$, showing that the constant $N\/2$ obtained in \\cite{Gutman} is optimal. \n\n\\medskip\n\nThe notion of metric mean dimension for a dynamical system $\\phi:(X,d)\\to (X,d)$ was   introduced in \\cite{lind},\nwhere $(X,d)$ is a compact metric space with metric $d$  and $\\phi$ is a continuous map. It refines the topological entropy for systems with infinite entropy,\nwhich, in the case of a manifold of dimension greater than one,  form a residual subset of the set consisting of   homeomorphisms   defined on the manifold (see \\cite{Yano}). In fact,\n every system with finite topological entropy has metric mean dimension equals to zero and\nfor any metric $d'$ equivalent to $d$ on $X$ one has $\\text{mdim}(X,\\phi)\\leq \\text{mdim}_M(X,\\phi,d')$, where $\\text{mdim}_M(X,\\phi,d')$\ndenotes the metric mean dimension of $(X,\\phi)$ with respect to  $d'$ (see \\cite{lind2}, \\cite{lind}). The metric mean dimension depends on the metric   $d$, therefore it is not a  topological invariant. However, for a metrizable topological space $X$,  $\\text{mdim}_M(X,\\phi)=\\inf _{d'}\\text{mdim}_M(X,\\phi,d')$ is invariant under topological conjugacy, where the infimum  is taken over all the metrics on $X$ which induce the topology on $X$. In  \\cite{lind2}, Theorem 4.3, the author  proved  that if $(X,\\phi)$  is an extension of a minimal system, then there exists a metric $d^{\\prime}$ on $X$, equivalent to $d$, \nsuch that $\\text{mdim}(X,\\phi) = \\text{mdim}_{M}(X,\\phi, d^{\\prime})$.  \n\n\\medskip\n\nB. Kloeckner (\\cite{BK}) studied the dynamical system $(\\mathcal{P}(\\mathbb S^1),\\Phi_{d\\sharp})$, where\n$\\mathcal{P}(\\mathbb S^1)$ is the space of probability measures on the circle $\\mathbb S^1$ and\n$\\Phi_{d\\sharp}$ is the push-forward map induced by a $d$-expanding map $\\Phi_d:\\mathbb S^1\\to\\mathbb S^1$.\nThe author shows   if we  take the Wasserstein metric with cost function\n$|\\cdot |^p$ ($p\\in[1,\\infty)$) on $\\mathcal{P}(\\mathbb S^1)$, denoted by $\\mathcal{W}_p$,   then\n$\\text{mdim}_M(\\mathcal{P}(\\mathbb S^1),\\Phi_{d\\sharp},\\mathcal{W}_p)\\geq p(d-1)$. H. Lee (in \\cite{HL})\nintroduced the mean dimension for continuous actions of countable sofic groups on compact metrizable spaces and proved that, in this setting, the  mean dimension is an important tool for distinguishing continuous actions of countable\nsofic groups with infinite entropy.\n\n\\medskip\n\nA \\emph{non-autonomous dynamical system} (or a sequential dynamical system) is a sequence $\\textit{\\textbf{f}}=(f_n)_{n=1}^{\\infty}$  of continuous maps $f_n : X_n \\to X_{n+1}$, where $X_n$ is a compact topological space for every $n\\in \\mathbb N$.\nIn the last two decades, several authors have tried to extend some results that are valid for autonomous systems for the non-autonomous case. Kolyada and Snoda in \\cite{K-S} introduced the notion  of topological  entropy for this setting and proved that, just as in the case of autonomous systems, it is an invariant under equiconjugacy and furthermore  that it is concentrated in the non-wandering set of the  dynamics   (see \\cite{K-S} and \\cite{JeoCTE}). In a more recent work, Freitas et al \\cite{freitas} have analyzed the existence of Extreme Value Laws in this setting. In \\cite{Sta} Stadlbauer guarantees, under appropriate conditions, the existence of a spectral gap for transference operators associated with sequential systems.\n\n\\medskip\n\nAs we said above, the set consisting of continuous maps with infinite topological entropy is residual. On the other hand, it is easy to build non-autonomous dynamical systems with infinite topological entropy (take $\\phi$ a continuous map with positive topological entropy, then $(\\phi,\\phi^{2},\\phi^{2^{2}},\\phi^{2^3},\\dots)$ is a non-autonomous dynamical systems with infinite topological entropy).  This is the main reason to extend the concepts of mean dimension  and metric mean dimension to non-autonomous systems, since these become  a tool to classify non-autonomous dynamical systems with infinite topological entropy (see Theorem \\ref{edee344}).\n\n\n\n \n\n\\medskip\n\n {In the next two sections we will extend  the mean dimension and the metric mean dimension for  a non-autonomous dynamical system $ \\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$, which will be denoted by \n$ \\text{mdim}(X,\\textit{\\textbf{f}}\\,)$. Furthermore, we will prove  some properties   which are valid for  the entropy of non-autonomous dynamical systems  (see \\cite{K-S} and \\cite{JeoCTE}).  An application of these properties is that, for any continuous maps $\\phi$ and $\\psi$ on $X$, the    compositions $\\phi\\circ \\psi$ and $\\psi\\circ \\phi$  have the same mean dimension (see Corollary \\ref{desdede}). Furthermore, Remark \\ref{dgrdgrw} proves the inequality $\\text{mdim}_M(X,\\phi^{p},d)\\leq p\\, \\text{mdim}_M(X,\\phi,d)$ can be strict.  Proposition    \\ref{jsffeke}  proves  if  $X=[0,1]$ or $\\mathbb{S}^{1},$ then for each $a\\in [0,1]$, there exists a continuous map $\\phi_{a}$ on $X$  with metric mean dimension equals to $a$.  In Theorem \\ref{thm:non-wond} we show that, as the topological entropy,  the metric mean dimension is concentrated in the non-wandering set of the  dynamics.}  \n\n\n\n\n\n\n\\medskip\n\nIn Section 5 we will discuss some upper bounds for the metric mean dimension of both autonomous and non-autonomous dynamical systems. \n\n\n\n\\medskip\n\nAs we said above,  the metric mean dimension for single continuous maps, and consequently for non-autonomous dynamical systems, \ndepends on the metric $d$.  In Section \\ref{section6} we will discuss some properties related to the invariance of the metric mean dimension under topological  equiconjugacy.\n\n\\medskip\n\nIn the last section we will present some results related to the continuity of the metric mean dimension. \n\n\n\n\\medskip\n\nSome  ideas given to proof the results  that  are well-known for the autonomous case work or can be adapted for the non-autonomous case. We will present these  proofs for the sake of  comprehensiveness.\n\n\n\\section{Mean dimension for non-autonomous dynamical systems}\n\nLet $X$ be a compact metric space. In this section we will suppose that  $\\textit{\\textbf{f}}=(f_n)_{n=1}^{\\infty}$  is a  non-autonomous\ndynamical system, where $f_n:X\\to X$\nis a continuous map for all $n\\geq 1$. We write $(X,\\textit{\\textbf{f}},d)$ to denote a non-autonomous dynamical system \\textit{\\textbf{f}} on $X$ endowed with the metric $d$. For $n,k\\in \\mathbb N$ define\n$$\nf_n^{(0)}:=I_X:= \\text{the identity on }X\\quad\\quad \\text{and} \\quad \\quad  f_n^{(k)}(x):=f_{n+k-1}\\circ \\dots \\circ f_n(x)\\quad\\text{for }k\\geq 1.\n$$\n\n Set\n\\[ \\mathcal{C}(X)=\\{ (f_{n})_{n=1}^{\\infty}: f_{n}:X\\rightarrow X \\text{ is a continuous map}\\}.\\]\n\n\n\\medskip\n\nGiven $\\alpha $ an open cover of $X$ define\n\\[\n\\alpha_{0}^{n-1}=\\alpha \\vee f^{-1}_1(\\alpha)\\vee (f_1^{(2)})^{-1}(\\alpha)\\vee \\dots \\vee (f_1^{(n-1)})^{-1}(\\alpha)\n\\]\nand set \n\\[\\text{ord}(\\alpha)=\\sup_{x\\in X}\\sum_{U\\in \\alpha }1_{U}(x)-1\\quad \\quad \\text{ and }\\quad \\quad \\mathcal D(\\alpha )\n                    =\\min_{\\beta  \\succ\\alpha}\\text{ord}(\\beta),\\]\n                   where $1_{U}$ is the indicator function and  $\\beta \\succ \\alpha$ means that $\\beta$ is an open cover  of $X$ finner than $\\alpha$.\n\\begin{definition} The \\textit{mean dimension} of $\\textit{\\textbf{f}}\\in\\mathcal{C}(X)$ is defined to be\n\\begin{equation*} \n\\text{mdim}(X,\\textit{\\textbf{f}}\\,)=\\sup_{\\alpha}\\lim_{n\\to\\infty}\\frac{\\mathcal D(\\alpha_{0}^{n-1})}{n}.\n\\end{equation*}\\end{definition}\n\nBy Corollary 2.5 of \\cite{lind} we have that $\\mathcal D(\\alpha\\vee \\beta)\\leq \\mathcal D(\\alpha)+\\mathcal D(\\beta)$, for\nany open covers $\\alpha$ and $\\beta$. It follows that the limit that defines the mean dimension is well defined.\n\n\\begin{remark}\\label{tete} We present \n  a list of   some important properties  about the mean dimension for both autonomous and non-autonomous dynamical systems: \n\\begin{enumerate}\n\\item For a non-autonomous dynamical system given by the iterates of a single continuous map $f:X\\to X$, i.e.,\n  $\\textit{\\textbf{f}}=(f)_{n=1}^{\\infty}$, the definition of mean dimension coincides with\n  the one presented in \\cite{lind}, that is,\n   \\(  \\text{mdim}(X,(f)_{n=1}^{\\infty})= \\text{mdim}(X,f).\\)\n\\item Recall that for a topological space $X$, the \\textit{topological dimension} is defined as\n$$\n\\text{dim}(X)=\\sup_{\\alpha}\\mathcal{D}(\\alpha)\n$$\nwhere $\\alpha $ runs the open covers of $X$. If $\\text{dim}(X)<\\infty$, then $\\mathcal{D}(\\alpha_{0}^{n-1})\\leq \\text{dim}(X)$\nfor all $n\\in \\mathbb N$ and therefore $\\text{mdim}(X,\\textit{\\textbf{f}}\\,)=0$ for any   $\\textit{\\textbf{f}}\\in  \\mathcal{C}(X)$.\n\\item  In  \\cite{lind}, Proposition 3.1, is proved that  $ \n\\text{mdim}(X^{\\mathbb{Z}},\\sigma)\\leq \\text{dim}( X),\n$\n where $\\sigma$ is the shift on $ X^{\\mathbb{Z}}$. Analogously we can prove  $ \n\\text{mdim}(X^{\\mathbb{N}},\\sigma)\\leq \\text{dim}( X).\n$\n\\item If $X=[0,1]^{m}$,  then $ \n\\text{mdim}(X^{\\mathbb{Z}},\\sigma)= m\n$ \n(see \\cite{lind},  Proposition 3.3).\n\\item It is clear that if $Y\\subseteq X$ is an invariant subset by a continuous map $\\phi:X\\rightarrow X$, then $\\text{mdim}(Y,\\phi)\\leq \\text{mdim}(X,\\phi)$. \nWe can define the mean dimension for any $Y\\subseteq X$ as follows: let $\\alpha$ be  an open cover  of\n$X$ and consider $\\alpha|_Y=\\{U\\cap Y:U\\in\\alpha\\}$, the open cover of $Y$ given by the restriction of $\\alpha$ to $Y$.\nThen define\n$$\n\\text{mdim}(Y,\\textit{\\textbf{f}}\\,|_Y)=\\sup_{\\alpha}\\lim_{n\\to\\infty}\\frac{\\mathcal D((\\alpha|_Y)_{0}^{n-1})}{n}.\n$$\nIt is clear   that $\\text{mdim}(Y,\\textit{\\textbf{f}}\\,|_Y)\\leq \\text{mdim}(X,\\textit{\\textbf{f}}\\,)$.\n\\item  A necessary condition for an invertible dynamical system  $\\phi:X\\rightarrow X$  to be imbeddable in $(([0, 1]^{m})^{\\mathbb{Z}},\\sigma )$ is that $\\text{mdim} (X,\\phi)\\leq  m$ (see \\cite{lind}, Corollary 3.4).\n\\item Any  nontrivial factor of $([0, 1]^{\\mathbb{Z}}, \\sigma)$ has positive mean dimension  (see \\cite{lind}, Theorem 3.6).\n\\end{enumerate}\n\\end{remark}\n \nWe will show some properties of the mean dimension which are   valid for the topological entropy. \nDenote by $h_{top}(\\textit{\\textbf{f}}\\,)$ the topological entropy of $\\textit{\\textbf{f}}$ (see \\cite{K-S}, \\cite{JeoCTE}).\n\n\\begin{definition}\\label{composision} For any $p\\geq 1$, set  $$ {\\textit{\\textbf{f}}^{\\,(p)}=\\{f_1^{(p)},f_{p+1}^{(p)},f_{2p+1}^{(p)},\\dots\\}=   \\{ f_{p}\\circ\\cdots \\circ f_{1}, f_{2p}\\circ\\cdots \\circ  f_{p+1},f_{3p}\\circ\\cdots \\circ  f_{2p+1},\\dots \\}}.$$\\end{definition}\n\nIt is well-known that    $  \n h_{top}(\\phi^{p})= p\\, h_{top}(\\phi)$ for any $p\\geq1 $, where $\\phi$ is any continuous map. For non-autonomous dynamical systems we have  $$\n h_{top}(\\textit{\\textbf{f}}^{\\,(p)})\\leq p\\, h_{top}(\\textit{\\textbf{f}}\\,)\\quad \\text{for any }p\\geq 1\n  $$ (see \\cite{K-S}, Lemma 4.2).  In general, the equality $\n h_{top}(\\textit{\\textbf{f}}^{\\,(p)})= p\\, h_{top}(\\textit{\\textbf{f}}\\,)$ is not valid, as we can see in the next example, which was given by Kolyada and Snoha in \\cite{K-S}.  \n \n \\begin{example} Take $\\psi: [0,1]\\rightarrow [0,1]$ defined by $ \\psi(x)=1-|2x-1|$ for any $x\\in [0,1]$. Consider $\\textbf{\\textit{f}}=(f_{n})_{n=1}^{\\infty}$, where  \n $$\nf_n(x)= \\begin{cases}\n    \\psi^{(n+1)\/2}(x), &  \\text{ if }n \\text{ is odd}, \\\\\n     x\/2^{n\/2}, & \\hbox{ if }n\\text{ is even},\n      \\end{cases}\n$$ for any $n\\in\\mathbb{N}$. \nThen $h_{top}(\\textbf{\\textit{f}}^{(2)})= 0 $ and   $h_{top}(\\textbf{\\textit{f}}) \\geq \\frac{\\log 2}{2}$.\n \\end{example}\n The equality $\n h_{top}(\\textit{\\textbf{f}}^{\\,(p)})= p\\, h_{top}(\\textit{\\textbf{f}}\\,)$ is valid  if the sequence $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ is equicontinuous (see \\cite{K-S}, Lemma 4.4). On the other hand, the equality always holds for the mean dimension. \n\n\n\n\\begin{proposition}\\label{pro:meandimension-periodic}\n  For any  $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}\\in  \\mathcal{C}(X)$ and $p\\in \\mathbb{N}$   we have\n  $$\n  \\emph{mdim}(X,\\textit{\\textbf{f}}^{\\,(p)})=p\\, \\emph{mdim}(X,\\textit{\\textbf{f}}\\,).\n  $$\n\\end{proposition}\n\n\\begin{proof}\nLet  $\\alpha$ be an open cover of $X$. Note that, for $k\\in \\mathbb N$,\n\\[\n\\lim_{k\\to\\infty}\\frac{\\mathcal D(\\alpha\\vee (f_1^{(p)})^{-1}(\\alpha)\\vee\\dots \\vee (f_1^{((k-1)p)})^{-1}(\\alpha))}{k}\\leq\n\\lim_{k\\to\\infty} \\frac{\\mathcal D(\\alpha_0^{(k-1)p})}{k}\\leq p  \\lim_{k\\to\\infty} \\frac{\\mathcal D(\\alpha_0^{(kp-1)})}{kp},\n\\]\nwhich implies that  $\\text{mdim}(X,\\textit{\\textbf{f}}^{\\, (p)})\\leq p\\ \\text{mdim}(X,\\textit{\\textbf{f}}\\,)$. \nFor the converse, note   that\n\\begin{align*}\n  \\alpha_{0}^{kp-1} & = \\alpha_{0}^{p-1}\\vee (f_1^{(p)})^{-1}(\\alpha_{0}^{p-1})\\vee (f_1^{(2p)})^{-1}(\\alpha_{0}^{p-1})\\vee\\dots\\vee (f_1^{((k-1)p)})^{-1}(\\alpha_{0}^{p-1}) ,\n\\end{align*}\nand therefore\n\\[\n\\text{mdim}(X,\\textit{\\textbf{f}}\\,)=\\sup_{\\alpha}\\lim_{k\\to\\infty}\\frac{\\mathcal D(\\alpha_0^{(k-1)p})}{kp}\\leq  \\frac{\\text{mdim}(X,\\textit{\\textbf{f}}^{\\,(p)})}{p},\n\\]\nwhich proves the proposition.\n\\end{proof}\n\n \n \n  In  \\cite{K-S}, Lemma 4.5, Kolyada and Snoha proved that $$ h_{top}(\\sigma^{i}(\\textit{\\textbf{f}}\\,))\\leq h_{top}(\\sigma^{j}(\\textit{\\textbf{f}}\\,))\\quad \\text{for any }i\\leq j,$$  where $\\sigma$ is the left shift   $\\sigma ((f_n)_{n=1}^{\\infty})=(f_{n+1})_{n=1}^{\\infty}$.  Furthermore, in \\cite{JeoCTE},  Corollary 5.6, the author showed that if each $f_{n}$ is an homeomorphism then the equality holds, that is, the topological entropy for non-autonomous dynamical systems is independent on the   first maps on a sequence of homeomorphisms  $\\textit{\\textbf{f}}=(f_{n})_{n\\in\\mathbb{Z}}$.\nNext proposition shows that these properties also hold for the mean dimension.\n \n\n\\begin{proposition}\\label{pro:shift}\nLet $i,j$ be two positive integers with $i\\leq j$. Then\n $$ \\emph{mdim}(X,\\sigma^i(\\textit{\\textbf{f}}\\,))\\leq  \\emph{mdim}(X,\\sigma^j(\\textit{\\textbf{f}}\\,)).$$ If each $f_{n}$ is a homeomorphism then the equality holds. \n\\end{proposition}\n\n\\begin{proof}\n It is enough to prove the proposition for $i=0$ and $j=1$. For  any open cover $\\alpha $ of $X$ we have\n\\begin{align*}\n  \\mathcal D(\\alpha_{0}^{n-1})&\\leq \\mathcal D(\\alpha) +\\mathcal D(f_1^{-1}(\\alpha\\vee (f_2)^{-1}(\\alpha)\\vee (f_2^{(2)})^{-1}(\\alpha)\\vee\\dots\\vee (f_2^{(n-2)})^{-1}(\\alpha)))\\\\\n                              &=\\mathcal D(\\alpha)+\\mathcal D(\\alpha\\vee (f_2)^{-1}(\\alpha)\\vee (f_2^{(2)})^{-1}(\\alpha)\\vee\\dots \\vee(f_2^{(n-2)})^{-1}(\\alpha)).\n\\end{align*}\nThus\n\\begin{align*}\n\\lim_{n\\to\\infty}\\frac{ \\mathcal{D}(\\alpha_{0}^{n-1})}{n}\n                     &\\leq \\lim_{n\\to\\infty}\\frac{ \\mathcal{D}(\\alpha)}{n}+ \\lim_{n\\to\\infty}\\frac{\\mathcal D(\\alpha\\vee f_2^{-1}(\\alpha)\\vee (f_2^{(2)})^{-1}(\\alpha)\\vee\\dots\\vee (f_2^{(n-2)})^{-1}(\\alpha))}{n}\\\\\n                     &= \\lim_{n\\to\\infty}\\frac{n-1}{n}\\frac{\\mathcal D(\\alpha\\vee f_2^{-1}(\\alpha)\\vee (f_2^{(2)})^{-1}(\\alpha)\\vee\\dots\\vee (f_2^{(n-2)})^{-1}(\\alpha))}{n-1}\\\\\n                     &\\leq\\text{mdim}(X,\\sigma(\\textit{\\textbf{f}}\\,)),\n\\end{align*}\nand therefore  $\\text{mdim}(X,\\textit{\\textbf{f}}\\,)\\leq \\text{mdim}(X,\\sigma(\\textit{\\textbf{f}}\\,)).$\n\n\nNext, suppose that each $f_{n}$ is a homeomorphism.  Note that if $\\beta $ refines $\\alpha$ then $ \\mathcal{D}(\\beta)\\geq  \\mathcal{D}(\\alpha)$.  Therefore, we have\n\\begin{align*}\n \\mathcal D(\\alpha\\vee (f_2)^{-1}(\\alpha)\\vee (f_2^{(2)})^{-1}(\\alpha)\\vee\\dots ) &=  \\mathcal D(f_1^{-1}(\\alpha\\vee (f_2)^{-1}(\\alpha)\\vee (f_2^{(2)})^{-1}(\\alpha)\\vee\\dots ))\\\\\n                              &= \\mathcal D((f_{1})^{-1}(\\alpha)\\vee (f_1^{(2)})^{-1}(\\alpha)\\vee (f_1^{(3)})^{-1}(\\alpha)\\vee\\dots )\\\\\n                              &\\leq \\mathcal D(\\alpha \\vee(f_{1})^{-1}(\\alpha)\\vee (f_1^{(2)})^{-1}(\\alpha)\\vee (f_1^{(3)})^{-1}(\\alpha)\\vee\\dots ).\n\\end{align*}\nHence $\\text{mdim}(X,\\sigma(\\textit{\\textbf{f}}\\,))\\leq \\text{mdim}(X, \\textit{\\textbf{f}}\\,)$.\n\\end{proof}\n\nIf some $f_{n}$ is not a  homeomorphism, then the inequality above can be strict. In fact, take $f_{n}=f:X\\rightarrow X$ for any $n\\geq 2$, where $f$ is any continuous  map with positive mean dimension and $f_{1}:X\\rightarrow X$ a constant map. Then  $\\text{mdim}(X, \\textit{\\textbf{f}}\\,)=0$ and $ \\text{mdim}(X, \\sigma(\\textit{\\textbf{f}}\\,))= \\text{mdim}(X, f)$. \n\n\\medskip\n\n\nNext corollary follows from Propositions \\ref{pro:meandimension-periodic} and   \\ref{pro:shift}:\n\n\\begin{corollary}\\label{desdede}\n  Let $\\textit{\\textbf{f}}=(f,g,f,g,\\dots)$ and $\\textit{\\textbf{g}}=(g,f,g,f,\\dots)$, where $f,g:X\\to X$ are continuous maps.\n  Then\n  \\[\n   \\emph{mdim}(X,\\textit{\\textbf{f}}\\,)=\\emph{mdim}(X,\\textit{\\textbf{g}}).\n  \\]\nTherefore,  $$\\emph{mdim}(X,f\\circ g)=\\emph{mdim}(X,g\\circ f).$$\\end{corollary}\n\\begin{proof}\nIt follows directly from Proposition \\ref{pro:shift} that $\\text{mdim}(X,\\textit{\\textbf{f}}\\,)=\\text{mdim}(X,\\textit{\\textbf{g}})$. Now, by Proposition \\ref{pro:meandimension-periodic}   we have\n\\begin{align*}\n  \\text{mdim}(X,f\\circ g)&=\\text{mdim}(X,\\textit{\\textbf{f}}^{\\,(2)})=2\\,\\text{mdim}(X,\\textit{\\textbf{f}}\\,)=2\\,\\text{mdim}(X,\\textit{\\textbf{g}})\\\\\n  &=\\text{mdim}(X,\\textit{\\textbf{g}}^{\\,(2)})= \\text{mdim}(X,g\\circ f),\n  \\end{align*}\n  which proves the corollary.\n\\end{proof}\n\n It follows directly from Corollary \\ref{desdede} that if  $f$ and $g$  are topologically conjugate continuous maps, then\n  \\[\\text{mdim}(X,f)=\\text{mdim}(X,g),\\]\n since if $\\phi$ is a topological conjugacy between $f$ and $g$, that is, $ \\phi$ is a homeomorphism and $ \\phi \\circ f=g\\circ \\phi$, then\n \\[\\text{mdim}(X,f)=\\text{mdim}(X,\\phi^{-1}\\circ \\phi \\circ f)=\\text{mdim}(X, \\phi \\circ f\\circ \\phi^{-1})=\\text{mdim}(X,g).\\]\n\n\n\n \n\n\nFor any $\\textit{\\textbf{f}}=(f_n)_{n=1}^{\\infty}\\in \\mathcal{C}(X)$, the \\emph{asymptotic  mean dimension} is defined by the limit \n\\begin{align*}\\label{eq:assymp1}\n\\text{mdim}(X,\\textit{\\textbf{f}}\\,)^*=\\lim_{n\\to\\infty} \\text{mdim}(X,\\sigma^n(\\textit{\\textbf{f}}\\,)).\n\\end{align*}\nIt follows from Proposition \\ref{pro:shift} that the asymptotic  mean dimension always  exists. \n\n\\begin{theorem}\\label{prop:unif-limit}\n Let $\\textit{\\textbf{f}}=(f_n)_{n=1}^{\\infty}\\in  \\mathcal{C}(X)$. If $\\textit{\\textbf{f}}$\n  converges uniformly to a continuous map $f:X\\to X$, then\n\\[\n  \\emph{mdim}(X,\\textit{\\textbf{f}}\\,)^*\\leq \\emph{mdim}(X,f).\n  \\]\nIn particular, $\\emph{mdim}(X,\\textit{\\textbf{f}}\\,)\\leq \\emph{mdim}(X,f).$ \n\\end{theorem}\n\n\\begin{proof}\n  Let $(x_n)_{n\\in\\mathbb N}$ be a sequence of mutually different point converging to\n  a point $x_0$. Define the map $F:\\{x_n:n=0,1,\\dots\\}\\times X\\to\\{x_n:n=0,1,\\dots\\}\\times X$\n  by $F:(x,y)\\mapsto(\\phi(x),\\psi(x,y))$, where\n\\begin{align*}\n  \\phi(x_n)=\\left\\{\n         \\begin{array}{ll}\n           x_0, & \\hbox{ if }n=0 \\\\\n           x_{n+1}, & \\hbox{ if }n>0\n         \\end{array}\\;\\; \\text{ and }\\;\\;\n\\psi(x_n,y)=\\left\\{\n  \\begin{array}{ll}\n    f(y), & \\hbox{ if }n=0 \\\\\n    f_n(y), & \\hbox{ if }n>0.\n  \\end{array}\n\\right.\n       \\right.\n\\end{align*}\nNote that the non wandering set of $F$, $\\Omega(F)$, is a subset  of the fix fiber $x_0\\times X$. Since \n$$\\text{mdim}(\\{x_n:n=0,1,\\dots\\}\\times X,F)=\\text{mdim}(\\Omega(F) ,F)$$ (by \\cite[Lemma 7.2]{GTM}), we have that\n\\[\n\\text{mdim}(\\{x_n:n=0,1,\\dots\\}\\times X,F)= \\text{mdim}(\\{x_0\\}\\times X,F).\n\\]\n Therefore, $$\\text{mdim}(\\{x_m:m\\geq k\\}\\times X,F)\\leq \\text{mdim}(\\{x_0\\}\\times X,F)=\\text{mdim}(\\{x_n:n=0,1,\\dots\\}\\times X,F), $$ for all $k>0$ (see Remark \\ref{tete}, item (3)).\nNext, note that by the definition of   $F$\n we have that $$\\text{mdim}(\\{x_m:m\\geq k\\}\\times X,F)=\\text{mdim}(X,\\sigma^k(\\textit{\\textbf{f}}\\,)),\\quad \\text{ for }k>0,$$ and\n$\\text{mdim}(\\{x_0\\}\\times X,F)=\\text{mdim}(X,f)$. Hence, $\\text{mdim}( X,\\sigma^k(\\textit{\\textbf{f}}\\,))\n\\leq \\text{mdim}(X,f)$, for all $k$.\n\\end{proof}\n\nNext example proves that the inequality above can be strict. \n \n \\begin{example}\\label{egre}\nLet $\\phi: I^{\\mathbb{N}}\\rightarrow I^{\\mathbb{N}}$ be a continuous map with positive  mean dimension. For each $n\\geq 1$, set $f_{n}:I^{\\mathbb{N}}\\times I^{\\mathbb{N}}\\rightarrow I^{\\mathbb{N}}\\times I^{\\mathbb{N}}$ defined by $$f_{n}((x_{i})_{i\\in\\mathbb{N}},(y_{i})_{i\\in\\mathbb{N}})=((\\lambda_{n} x_{i})_{i\\in\\mathbb{N}},(x_{i}(\\phi(y))_{i})_{i\\in\\mathbb{N}} ),$$  where $\\lambda _{n}\\rightarrow 1$ and $\\lambda_{n} \\cdots \\lambda_{1}\\rightarrow 0$ as $n\\rightarrow \\infty$.\nNote that $f_{n}$ converges uniformly on $I^{\\mathbb{N}}\\times I^{\\mathbb{N}}$  to $f((x_{i})_{i\\in\\mathbb{N}},(y_{i})_{i\\in\\mathbb{N}})=((x_{i})_{i\\in\\mathbb{N}},(x_{i}(\\phi(y))_{i})_{i\\in\\mathbb{N}})$ as $n\\rightarrow \\infty$ and \\[ \\text{mdim}(I^{\\mathbb{N}}\\times I^{\\mathbb{N}},f)\\geq \\text{mdim}(\\{(\\dots,1,1,1,\\dots)\\}\\times I^{\\mathbb{N}},f)=\\text{mdim}(I^{\\mathbb{N}},\\phi)>0. \\]\nOn the other hand, note that $f_{k}^{n}(\\bar{x},\\bar{y})\\rightarrow (\\bar{0},\\bar{0})$ as $n\\rightarrow \\infty$ for any $(\\bar{x},\\bar{y})\\in I^{\\mathbb{N}}\\times I^{\\mathbb{N}}$ and $k\\geq 1$.  Hence $\\text{mdim}(I^{\\mathbb{N}}\\times I^{\\mathbb{N}},\\sigma^{k}( \\textit{\\textbf{f}}\\,))=0$\n for any $k\\geq 1$, where $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ and therefore  $\\text{mdim}(I^{\\mathbb{N}}\\times I^{\\mathbb{N}}, \\textit{\\textbf{f}}\\,)^{\\ast}=0$. \\end{example}\n\n\n\\section{Metric mean dimension for non-autonomous dynamical systems}\\label{section3}\nThroughout this  section,   we will fix $\\textit{\\textbf{f}}=(f_n)_{n=1}^{\\infty}\\in  \\mathcal{C}(X)$ where $X$ is a compact metric space with metric $d$.   For any  $n\\in\\mathbb{N}$ let   $d_n:X\\times X\\to [0,\\infty)$ defined by\n$$\nd_n(x,y)=\\max\\{d(x,y),d(f_1(x),f_1(y)),\\dots,d(f_1^{(n-1)}(x),f_1^{(n-1)}(y))\\}.\n$$ \nThus $d_n$ is a metric on $X$ for all $n$ and generates the same topology induced by  $d$. Fix $\\varepsilon>0$. We say that $A\\subset X$ is an $(n,\\textit{\\textbf{f}},\\varepsilon)$-\\textit{separated} set\nif $d_n(x,y)>\\varepsilon$, for any two  distinct points  $x,y\\in A$. We denote by $\\text{sep}(n,\\textit{\\textbf{f}},\\varepsilon)$ the maximal cardinality of an $(n,\\textit{\\textbf{f}},\\varepsilon)$-separated\nsubset of $X$.  {Given an open cover $\\alpha$ of $X$, we say that $\\alpha$ is an \n$(n,\\textit{\\textbf{f}},\\varepsilon)$-\\textit{cover} if the $d_n$-diameter of any element of $\\alpha$ is less than $\\varepsilon$.} Let $\\text{cov}(n,\\textit{\\textbf{f}},\\varepsilon)$  be the minimum number of elements in an  $(n,\\textit{\\textbf{f}},\\varepsilon)$-cover of $X$. We say that $E\\subset X$ is an $(n,\\textit{\\textbf{f}},\\varepsilon)$-\\textit{spanning} set for $X$ if \nfor any $x\\in X$ there exists $y\\in E$ such  that $d_n(x,y)<\\varepsilon$. Let $\\text{span}(n,\\textit{\\textbf{f}},\\varepsilon)$ be the minimum cardinality\nof any $(n,\\textit{\\textbf{f}},\\varepsilon)$-spanning subset of $X$.   By the compactness of $X$, $\\text{sep}(n,\\textit{\\textbf{f}},\\varepsilon)$, $\\text{span}(n,\\textit{\\textbf{f}},\\varepsilon)$  and $\\text{cov}(n,\\textit{\\textbf{f}},\\varepsilon)$ are finite real numbers.\n\n \\begin{definition}\n  We define the \\emph{lower  metric mean dimension} of $(X,\\textit{\\textbf{f}},d)$  and the \\emph{upper metric mean dimension} of $(X,\\textit{\\textbf{f}},d)$ by\n\\begin{align*}\n \\underline{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)=\\liminf_{\\varepsilon\\to0} \\frac{\\text{sep}(\\textit{\\textbf{f}},\\varepsilon)}{|\\log \\varepsilon|}\\quad \\text{ and }\\quad\\overline{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)=\\limsup_{\\varepsilon\\to0} \\frac{\\text{sep}(\\textit{\\textbf{f}},\\varepsilon)}{|\\log \\varepsilon|},\n\\end{align*}\nrespectively, where $\\text{sep}(\\textit{\\textbf{f}},\\varepsilon)=\\underset{n\\to\\infty}\\limsup \\frac{1}{n}\\log \\text{sep}(n,\\textit{\\textbf{f}},\\varepsilon)$. \\end{definition}\n\n\nIt is not difficult to see that\n$$\n\\underline{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)=\\liminf_{\\varepsilon\\to0} \\frac{\\text{span}(\\textit{\\textbf{f}},\\varepsilon)}{|\\log \\varepsilon|}=\\liminf_{\\varepsilon\\to0} \\frac{\\text{cov}(X,\\varepsilon)}{|\\log \\varepsilon|},\n$$\nwhere $ \\text{span}(\\textit{\\textbf{f}},\\varepsilon)=\\underset{n\\to\\infty}\\limsup\\frac{1}{n}\\log \\text{span}(n,\\textit{\\textbf{f}},\\varepsilon)$ and  $\\text{cov}(\\textit{\\textbf{f}},\\varepsilon)=\\underset{n\\to\\infty}\\limsup\\frac{1}{n}\\log \\text{cov}(n,\\textit{\\textbf{f}},\\varepsilon).$\nThis fact holds for the upper metric mean dimension.\nWe will write $ {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)$ to refer to both  $ \\overline{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)$  and $ \\underline{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)$. \n\\medskip\n\nTopological entropy for non-autonomous dynamical systems is invariant under uniform equi\\-conjugacy (see \\cite{K-S} and \\cite{JeoCTE}). Metric mean dimension for single dynamical systems depends on the metric $d$ on $X$. Consequently, it is not an invariant under conjugacy and therefore it is not an invariant under uniformly equiconjugacy between   non-autonomous dynamical systems. Set $$\\mathcal{B}=\\{\\rho: \\rho \\text{ is a metric on }X\\text{ equivalent to }d \\}$$ and take  \n\\begin{equation}\\label{infmean}   {\\text{mdim}_M}(X, \\textit{\\textbf{f}}\\,)=\\inf_{\\rho\\in \\mathcal{B}}  {\\text{mdim}_M}(X,\\textit{\\textbf{f}},\\rho).\\end{equation}\n\nFor single maps, ${\\text{mdim}_M}(X, \\phi)$ is an  invariant under topological conjugacy. In Proposition \\ref{edee344} we will prove an analogous result for non-autonomous dynamical systems. \n\n\n\\begin{remark}\\label{nmvnrit}\nIt follows from  the definition of the topological entropy for non-autonomous dynamical systems introduced in \\cite{K-S} that \nif the topological entropy of the non-autonomous system $(X,\\textit{\\textbf{f}},d)$ is finite then its metric mean dimension is zero.\n\\end{remark}\n \n Next, we will present some examples of the  the metric mean dimension for  both  autonomous and non-autonomous dynamical systems. In Section \\ref{Section5} we will show more examples. \n \n \\medskip\n \n\nTake  $\\mathbb{K}=\\mathbb{N}$ or $ \\mathbb{Z}$.  Consider  the metric $\\tilde{d}$ on $X^{\\mathbb{K}}$ defined by \\begin{equation}\\label{mnvc}\\tilde{d}(\\bar{x},\\bar{y})= \\sum_{i\\in\\mathbb{K}}\\frac{1}{2^{|i|}}d(x_{i},y_{i}) \\quad\\text{ for }\\bar{x}=(x_{i})_{i\\in\\mathbb{K}}, \\bar{y}=(y_{i})_{i\\in\\mathbb{K}} \\in X^{\\mathbb{K}}.\\end{equation} \n \n  Take $X=[0,1]$, endowed with the metric $d(x,y)=|x-y|$ for $x,y\\in X.$ In \\cite{lind3}, Example E, is proved that  $    \\text{mdim}(X^{\\mathbb{Z}}, \\sigma, \\tilde{d}) = 1.$\nAnalogously,   we can prove that    $   \\text{mdim}(X^{\\mathbb{N}}, \\sigma, \\tilde{d}) = 1:$\n\n\\begin{lemma}\\label{bcbcbcbc}  Take $X=[0,1]$ endowed with the metric $d(x,y)=|x-y|$ for $x,y\\in X.$ Thus  $$   \\emph{mdim}(X^{\\mathbb{N}}, \\sigma, \\tilde{d}) = 1.$$ \\end{lemma}\n\\begin{proof}   Fix $\\varepsilon>0$ and take $l = \\lceil\\log(4\/\\varepsilon)\\rceil$, where $\\lceil x\\rceil =\\min\\{k\\in \\mathbb{Z}: x\\leq k\\}$. Note that $\\sum_{n>l} 2^{-n}\\leq \\varepsilon\/2$. Consider the open cover of $X$ given by \n$$I_{k}=\\left(\\frac{(k-1)\\varepsilon}{12},\\frac{(k+1)\\varepsilon}{12} \\right),\\quad\\text{for }0\\leq k\\leq \\lfloor  12\/\\varepsilon\\rfloor. $$\n  Note that $I_{k}$ has length \n$\\varepsilon\/6$. Let $n\\geq 1$. Next,  consider the following open cover  of $X^{\\mathbb{N}}$: \n$$I_{k_{1}}\\times I_{k_{2}}\\times \\cdots \\times I_{k_{n+l}}\\times X\\times X\\times\\cdots,\\quad  \\text{ where }0\\leq k_{1}, k_{2},\\dots, k_{n+l}\\leq \\lfloor  12\/\\varepsilon\\rfloor. $$ \nEach open set has diameter less than $\\varepsilon$ \n with respect to the\ndistance $\\tilde{d}_{n}$ (see \\eqref{mnvc}). Therefore  $$\\text{cov}(n,\\sigma,\\varepsilon)\\leq (1+ \\lfloor  12\/\\varepsilon\\rfloor)^{n+l}\\leq(2+12\/\\varepsilon)^{n+1 +  12\/\\varepsilon}.$$ \nHence \n$$\\text{cov}(\\sigma,\\varepsilon)=\\lim_{n\\rightarrow \\infty}\\frac{\\log\\text{cov}(n,\\sigma,\\varepsilon)}{n}\\leq \\lim_{n\\rightarrow \\infty}\\frac{(n+1 +  12\/\\varepsilon)\\log (2+12\/\\varepsilon)}{n}=\\log (2+12\/\\varepsilon).   $$ \nThus $$ \\text{mdim}(X^{\\mathbb{N}}, \\sigma, \\tilde{d}) =\\lim_{\\varepsilon \\rightarrow \\infty} \\frac{\\text{cov}(\\sigma,\\varepsilon)}{|\\log \\varepsilon|}\\leq 1. $$\n\nOn the other hand, any two distinct points in the sets \n$$ \\{(x_{i})_{i\\in\\mathbb{N}}\\in X^{\\mathbb{N}} : x_{i} \\in \\{0,\\varepsilon, 2\\varepsilon, \\dots ,\\lfloor  1\/\\varepsilon\\rfloor\\varepsilon\\} \\text{ for all }0\\leq i<n\\} $$ \nhave distance $\\geq \\varepsilon$ with respect to $d_{n}$. It follows that\n$$\\text{cov}(n,\\sigma,\\varepsilon)\\geq (1+\\lfloor  1\/\\varepsilon\\rfloor)^{n}\\geq (1\/\\varepsilon)^{n}.$$ Therefore\n$$\\text{cov}(\\sigma,\\varepsilon)\\geq \\lim_{n\\rightarrow \\infty}\\frac{\\log\\text{cov}(n,\\sigma,\\varepsilon)}{n}=|\\log \\varepsilon| .$$ \nHence  $\\text{mdim}_{M}(X^{\\mathbb{N}},\\sigma,d)=1.$\n\\end{proof}\n\n \n  {Next example proves that there exist dynamical systems on the interval with positive metric mean dimension (see also \\cite{VV}). } \n\n\\begin{example}\\label{exfagner} Take $g:[0,1]\\rightarrow [0,1]$, defined by $x\\mapsto |1-|3x-1||$, and  $0=a_{0}<a_{1}< \\cdots <a_{n}<\\cdots$, where $a_{n}=\\sum_{k=1}^{n}6\/\\pi ^{2}k^{2}$ for $n\\geq 1$. For each $n\\geq 1$, let \n $T_{n}: J_{n}:=[a_{n-1},a_{n}] \\rightarrow [0,1] $ be the unique increasing affine map from $J_{n}$ (which has length $6\/\\pi ^{2}n^{2}$)  onto $[0,1]$ and take any strictly increasing sequence of natural numbers $m_{n}$.\n Consider the continuous map  $\\phi:[0,1]\\rightarrow [0,1]$ such that, for each $n\\geq 1$, $\\phi|_{J_{n}}= T_{n}^{-1}\\circ g^{m_{n}}\\circ T_{n}$.  \n \\medskip\n \n Fix $n\\geq 1$. Note that $J_{n}$ can be divided into $3^{m_{n}}$ intervals with the same length  $J_{n}(1),\\dots,$ $  J_{n}({3^{m_{n}}})$, such that \n\\[ \\phi(J_{n}({i}))=J_{n} \\quad\\text{ for each } i\\in\\{1,\\dots ,3^{m_{n}}\\}.\\] \n\nNext, $J_{n}({i})$ can be divided into $3^{n}$ intervals with the same length  $J_{n}({i},1),\\dots,$ $  J_{n}({i},3^{n})$\nsuch that $$\\phi^{2}(J_{n}({i},s))=J_{n}\\quad \\text{for }i=1,\\dots,3^{n}\\quad\\text{ and } s=1,\\dots, 3^{n}.$$\nInductively, we can prove that for all $k\\geq 1$ and    $(i_{1},\\dots,i_{k})$, where  $i_{j}\\in \\{1,\\dots, 3^{n}\\}$,  we can divide  $J_{n}(i_{1},\\dots,i_{k})$   into $3^{n}$ intervals with the same length $J_{n}(i_{1},\\dots,i_{k},1),\\dots,$ $  J_{n}(i_{1},\\dots,i_{k},3^{n})$ such that \n$$\\phi^{k+1}(J_{n}(i_{1},\\dots,i_{k},i))=J_{n} \\quad \\text{for }i=1,\\dots,3^{n}.$$\nEach $J_{n}(i_{1},\\dots,i_{k})$ has length $|J_{n}|\/3^{kn}$ for each $k\\geq 1$. Furthermore, each $J_{n}(i_{1},\\dots,i_{k})$ has length $|J_{n}|\/3^{km_{n}}$ for each $k\\geq 1$.\n\n\\medskip\n\n Take $\\varepsilon_n= |J_{n}|\/3^{m_n}= 3\/\\pi ^{2}n^{2}3^{m_{n}}$ for each $n\\geq 1$. If $x\\in J_{n}(i_{1},\\dots,i_{k})$ and $y\\in J_{n}(j_{1},\\dots,j_{k})$ where $(i_{1},\\dots,i_{k})\\neq (j_{1},\\dots,j_{k})$ and each $i_{1},\\dots,i_{k},j_{1},\\dots,j_{k}$ is odd, then\n \\[ d_{n+k}(x,y)\\geq \\varepsilon_{n}.\\] \n For each $k\\geq1,$ there are more than $(3^{m_{n}}\/2)^{k}$ intervals $J_{n}(i_{1},\\dots,i_{k})$ with $i_{s}$ odd, $s=1,\\dots,k$. \n Hence    $\\text{sep}(n+k,\\phi, \\varepsilon_{n})\\geq (3^{m_{n}}\/2)^{k}$ and then\n$$\n\\text{sep}(\\phi,\\varepsilon_{n})\\geq \\lim_{k\\to\\infty}\\frac{\\log \\text{sep}(n+k,\\phi,\\varepsilon_{n})}{k} \\geq \\log (3^{m_{n}}\/2).$$ Therefore \n\\begin{align*}\n\\overline{\\text{mdim}_M}([0,1],\\phi,| \\cdot |)&\\geq \\lim_{n\\to\\infty}\\frac{\\log (3^{m_{n}}\/2)}{-\\log \\varepsilon_{n}}= \\lim_{n\\to\\infty}\\frac{\\log (3^{m_{n}}\/2)}{-\\log(3\/\\pi ^{2}n^{2}3^{m_{n}})}\\\\\n&= \\lim_{n\\to\\infty}\\frac{\\log (3^{m_{n}})+\\log2}{\\log(\\pi ^{2}n^{2}\/3)+\\log (3^{m_{n}})}=1,\\end{align*}\nhence $ \\overline{\\text{mdim}_M}([0,1],\\phi,|\\cdot |)\\geq1.$  We will obtain  from Proposition \\ref{erfdy}  that $ \\overline{\\text{mdim}_M}([0,1],\\phi,|\\cdot |)\\leq 1$. Therefore $  \\overline{\\text{mdim}_M}([0,1],\\phi,|\\cdot |)= 1.\n$  \\end{example}\n\nSince $\\phi(0)=0$ and $\\phi(1)=1$, the map $\\phi$ induces a continuous map on $\\mathbb{S}^{1}$ with metric mean dimension equal to 1. More generally, we have:\n  \n \\begin{proposition}\\label{jsffeke}  {Take $X=[0,1]$ or $\\mathbb{S}^{1}.$ For each $a\\in [0,1]$, there exists $\\phi_{a}\\in C^{0}(X)$ with $\\emph{mdim}_{M}(\\phi_{a})=a$.}\\end{proposition}\n \n\\begin{proof} Any constant map has metric mean dimension equal to 0. On the other hand, Example \\ref{exfagner} proves that there exist continuous maps on $X$ with metric mean dimension equal to 1.  Fix $a\\in (0,1)$ and take  $r=\\frac{1}{a}$.  Set $a_{0}=0$ and $a_{n}= \\sum_{i=1}^{n}C(3^{-ir})$ for $n\\geq 1$, where $C=1\/ \\sum_{i=1}^{\\infty}3^{-ri}=1\/(3^{r}-1)$. For each $n\\geq 1$, take $J_{n}$,   \n $T_{n}$ and $g$ as in Example \\ref{exfagner}.  \n Consider the continuous map  $\\phi_{a}:[0,1]\\rightarrow [0,1]$ such that, for each $n\\geq 1$, $\\phi_{a}|_{J_{n}}= T_{n}^{-1}\\circ g^{n}\\circ T_{n}$ (note that $\\phi_{a}(0)=0$ and $\\phi_{a}(1)=1$, consequently $\\phi_{a}$ induces a continuous map on $\\mathbb{S}^{1}$).    Fix $n\\geq 1$. Each $J_{n}$ can be divided into $3^{n}$ intervals with the same length  $J_{n}(1),\\dots,$ $  J_{n}({3^{n}})$, such that \n\\[ \\phi_{a}(J_{n}({i}))=J_{n} \\quad\\text{ for each } i\\in\\{1,\\dots ,3^{n}\\}.\\] \nNext, $J_{n}({i})$ can be divided into $3^{n}$ intervals with the same length  $J_{n}({i},1),\\dots,$ $  J_{n}({i},3^{n})$\nsuch that $$\\phi_{a}^{2}(J_{n}({i},s))=J_{n}\\quad \\text{for }i=1,\\dots,3^{n}\\quad\\text{ and } s=1,\\dots, 3^{n}.$$\nInductively, we can prove that for all $k\\geq 1$ and    $(i_{1},\\dots,i_{k})$, where  $i_{j}\\in \\{1,\\dots, 3^{n}\\}$,  we can divide  $J_{n}(i_{1},\\dots,i_{k})$   into $3^{n}$ intervals with the same length $J_{n}(i_{1},\\dots,i_{k},1),\\dots,$ $  J_{n}(i_{1},\\dots,i_{k},3^{n})$ such that \n$$\\phi_{a}^{k+1}(J_{n}(i_{1},\\dots,i_{k},i))=J_{n} \\quad \\text{for }i=1,\\dots,3^{n}.$$\nEach $J_{n}(i_{1},\\dots,i_{k})$ has length $|J_{n}|\/3^{kn}$ for each $k\\geq 1$.\n\n\\medskip\n\nTake $\\varepsilon_n= |J_{n}|= C\/3^{rn}$ for each $n\\geq 1$. Each $  J_{n}(i_{1},\\dots,i_{k})$ has $d_{k+1}$-diameter equal to  $\\varepsilon_n$. Consequently,   $\\text{cov}(k+1,\\phi_{a}, \\varepsilon_{n})\\geq 3^{{n}k}$ and then $$ \\text{cov}(\\phi_{a},\\varepsilon_{n})\\geq \\lim_{k\\to\\infty}\\frac{\\log \\text{cov}(k+1,\\phi_{a},\\varepsilon_{n})}{k+1} \\geq \\log 3^{{n}}.$$     Therefore \\begin{align*} {\\text{mdim}_M}([0,1],\\phi_{a},| \\cdot |)&\\geq \\lim_{n\\to\\infty}\\frac{\\log 3^{n}}{-\\log \\varepsilon_{n}}= \\lim_{n\\to\\infty}\\frac{\\log 3^{n}}{-\\log(C\/3^{nr})} = \\lim_{n\\to\\infty}\\frac{\\log 3^{n}}{\\log 3^{nr}}\\\\&=\\lim_{n\\to\\infty}\\frac{n\\log 3}{nr\\log3} = \\frac{1}{r}=a.\\end{align*}\n\nOn the other hand, fix $n\\geq 1 $.  Let $m\\geq n$ be such that $\\sum_{i=m}^{\\infty}C(3^{-ir})< \\varepsilon_{n}$. Therefore \\begin{equation}\\label{bdhekn3rf}\\text{cov}(\\cup_{i=m}^{\\infty}J_{i},k,\\phi_{a}, \\varepsilon_{n})=1\\quad \\text{ for any }k\\geq 1.\\end{equation}   Note that for each $k\\geq 1$ and    $(i_{1},\\dots,i_{k})$, where  $i_{j}\\in \\{1,\\dots, 3^{n}\\}$, the subintervals $J_{n}(i_{1},\\dots,i_{k})$ have diameter less than $\\varepsilon_n$ with the metric $d_{k}$ for any $k\\geq 1$.  \nConsequently, we have      \\begin{equation}\\label{bdheknrf}\\text{cov}({J}_{n},k,\\phi_{a}, \\varepsilon_{n})\\leq (3^{{n}})^{k} \\quad \\text{ for any }k\\geq 1.\\end{equation}\nFor each $i\\in\\{1,\\dots,n-1\\}$, divide each  interval $J_{i}$ into \n $(3^{n})^{k+1}\\lceil {|J_{i}|}\/{|J_{n}|}\\rceil$ subintervals with the same length, where $\\lceil x\\rceil =\\min\\{{j}\\in \\mathbb{Z}: x\\leq {j}\\}$. \nEach  subinterval  has $d_{k}$-diameter less than $\\varepsilon_{n}$, thus \\begin{equation}\\label{bdheknrf2}\\text{cov}(\\cup_{i=1}^{n-1}{J}_{i},k,\\phi_{a}, \\varepsilon_{n})\\leq \\sum_{i=1}^{n-1} (3^{{n}})^{k+1}\\lceil {|J_{i}|}\/{|J_{n}|}\\rceil.\\end{equation}\nFor  $i\\in\\{n+1,\\dots,m-1\\}$, each $J_{i}$    has $d_{k}$-diameter less than $\\varepsilon_{n}$, thus \\begin{equation}\\label{bdheknderf2}\\text{cov}(\\cup_{i=n+1}^{m-1}{J}_{i},k,\\phi_{a}, \\varepsilon_{n})\\leq m-n-1\\quad\\text{ for any }k\\geq 1.\\end{equation}\nBy \\eqref{bdhekn3rf}-\\eqref{bdheknderf2}, we have \n\\begin{align*}\\text{cov}(\\phi_{a}, \\varepsilon_{n})&\\leq \\lim_{k\\rightarrow\\infty}\\frac{\\log[1+(3^{{n}})^{k}+\\sum_{i=1}^{n-1} (3^{{n}})^{k+1}\\lceil {|J_{i}|}\/{|J_{n}|}\\rceil+ m-n-1]}{k-1}\\\\\n&\\leq  \\lim_{k\\rightarrow\\infty}\\frac{\\log[(\\sum_{i=1}^{n-1} \\lceil {|J_{i}|}\/{|J_{n}|}\\rceil+ m-n+1) (3^{{n}})^{k+1}]}{k-1}=\\lim_{k\\rightarrow\\infty}\\frac{\\log (3^{n})^{k+1}}{k-1}\\\\\n&=\\log(3^{{n}}).\\end{align*}\nHence  \n\\begin{align*}\n {\\text{mdim}_M}([0,1],\\phi_{a},| \\cdot |)&\\leq \\lim_{n\\to\\infty}\\frac{\\log (3^{{n}})}{-\\log \\varepsilon_{n}}=  \\lim_{n\\to\\infty}\\frac{\\log (3^{{n}})}{\\log(3^{{rn}})} =\\lim_{n\\to\\infty}\\frac{n\\log (3)}{(rn)\\log(3)}  =a.\\end{align*} \n Therefore $   {\\text{mdim}_M}([0,1],\\phi_{a},|\\cdot |)= a.$  \\end{proof}\n  \n\n\\begin{example}\\label{lkjhfg}\nLet $X=\\{0,1\\}^{\\mathbb N}$ with its usual metric  and consider $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$, where $f_n:\\{0,1\\}^{\\mathbb N}\\to\\{0,1\\}^{\\mathbb N}$ is given by $f_n(\\omega)=\\sigma^{2^n}(\\omega)$, for any  $n\\in\\mathbb{N}$.\nNote that $f_1^{(n)}(\\omega)=\\sigma^{2^{n+1}-2}(\\omega)$. We claim that $\\text{mdim}_{M}(X,\\textit{\\textbf{f}},d)=\\infty$. Fix $\\varepsilon>0$.  Take a  positive integer $k$\nso that $2^{-(k+1)}\\leq\\varepsilon<2^{-k}$.\n Now consider $A\\subset \\{0,1\\}^{\\mathbb N}$ a $ (2^{n+1}-2,\\varepsilon)$-separated set for the shift map $\\sigma$ of maximum cardinality and note that $A$ is\nan $(n,\\varepsilon)$-separated set for $\\textit{\\textbf{f}}$.\nTherefore,\n$  \\text{sep}(n,\\textit{\\textbf{f}},\\varepsilon) \\geq 2^{2^{n+1}-2+k}$ and then\n\\begin{align*}\n \\frac{\\log\\text{sep}(n,\\textit{\\textbf{f}},\\varepsilon) }{n\\log \\varepsilon}& \\geq \\frac{(2^{n+1}-2+k)\\log2}{nk}.\n\\end{align*}\nHence, by the definition of the upper metric mean dimension, we have\n$$\n {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)=\\limsup_{\\varepsilon\\to0}\\limsup_{n\\to\\infty}\\frac{\\log\\text{sep}(n,\\textit{\\textbf{f}},\\varepsilon)}{n|\\log\\varepsilon|}=\\infty.\n$$\n\\end{example} \n\nIn \\cite{Zhu}, Zhu,   Liu,   Xu, and Zhang  showed that if $X$ is a  $k$-dimensional Riemannian manifold and $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ is \na sequence of $C^{1}$-maps on $X$   such that $a_{n}= \\underset{x\\in M}{\\sup}\\Vert D_{x}f_{n}\\Vert<\\infty$ for all $n\\in \\mathbb{N}$, then \n$$ h_{top}(\\textit{\\textbf{f}}\\, ) \\leq \\max\\left\\{0,  \\limsup_{n\\rightarrow \\infty}\\frac{k}{n}\\sum_{i=1}^{n-1}\\log a_{i} \\right\\}. $$\nHence, by Remark \\ref{nmvnrit}, we have: \n\\begin{proposition} If $\\limsup_{n\\rightarrow \\infty}\\frac{k}{n}\\sum_{i=1}^{n-1}\\log a_{i} <\\infty$, we have $ {\\emph{mdim}_M}(M,\\textit{\\textbf{f}},d)=0.$\\end{proposition}\n \n \n  Any  sequence of homeomorphisms on both the interval or   the circle has zero topological entropy (see \\cite{K-S}, Theorem D).   \nTherefore, the metric mean dimension of any $\\textbf{\\textit{f}}$ on both the interval or   the circle   is equal to zero.  In the next example we will see that there exist non-autonomous dynamical systems consisting of diffeomorphisms on a surface with infinite metric mean dimension.\n\n\\begin{example}\\label{hfkenrkflr} Let $\\phi:\\mathbb{T}^{2}\\rightarrow \\mathbb{T}^{2}$ be the diffeomorphism induced by a hyperbolic matrix $A$ with eigenvalue $\\lambda>1$, where $\\mathbb{T}^{2}$ is the torus endowed with the metric $d$ inherited from the plane.    Consider $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ where    $f_{n}=\\phi^{2^n} $ for each $i\\geq 1$.  We have $|\\text{Fix}(\\phi^{n})|=\\lambda^{n}+\\lambda^{-n}-2, $ where  $\\text{Fix}(\\psi)$ is the set consisting of fixed points of a continuous map $\\psi$ (see \\cite{Katok}, Proposition 1.8.1). Furthermore,  $$\\text{sep}(n,  \\textit{\\textbf{f}},1\/4)\\geq \\text{sep}(2^n,  \\phi,1\/4)\\geq \\text{Fix}(\\phi^{2^n})=\\lambda^{2^n}+\\lambda^{-2^n}-2$$\n(see \\cite{Katok}, \nChapter 3, Section 2.e). Therefore, \n$$\\lim_{n\\rightarrow \\infty}\\frac{\\text{sep}(n,  \\textit{\\textbf{f}},1\/4)}{n}\\geq \\lim_{n\\rightarrow \\infty}\\frac{\\log \\lambda^{2^n}}{n}=\\infty,$$\nand hence $\\text{mdim}_{M}(\\mathbb{T}^{2},\\textit{\\textbf{f}}, d)=\\infty$.\n\\end{example}\n \n \n  Suppose the   Hausdorff dimension of $X$ is finite. Let $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ be a non-autonomous dynamical system where each $f_{n}$ is a $C^{r}$-map on $X$.   We have that if $h_{top}(\\textit{\\textbf{f}}\\, )<\\infty$ then $  {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)=0.$ Therefore, if $\\sup_{n\\in\\mathbb{N}}  L (f_{n})  <\\infty$, where $L(f_{n})$ is the Lipschitz constant of $f_{n}$, we have that $h_{top}(\\textit{\\textbf{f}}\\, )  <\\infty$ and hence $   {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)=0.$ Thus if   $\\sup_{n\\in\\mathbb{N}}  L( f_{n})  <\\infty$, then $   {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)=0.$ \nIn particular, if $X$ is a compact Riemannian manifold and $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ is a sequence of differentiable maps  that  $\\sup_{n\\in\\mathbb{N}}\\Vert D f_{n}\\Vert <\\infty$, where $Df_{n}$ is the derivative of $f_{n},$ we have that $h_{top}(\\textit{\\textbf{f}}\\, )  <\\infty$ and hence $   {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)=0.$ \n \n \\section{{Some fundamental properties of the metric mean dimension}}\\label{section4}\n      \n  \nIn this section we show some properties which are well-known for topological entropy and metric mean dimension for  dynamical systems.  \n In the next proposition we will consider $\\textit{\\textbf{f}}^{\\,(p)}$, which was defined in  Definition  \\ref{composision}. \n \n \\medskip\n \n It is well-known  that  $h_{top}(\\textit{\\textbf{f}}^{\\, (p)})\\leq p\\, h_{top}(\\textit{\\textbf{f}}\\, )$ and if the sequence  $(f_{n})_{n=1}^{\\infty}$ is equicontinuous, then the equality holds (see \\cite{K-S}, Lemma 4.2).  For the case of the metric mean dimension, we always have that  $\\text{mdim}_M(X,\\textit{\\textbf{f}}^{\\,(p)},d)\\leq p\\, \\text{mdim}_M(X,\\textit{\\textbf{f}},d)$.   However we will present an example where the inequality can be strict even for single continuous maps (see Remark \\ref{dgrdgrw}). \n \n \\begin{proposition}\\label{propo211}\nFor any $\\textit{\\textbf{f}}=(f_n)_{n=1}^{\\infty}$ and  $p\\in \\mathbb N$, we have\n  $$\n  {\\emph{mdim}_M}(X,\\textit{\\textbf{f}}^{\\,(p)},d)\\leq p\\,  {\\emph{mdim}_M}(X,\\textit{\\textbf{f}},d).\n  $$\n  Consequently (see \\eqref{infmean}),  $$\n   {\\emph{mdim}_M}(X,\\textit{\\textbf{f}}^{\\,(p)})\\leq p\\, {\\emph{mdim}_M}(X,\\textit{\\textbf{f}}\\,).\n  $$\n\\end{proposition}\n\\begin{proof}\nNote that, for any positive integer $m$, we have\n$$\\max_{0\\leq j<m}d(f_1^{(jp)}(x),f_1^{(jp)}(y))\\leq \\max_{0\\leq j<mp}d(f_1^{(j)}(x),f_1^{(j)}(y)).$$\nThus $\\mbox{span}(m,\\textit{\\textbf{f}}^{\\,(p)},\\varepsilon)\\leq \\mbox{span}(mp,\\textit{\\textbf{f}},\\varepsilon)$ and therefore \n$$\n\\text{span}(\\textit{\\textbf{f}}^{(p)},\\varepsilon)=\\limsup_{m\\to \\infty}\\frac{1}{m}\\log \\mbox{span}(m, \\textit{\\textbf{f}}^{(p)},\\varepsilon)\n                                    \\leq p\\limsup_{m\\to \\infty}\\frac{1}{mp}\\log \\mbox{span}(m ,\\textit{\\textbf{f}},\\varepsilon)=p\\ \\text{span}(\\textit{\\textbf{f}},\\varepsilon). $$ \nHence ${\\text{mdim}_M}(X,\\textit{\\textbf{f}}^{\\,(p)},d)\\leq p\\,{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d).$\n\\end{proof}\n\n\\begin{remark}\\label{dgrdgrw}\n In Example \\ref{exfagner} we  prove that there exists a continuous map $\\phi:[0,1]\\rightarrow [0,1]$ such that  $ \\underline{\\text{mdim}_M}([0,1],\\phi,d)=1$, where $d(x,y)=|x-y|$ for $x,y\\in [0,1]$.  It follows from Proposition  \\ref{erfdy} that for any $f:[0,1]\\rightarrow [0,1]$  we have \n$\\overline{\\text{mdim}_{M}}([0,1],f,d)\\leq1. $ Consequently, $\\overline{\\text{mdim}_{M}}([0,1],\\phi^{n},d)\\leq1$ for any $n\\geq 1$, which proves that the inequality  in Proposition \\ref{propo211} can be strict for autonomous systems and therefore for non-autonomous   systems.\n\\end{remark} \n  \n  \n   If $A,B\\subseteq X $ are invariant subsets  under a continuous map $\\phi$, then   $$h_{top}(\\phi)=\\max \\{ h_{top}(\\phi|_{A})  ,h_{top}(\\phi|_{B})\\}.$$ \nIt is clear this property is also valid for the metric mean dimension. \n\\begin{proposition}\\label{invariant}\n  If $A,B\\subseteq X $ are invariant subsets  under $\\phi$, then   $$\\emph{mdim}_{M}(X,\\phi,d)=\\max \\{\\emph{mdim}_{M}(X,\\phi|_{A},d)  ,\\emph{mdim}_{M}(X,\\phi|_{B},d)\\}.$$\n\\end{proposition}\n\n\nIf $A_{1}$, $A_{2}$, $\\dots$ is a sequence of invariant subsets under $\\phi$, then $$\\max_{n\\in\\mathbb{N}} \\{\\text{mdim}_{M}(X,\\phi|_{A_{n}},d) \\}\\leq \\text{mdim}_{M}(X,\\phi,d).$$ Example \\ref{exfagner} proves that the inequality can be strict (the sets $J_{1}$, $J_{2}$,  $ \\dots$ are invariant under $\\phi$, however   $\\text{mdim}_{M}(X,\\phi|_{J_{n}},d)=0$ for each $n$). \n\n\n \\medskip\n  \n  Metric mean dimension can be defined on any subset $A$ of $X$.   Kolyada  and  Snoha in \\cite{K-S}, Lemma 4.1, proved that if $X= \\cup_{i}^{n} A_{i}$, then  $$ h_{top}(\\textit{\\textbf{f}}\\,)= \\max_{i=1,\\dots,n}h_{top}(\\textit{\\textbf{f}}\\,|_{A_{i}}).$$\n  Analogously we can prove that: \n  \\begin{proposition}\n  If $X= \\cup_{i}^{n} A_{i}$, then  $$ \\emph{mdim}_{M}(X, \\textit{\\textbf{f}}\\, , d)= \\max_{i=1,\\dots,n}\\emph{mdim}_{M}(X,\\textit{\\textbf{f}}\\,|_{A_{i}},d).$$\n  \\end{proposition}\n  \n  \n \n\n\\begin{definition}\nWe say that $x\\in X$ is a \\textit{nonwandering point} for $\\textit{\\textbf{f}}$ if for every neighbourhood $U$ of $x$ there exist positive integers $k$ and $n$ with\n$f_n^{(k)}(U)\\cap U\\not=\\emptyset$. We denote by $\\Omega({\\textit{\\textbf{f}}}\\,)$ the set consisting of the nonwandering points of $\\textit{\\textbf{f}}$.\n\\end{definition} \n\n It is well-known that for any continuous map $\\phi:X\\rightarrow X$ we have $h_{top}(\\phi)=h_{top}(\\phi|_{\\Omega(\\phi)}).$ This fact  was proved  for  non-autonomous dynamical systems by Kolyada and Snoha in  \\cite{K-S}.  For mean dimension of single continuous maps this fact was proved by Gutman in \\cite{GTM},  Lemma 7.2.   For the metric mean dimension of non-autonomous dynamical systems we also have:  \n \n\n\\begin{theorem}\\label{thm:non-wond}\nWe have \n  \\[\n   {\\emph{mdim}_M}(X,\\textit{\\textbf{f}},d)= {\\emph{mdim}_M}(\\Omega({\\textit{\\textbf{f}}\\,)},\\textit{\\textbf{f}},d).\n  \\]\n\\end{theorem}\n\n\\begin{proof}\nIt is clear that $ {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)\\geq {\\text{mdim}_M}(\\Omega({\\textit{\\textbf{f}}}\\,),\\textit{\\textbf{f}},d)$.\nFix $\\varepsilon>0$ and $n\\in\\mathbb N$. Let $\\alpha $ be an open $(n ,\\textit{\\textbf{f}},\\varepsilon)$-cover of $X$ with minimum cardinality. Take $\\beta$ a minimal finite open subcover of $\\Omega(\\textit{\\textbf{f}}\\,)$, chosen from $\\alpha$ (note that $\\beta$ is an $(n,\\textit{\\textbf{f}},\\varepsilon)$-cover of $\\Omega(\\textit{\\textbf{f}}\\,)$). By the minimality of $\\alpha$ we have that $\\beta $ is an $(n,\\textit{\\textbf{f}}, \\varepsilon)$-cover of $\\Omega(\\textit{\\textbf{f}}\\,)$  with minimum cardinality, which we denote by $ \\text{cov}(\\Omega(\\textit{\\textbf{f}}\\,),n,\\textit{\\textbf{f}}, \\varepsilon)$, i.e., $\\text{Card}(\\beta)=\\text{cov}(\\Omega(\\textit{\\textbf{f}}\\,),n,\\textit{\\textbf{f}}, \\varepsilon)$.\n\n\\medskip\n\nThe set $K=X\\backslash \\bigcup_{U\\in\\beta}U$ is compact and consists of wandering points. We can cover\n$K$ by a finite number of wandering  subsets, each of them contained in some element of  $\\alpha$. The sets defined before together\nwith $\\beta$ form a finite open cover $\\gamma(n)=\\gamma$ of $X$, finer than $\\alpha$. Consider, for each $k$, the open cover $\\gamma(k,\\textit{\\textbf{f}}^{(n)})$\nassociated to the sequence $\\textit{\\textbf{f}}^{(n)}$. Note that each element of $\\gamma(k,\\textit{\\textbf{f}}^{(n)})$ is of the form\n$$\nA_0\\cap (f_1^{(n)})^{-1}(A_1)\\cap (f_{1}^{(n)})\\circ (f_{n+1}^{(n)})^{-1}(A_2)\\cap\\dots\\cap(f_1^{(n)})^{-1}\\circ \\dots\\circ(f_{(k-2)n+1}^{(n)})^{-1}(A_{k-1}),\n$$\nwhere $A_i\\in\\gamma$, for $i=0\\dots,k-1$. It implies that $\\gamma(k,\\textit{\\textbf{f}}^{(n)})$ is a $(k,\\textit{\\textbf{f}}^{(n)},\\varepsilon)$-cover of $X$. Let $A_i$ and $A_j$ be  nonempty  open sets of $\\gamma(k,\\textit{\\textbf{f}}^{(n)})$  for\nsome $i<j$. If $A_i=A_j$, then $$(f_{(j-1)n+1}^{(n)}\\circ\\dots\\circ f_{in+1}^{(n)}) (A_i)=f_{in+1}^{(j-i)n}(A_i)$$ intersects $A_i=A_j$. In that case $A_i$ does not contain non-wandering points for $\\textit{\\textbf{f}}$ (and hence $A_i\\in \\beta$). Now we estimate the number of elements of $\\gamma(k,\\textit{\\textbf{f}}^{(n)})$.\nSetting \\[{j:=\\text{Card} \\{A_{i}:i=0,1,\\dots k-1\\}\\quad \\text{and}\\quad  m:=\\text{Card}(\\gamma(k,\\textit{\\textbf{f}}^{(n)})\\backslash \\beta)}, \\] we have \n $0\\leq j\\leq m$.\nIn this case we have $\\binom{m}{j}$ possibilities of the choice of a $j$-element subset of $\\gamma(k,\\textit{\\textbf{f}}^{(n)})\\backslash \\beta$ and then these sets\ncan appear as various $A_i'$s in $k\\cdot(k-1)\\cdots(k-j+1)=k!\\slash(k-j)!$ ways. For the rest of $A_i'$s we can choice any element of $\\beta$.\nSo, the number of elements of $\\gamma(k,\\textit{\\textbf{f}}^{(n)})$ is bounded by\n$$\n\\sum_{j=0}^m\\binom{m}{j}\\frac{k!}{(k-j)!}\\cdot (\\text{Card}(\\beta))^{k-j}.\n$$\nSince $k!\\slash(k-j)!\\leq k^m$ and $\\binom{m}{j}\\leq m!$, this number is not larger than $(m+1)\\cdot m!\\cdot k^m\\cdot (\\text{Card}(\\beta))^{k}$.\nThus, using the fact that $\\text{cov}(k,\\textit{\\textbf{f}}^{(n)},\\varepsilon)\\leq \\text{Card}(\\gamma(k,\\textit{\\textbf{f}}^{(n)}))$, we have\n\\begin{align*}\n\\limsup_{k\\to\\infty}\\frac{1}{k}\\log \\text{cov}(k,\\textit{\\textbf{f}}^{(n)},\\varepsilon)\\leq \\limsup_{k\\to\\infty}\\frac{1}{k}\\log(m+1)\\cdot m!\\cdot k^m\\cdot (\\text{Card}(\\beta))^{k}=\\log(\\text{Card}(\\beta)).\n\\end{align*}\nAs\n\\begin{align*}\n\\limsup_{k\\to\\infty}\\frac{1}{k}\\log \\text{cov}(k,\\textit{\\textbf{f}}^{(n)},\\varepsilon)=n  \\limsup_{k\\to\\infty}\\frac{1}{k}\\log \\text{cov}(k,\\textit{\\textbf{f}},\\varepsilon),\n\\end{align*}\nit follows that\n\\begin{align*}\n\\limsup_{k\\to\\infty}\\frac{1}{k} \\log \\text{cov}(k,\\textit{\\textbf{f}},\\varepsilon)\n                                     &\\leq\\frac{1}{n} \\log \\text{cov}(\\Omega(\\textit{\\textbf{f}}\\,),n,\\textit{\\textbf{f}},\\varepsilon).\n\\end{align*}\nTaking the limsup as $n\\to\\infty$  we obtain $$\\text{cov}(\\textit{\\textbf{f}},\\varepsilon)\\leq \\limsup_{n\\rightarrow\\infty}\\frac{1}{n} \\log \\text{cov}(\\Omega(\\textit{\\textbf{f}}\\,),n,\\textit{\\textbf{f}},\\varepsilon):= \\text{cov}(\\Omega(\\textit{\\textbf{f}}\\,),\\textit{\\textbf{f}},\\varepsilon).$$ \nSo,\n\\begin{align*}\n {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)&=\\liminf_{\\varepsilon\\to0}\\frac{\\text{cov}(\\textit{\\textbf{f}},\\varepsilon)}{|\\log\\varepsilon|}\n                                         \\leq \\liminf_{\\varepsilon\\to0}\\frac{ \\text{cov}(\\Omega({\\textit{\\textbf{f}}}\\,),\\textit{\\textbf{f}},\\varepsilon)}{|\\log\\varepsilon|}\n                                         = {\\text{mdim}_M}(\\Omega(\\textit{\\textbf{f}}\\,),\\textit{\\textbf{f}},d),\n\\end{align*}\nwhich proves  the theorem.\n\\end{proof}\n\n \n \n\n \n\\begin{definition}\nA continuous map $\\psi:X\\to Y$ will be called $\\alpha$-\\textit{compatible} if it is possible to find a finite open\ncover $\\beta $ of $\\psi(X)$ such that $\\psi^{-1}(\\beta)\\succ\\alpha$.\n\\end{definition}\n\n\n\nLindenstrauss  and  Weiss in \\cite{lind}, Theorem 4.2, proved that for any metric $d$ compatible with the topology of $X$, we have \\[\n \\text{mdim}(X,\\phi)\\leq  \\underline{\\text{mdim}_M}(X,\\phi,d) \n\\] for any continuous map $\\phi:X\\rightarrow X$. These ideas work in order to show  the non-autonomous case: metric mean dimension is an upper bound for the mean dimension of non-autonomous dynamical systems. We will need the next proposition, whose proof  can be found in \\cite{lind}, Proposition 2.4. \n\\begin{proposition}\\label{alpha-comp}\nIf $\\alpha $ is an open cover of $X$, then $\\mathcal D(\\alpha)\\leq k$ if and only if\nthere exists an $\\alpha$-compatible continuous map $\\psi:X\\to K$, where $K$ has topological dimension $k$.\n\\end{proposition}\n\n \n \n\n\n\\begin{theorem}\\label{efgetrr}\nFor any metric $d$ on $X$  compatible with the topology of $X$ we have that\n\\[\n \\emph{mdim}(X,\\textit{\\textbf{f}}\\,)\\leq  \\underline{\\emph{mdim}_M}(X,\\textit{\\textbf{f}},d).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nLet $\\alpha $ be an open cover of $X$. We can assume that $\\alpha $ is of the form\n\\[\\alpha=\\{U_1,V_1\\}\\vee\\dots \\vee\\{U_\\ell,V_\\ell\\},\\]\nwhere each $\\{U_i,V_i\\}$ is  an open cover of $X$ with two elements. For each $1\\leq i\\leq \\ell$ define\n$\\omega_i:X\\to [0,1]$ by\n\\[\n\\omega_i(x)=\\frac{d(x, X\\backslash V_i)}{d(x, X\\backslash U_i)+d(x, X\\backslash V_i)}.\n\\]\nIt is not difficult to see that $\\omega_i$ is Lipschitz, $U_i=\\omega_i^{-1}([0,1))$ and $V_i=\\omega_i^{-1}((0,1])$.\n\nLet $C$ be a common bound for the Lipschitz constants of all $\\omega_i$. For each positive integer\n$N$ define $F(N,\\cdot ):X\\to[0,1]^{\\ell N}$ by\n\\[\nF(N,x)=(\\omega_1(x),\\dots, \\omega_\\ell (x),\\omega_1(f_1(x)),\\dots,\\omega_\\ell(f_1(x)),\\dots,\\omega_1(f_1^{(N)}(x)),\\dots,\\omega_\\ell(f_1^{(N)}(x))).\n\\]\nAs $U_i=\\omega_i^{-1}([0,1))$ and $V_i=\\omega_i^{-1}((0,1])$ we have that  $F(N,\\cdot)\\succ\\alpha_0^{N-1}$.\n\nNow for each $S\\subset \\{1,\\dots,\\ell N\\}$, for $x\\in X$, denote by $F(N,x)_S$ the projection of $F(N,x)$ to the coordinates of\nthe index set $S$.\n\n\\noindent \\textbf{Claim.} Let $\\varepsilon>0$ and $D=\\underline{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)$. There exists $N(\\varepsilon)>0$\nso that, for all $N>N(\\varepsilon)$ there exists $\\xi\\in (0,1)^{\\ell N}$ which satisfies\n$$\n\\xi_S\\notin F(N,X)_S,\n$$\nfor any subset $S\\subset\\{1,\\dots,\\ell N\\}$ that satisfies $|S|>(D+\\varepsilon)N$.\n\\begin{proof}\nLet $\\delta>0$ such that \\[\\delta<(2^\\ell(2C)^{2D})^{-2\\varepsilon} \\quad\\quad\\text{ and }\\quad \\quad\\frac{\\text{sep}(\\textit{\\textbf{f}},\\delta)}{\\log \\delta}=\\underline{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)+\\frac{\\varepsilon}{4}.\n\\]\nWe notice that for $N$ sufficiently large we can cover $X$ by $\\delta^{-(D+\\varepsilon\\slash2)N}$ dynamical balls\n$B(x,N,\\delta)=\\{y\\in X:d_N(x,y)<\\delta\\}$. Since $C$ is the common Lipschitz constant for all $\\omega_i$, \nwe conclude that\n\\[\nF(N,B(x,N,\\delta))\\subset\\{a\\in [0,1]^{\\ell N}:\\|F(N,x)-a\\|_\\infty<C\\delta\\},\n\\]\nwhere $\\|(a_1,\\dots,a_{\\ell N})-(b_1,\\dots,b_{\\ell N})\\|_\\infty=\\sup_i|a_i-b_i|$. This fact implies that $F(N,X)$ can\nbe covered by $\\delta^{-(D+\\varepsilon\\slash2)N}$ balls in the $\\| \\; \\cdot\\|_\\infty$ norm of radius $C\\delta$. Let\n$B(1),\\dots, B(K)$ be these balls, with $K=\\delta^{-(D+\\varepsilon\\slash2)N}$.\n\nChoose $\\xi \\in[0,1]^{\\ell N}$ with uniform probability and notice that\n\\begin{align*}\n  \\mathbb P(\\xi\\in F(N,X)_S) & \\leq\\sum_{j=1}^{K}\\mathbb{P}(\\xi\\in B(j)_S)\\leq  \\delta^{-(D+\\varepsilon\\slash2)N}(2C\\delta)^{|S|},\n\\end{align*}\nand so\n\\begin{align*}\n  \\mathbb P(\\exists S:|S|>(D+\\varepsilon)N \\text{ and }\\xi_S\\in F(N,X)_S)\n                & \\leq \\sum_{|S|>(D+\\varepsilon)N} \\mathbb{P}(\\xi_S\\in F(N,X)_S)\\\\\n                & \\leq (\\sharp \\text{ of such }S)\\delta^{-(D+\\varepsilon\\slash2)N}(2C\\delta)^{D+\\varepsilon}N \\\\\n                & \\leq 2^{\\ell N}((2C)^{2D}\\delta^{\\varepsilon\\slash2})^N\\ll1.\n\\end{align*}\nHence, with high probability, a random $\\xi$ will satisfies the requirements.\n\\end{proof}\n\n\\noindent \\textbf{Claim.} If $\\pi: F(N,X)\\to [0,1]^{\\ell N}$ satisfies for both $a=0$ and $a=1$, and all $\\xi\\in[0,1]^{\\ell N}$,\n\\[\n    \\{1\\leq k\\leq \\ell N:\\xi_k=a\\}\\subset \\{1\\leq k\\leq \\ell N:\\pi(\\xi)_S=a\\},\n\\]\nthen $\\pi\\circ F(N,X)$ is compatible with $\\alpha_0^{N-1}$.\n\n\\begin{proof}\n  Given  $\\xi\\in [0,1]^{\\ell N}$, define for $0\\leq j< N$ and $1\\leq i<\\ell$\n\\[\nW_{i,j}=\\left\\{\n         \\begin{array}{ll}\n           (f_1^{(j)})^{-1}(U_i), & \\hbox{ if }\\xi_{j\\ell+i}=0, \\\\\n           (f_1^{(j)})^{-1}(V_i), & \\hbox{ otherwise.}\n         \\end{array}\n       \\right.\n\\]\nBy the definition of $W_{i,j}$ we have that $(\\pi\\circ F(N,\\cdot))^{-1}(\\xi)\\subset \\displaystyle\\bigcap_{1\\leq i\\leq \\ell,\\\\ 0\\leq j<N}W_{i,j}\\in \\alpha_0^{N-1}$. It follows that $\\pi\\circ F(N,X)$ is compatible with $\\alpha_0^{N-1}$.\n\\end{proof}\nFor a fixed  $\\varepsilon>0$, consider $\\bar\\xi$ and $N$ as in the first Claim. Set\n\\[\n\\Phi=\\{\\xi\\in[0,1]^{\\ell N}:\\xi_k=\\bar{\\xi}_k \\text{ for more than }(D+\\varepsilon)N \\text{ indexes }k\\}.\n\\]\nThen, $F(N,X)\\subset \\Phi^C=[0,1]^{\\ell N}\\backslash \\Phi$.\n\nNow, for each $m=1,2,\\dots$, denote by $J_m$ the set\n$$\nJ_m=\\{\\xi\\in [0,1]^{\\ell N}:\\xi_i\\in\\{0,1\\} \\text{ for at least }m \\text{ indexes }1\\leq i\\leq\\ell N\\}.\n$$\n\nSince $\\bar\\xi $ is in the interior of $[0,1]^{\\ell N}$, one can define $\\pi_1:[0,1]^{\\ell N}\\backslash \\{\\bar\\xi\\}\\to J_1$\nby mapping each $\\xi$ to the intersection of the ray starting at $\\bar\\xi$ and passing through $\\xi$ and $J_1$.\nFor each of the $(\\ell N-1)$-dimensional cubes $I^t$ that comprises $J_1$ we can define a retraction on $I^t$ in a similar fashion\nusing as a center the projection of $\\bar\\xi$ on $I^t$. This will define a continuous retraction $\\pi_2$ of $\\Phi^C$ onto $J_2$.\nAs long as there is some intersection of $\\Phi$ with the cubes in $J_m$ this process can be continued, thus we finally get\na continuous projection $\\pi$ of $\\Phi^C$ onto $J_{m_0}$, a space of topological dimension equals to $m_0$, with\n$$\nm_0\\leq \\lfloor D+\\varepsilon\\rfloor N+1,\n$$\nwhere $\\lfloor x\\rfloor =\\max\\{k\\in \\mathbb{Z}: k\\leq x\\}$. \nBy construction,  $\\pi$ satisfies the hypotheses of the second claim.\n Thus $\\pi\\circ F(N,\\cdot)\\succ\\alpha_0^{N-1}$.\nMoreover, since $F(N,X)\\subset \\Phi^C$, we have   $\\pi(F(N,X))\\subset J_{m_0}$.\n\nPutting all together, we have constructed a $\\alpha_0^{N-1}$ compatible function from $X$ to a space of\ntopological dimension less or equal to $\\lfloor D+\\varepsilon\\rfloor N+1$, and so\n$$\n\\frac{D(\\alpha_0^{N-1})}{N}\\leq \\frac{\\lfloor D+\\varepsilon\\rfloor N+1}{N}.\n$$\nAs $\\varepsilon$ goes to zero we get that $\\text{mdim}(X,\\textit{\\textbf{f}}\\,)\\leq D$. \n\\end{proof}\n\n\n\n\n\n\n \n\nThe inequality in the  theorem above can be strict for single maps and therefore for non-autonomous dynamical systems. \n In \\cite{lind2}, Theorem 4.3, is proved that if a continuous map $\\phi:X\\rightarrow X$ is an extension of a minimal system, then there is a metric $ d^{\\prime}$ on $X$, equivalent to $d$, such that   $$\\text{mdim}(X,\\phi) = \\underline{\\text{mdim}_{M}}(X,\\phi, d^{\\prime}).$$\n \n\n \n\n \n   \n \n\\section{Upper bound for the metric mean dimension}\\label{Section5}\n \n As we saw in Remark \\ref{tete}, we have  $ \n\\text{mdim}(X^{\\mathbb{K}},\\sigma)\\leq \\text{dim}( X)\n$, where $\\mathbb{K}= \\mathbb{Z}$ or $\\mathbb{N}$. Furthermore,  if $X=I^{k}$, then $ \n\\text{mdim}(X^{\\mathbb{Z}},\\sigma)= k$. \n In this section we will prove that the metric mean dimension of the shift on  $X^{\\mathbb{K}}$ is  equal to the box dimension of $X$ with respect to the metric $d$, which will be defined below. This fact implies that the metric mean dimension of any continuous map $\\phi: X\\rightarrow X$ is less or equal to the box dimension of $X$ with respect to the metric $d$ (see Proposition \\ref{erfdy}). \n\n \n \\begin{definition} For $\\varepsilon>0,$  let $N(\\varepsilon)$ be the minimum number of closed balls of radious $\\varepsilon$ needed\nto cover $X$. The numbers \n$$\\overline{\\text{dim}_{B}}(X,d)=\\limsup_{\\varepsilon\\rightarrow \\infty}\\frac{\\log N(\\varepsilon)}{|\\log\\varepsilon|} \\quad \\text{and}\\quad \\underline{\\text{dim}_{B}}(X,d)=\\liminf_{\\varepsilon\\rightarrow \\infty}\\frac{\\log N(\\varepsilon)}{|\\log\\varepsilon|}$$\n are called, respectively, the  \\textit{upper Minkowski dimension} (or \\textit{upper box\ndimension}) of $X$ and the  \\textit{lower Minkowski dimension} (or \\textit{lower box\ndimension}) of $X$, with respect to $d$. \\end{definition}\n\nFor any metric space $(X,d)$ we have $$\\text{dim}(X)\\leq \\text{dim}_{H}(X,d)\\leq   \\underline{\\text{dim}_{B}}(X,d), $$\nwhere $\\text{dim}_{H}(X,d)$ is the Hausdorff dimension of $X$ with respect to $d$ (see \\cite{Kawabata}, Section II, A). If $X=[0,1]$, then $\\text{dim}(X)=\\text{dim}_{H}(X,d)=   \\underline{\\text{dim}_{B}}(X,d)=1$. However, there exist sets such that the   inequalities above can be strict, as we will see in the next  example, which also proves  that neither $\\text{dim}(X)$ nor  $\\text{dim}_{H}(X,d)$ are    upper bounds for  $ \\overline{\\text{mdim}_{M}}(X^{\\mathbb{Z}},\\sigma,\\tilde{d})$. \n \n\n\n\\begin{example}\nLet  $A = \\{0\\} \\cup \\{1\/n: n\\geq 1\\}$ endowed with the metric $d(x,y)=|x-y|$ for $x,y\\in A$. In \\cite{Kawabata}, Lemma 3.1, is proved that $\\text{dim}_{H}(A) = 0$  while $\\underline{\\text{dim}_{B}}( A ) = 1\/2.$  Furthermore, we have $$   \\underline{\\text{mdim}_{M}}(A^{\\mathbb{Z}},\\sigma,\\tilde{d})= \\underline{\\text{dim}_{B}}( A ) = 1\/2$$ (see \\cite{lind3}, Section VII). \n\\end{example}\n\n\nUsing the \\textit{Classical Variational Principle}, in \\cite{VV}, Theorem 5, the authors claim to have proven that for any $(X,d)$ \n \\begin{equation*} \\overline{\\text{mdim}_{M}}(X^{\\mathbb{Z}},\\sigma,\\tilde{d})=\\overline{\\text{dim}_{B}}(X,d) .\\end{equation*}   \n \n \nThis fact can be proved generalizing the ideas given  in  \\cite{lind3}, Example E:  \n\\begin{theorem}\\label{bcbcbcbc1} For $\\mathbb{K}=\\mathbb{Z}$ or $\\mathbb{N}$ we have    $$ \\overline{\\emph{mdim}_{M}}(X^{\\mathbb{K}}, \\sigma, \\tilde{d}) =  \\overline{\\emph{dim}_{B}} (X,d) \\quad \\quad \\text{and}\\quad \\quad \\underline{\\emph{mdim}_{M}}(X^{\\mathbb{K}}, \\sigma, \\tilde{d})= \\underline{\\emph{dim}_{B}} (X,d). $$\n\\end{theorem}\n\\begin{proof} We will prove the case $\\mathbb{K}=\\mathbb{Z}$ (the case $\\mathbb{K}=\\mathbb{N}$ can be proved analogously as in Lemma \\ref{bcbcbcbc}).   Fix $\\varepsilon>0$ and take $l$ big enough such that   $\\sum_{n>l} 2^{-n}\\text{diam} (X)\\leq \\varepsilon\/2 $. Let $m=N(\\varepsilon)$ be the minimum   number of closed $\\varepsilon$-balls $X_{1},\\dots ,X_{m}$ needed\nto cover $X$. Consider the open cover of $X^{\\mathbb{Z}}$ given by the open sets\n$$\\cdots \\times X \\times X_{k_{-l}}\\times X_{k_{-l+1}}\\times \\cdots \\times X_{k_{n+l}}\\times X\\times\\cdots,\\quad  \\text{ where }1\\leq k_{-l}, k_{-l+1},\\dots, k_{n+l}\\leq    m. $$ \nNote that each one of these open sets has diameter less than $4\\varepsilon$ \n with respect to the\ndistance $\\tilde{d}_{n}$ on $X^{\\mathbb{Z}}$. Therefore  $ \\text{cov}(n,\\sigma,4\\varepsilon)\\leq m ^{n+2l+1}$ and hence \n$$\\text{cov}(\\sigma,4\\varepsilon)=\\lim_{n\\rightarrow \\infty}\\frac{\\log\\text{cov}(n,\\sigma,4\\varepsilon)}{n}\\leq \\lim_{n\\rightarrow \\infty}\\frac{(n+2l+1)\\log (m)}{n}=\\log N(\\varepsilon),  $$ \nwhich implies that $$ \\overline{\\text{mdim}_{M}}(X^{\\mathbb{Z}}, \\sigma, \\tilde{d}) =\\limsup_{\\varepsilon \\rightarrow \\infty} \\frac{\\text{cov}(\\sigma,4\\varepsilon)}{|\\log 4\\varepsilon|}\\leq \\limsup_{\\varepsilon \\rightarrow \\infty} \\frac{\\log N(\\varepsilon)}{|\\log 4\\varepsilon|}=  \\limsup_{\\varepsilon \\rightarrow \\infty} \\frac{\\log N(\\varepsilon)}{|\\log4 + \\log \\varepsilon|}=\\overline{\\text{dim}_{B}} (X,d) $$ and\n$$ \\underline{\\text{mdim}_{M}}(X^{\\mathbb{Z}}, \\sigma, \\tilde{d}) =\\liminf_{\\varepsilon \\rightarrow \\infty} \\frac{\\text{cov}(\\sigma,4\\varepsilon)}{|\\log 4\\varepsilon|}\\leq  \\underline{\\text{dim}_{B}} (X,d), $$\n\nTo prove the converse inequality, for $\\varepsilon>0$  let  $\\{x_1,x_2,\\dots,x_{N(\\varepsilon)}\\}$ be a maximal set of  points in $X$ which are $\\varepsilon$-separated.\n For $n\\geq 1$, consider the set\n$$ \\{(y_{i})_{i\\in\\mathbb{Z}}\\in X^{\\mathbb{Z}} : y_{i} \\in \\{x_1,x_2,  \\dots ,x_{N(\\varepsilon)}\\} \\text{ for all }-l\\leq i\\leq n+l\\} $$ \nand notice that it is $(\\sigma,n,\\varepsilon)$-separated and its cardinality is bounded from below by $N(\\varepsilon)^{n+2l+1}$. So \n$$\n\\text{sep}(\\sigma,\\varepsilon)\\geq \\lim_{n\\to\\infty}\\frac{\\log N(\\varepsilon)^{n+2l+1}}{n}=\\log N(\\varepsilon),\n$$\nand it implies that $$\\underline{\\text{mdim}_{M}}(X^{\\mathbb{Z}}, \\sigma, \\tilde{d}) \\geq \\underline{\\text{dim}_{B}} (X,d),$$\nwhich proves the theorem. \n\\end{proof}\n\n\n \n Next proposition proves  the metric mean dimension of any dynamical system is bounded by the box dimension of the space (see \\cite{VV}, Remark 4).    \n \n \n \n\\begin{proposition}\\label{erfdy}  For any continuous map $\\phi:X\\rightarrow X$  we have \n$$\\overline{\\emph{mdim}_{M}}(X,\\phi,d) \\leq \\overline{\\emph{dim}_{B}} (X,d) \\quad \\text{ and }\\quad \\underline{\\emph{mdim}_{M}}(X,\\phi,d) \\leq \\underline{\\emph{dim}_{B}} (X,d) .$$  In particular, if $X=[0,1]$, then  $$\\underline{\\emph{mdim}_{M}}(X,\\phi,d)\\leq \\overline{\\emph{mdim}_{M}}(X,\\phi,d)\\leq 1.$$ \n\\end{proposition}\n\\begin{proof}\nConsider the embedding  $\\psi: X\\rightarrow X^{\\mathbb{N}}$, defined by  $x\\mapsto\\psi(x)= (x,\\phi(x),\\phi^{2}(x),\\dots)$. We have $\\sigma\\circ \\psi=\\psi \\circ \\phi$.   Therefore, $Y=\\psi(X)$ is a closed subset of  $ X^{\\mathbb{N}}$ invariant by $\\sigma$.  Take the metric $d_{\\psi}$ on $X$ defined by $ d_{\\psi}(x,y)= \\tilde{d}(\\psi(x),\\psi(y)),$  for any  $x,y\\in X.$  Clearly $d(x,y)\\leq d_{\\psi}(x,y)$ for any $x,y\\in X$, therefore any $ (n,\\phi,\\varepsilon)$-separated subset of $X$ with respect to $d$ is a $ (n,\\phi,\\varepsilon)$-separated subset of $X$ with respect to $d_{\\psi}$. Hence $$\\overline{\\text{mdim}_{M}}(X,\\phi,d)\\leq  \\overline{\\text{mdim}_{M}}(X,\\phi,d_{\\psi}) =\\overline{\\text{mdim}_{M}}(Y,\\sigma|_{Y},\\tilde{d})\\leq \\overline{\\text{mdim}_{M}}(X^{\\mathbb{N}},\\sigma,\\tilde{d}) \\leq \\overline{\\text{dim}_{B}} (X,d) $$\nand, analogously, $ \\underline{\\text{mdim}_{M}}(X,\\phi,d)\\leq   \\underline{\\text{dim}_{B}} (X,d). $\n\\end{proof}\n \n Example \\ref{exfagner} proves that there exist dynamical systems $\\phi:X\\rightarrow X$ such that  $$\\overline{\\text{mdim}_M}(X,\\phi,d)=\\overline{\\text{dim}_{B}} (X,d)\\quad \\text{ and }\\quad \\underline{\\text{mdim}_M}(X,\\phi,d)=\\underline{\\text{dim}_{B}} (X,d).$$\n \n \n  \\medskip\n\n We can consider the \\emph{asymptotic metric mean dimension} as the   limit\n\\begin{align*}\\label{eq:assymp}\n {\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)^*=\\limsup_{i\\to\\infty}{\\text{mdim}_M}(X,\\sigma^i(\\textit{\\textbf{f}}\\,),d).\n\\end{align*}\n   \n  \n\\begin{theorem}\\label{maintheoremE} If $\\textit{\\textbf{f}}=(f_n)_{n=1}^{\\infty}$   converges uniformly to a continuous map $f:X\\to X$, then, for any $k\\geq 1$, \n\\begin{equation}\\label{tergss}  {\\emph{mdim}_M}(X,\\sigma^{k}(\\textit{\\textbf{f}}\\,),d)\\leq  {\\emph{mdim}_M}(X,f,d).\n  \\end{equation}\n  Consequently,  \\[ {\\emph{mdim}_M}(X,\\textit{\\textbf{f}},d)^*\\leq  {\\emph{mdim}_M}(X,f,d). \\]\n\\end{theorem}\n\\begin{proof}\nSee the proof of  Theorem \\ref{prop:unif-limit} and use Theorem \\ref{thm:non-wond}. \n\\end{proof}\n\nWe can prove, as in Example \\ref{egre}, that the  inequality above  can be strict. \n\n\n\\medskip\n\n Theorem \\ref{maintheoremE} and Proposition \\ref{erfdy} imply  that: \n\\begin{corollary}\\label{efrrttt}  \n If  $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$\n  converges uniformly to a continuous map on $X$, then \\begin{equation*}  \\overline{\\emph{mdim}_M}(X, \\textit{\\textbf{f}},d)\\leq\\overline{\\emph{dim}_{B}} (X,d)\\quad\\text{and}\\quad \\underline{\\emph{mdim}_M}(X, \\textit{\\textbf{f}},d)\\leq\\underline{\\emph{dim}_{B}} (X,d).\n  \\end{equation*}  and therefore  \\[\\overline{\\emph{mdim}_M}(X,\\textit{\\textbf{f}},d)^*\\leq\\overline{\\emph{dim}_{B}} (X,d) \\quad \\text{and}\\quad \\underline{\\emph{mdim}_M}(X,\\textit{\\textbf{f}},d)^*\\leq\\underline{\\emph{dim}_{B}} (X,d) . \\]  In particular, if $X=[0,1],$ then $\\overline{\\emph{mdim}_M}(X,\\textit{\\textbf{f}},d)^*\\leq 1$. \\end{corollary}\n \n \n\n  Example \\ref{lkjhfg}  proves that the box dimension is not an upper bound for the metric mean dimension of  sequences that are not convergent. Next example shows the inequality  in  Corollary \\ref{efrrttt}   can be strict. \n \n \n \n\\begin{example} For each $n\\geq 1,$ take $m_{n}=n$ and\n$$\nf_n(x)= \\begin{cases}\n    \\phi(x), &  \\text{ if }x\\in[0,a_{n+1}], \\\\\n    a_{n+1}, & \\hbox{ if }x\\in [a_{n+1},1],\n      \\end{cases}\n$$ where $ \\phi$ is the map  in Example \\ref{exfagner}.  \nThus $f_n$ converges uniformly to $\\phi$ as $n\\rightarrow \\infty$. In \\cite{K-S}, Figure 3, is proved that the topological entropy $h_{top}((f_{n+k})_{n=1}^{\\infty})=k\\log 3$ for each $k\\geq1$. Hence, $ \\overline{\\text{mdim}_M}([0,1],(f_{n+k})_{n=1}^{\\infty},|\\cdot |)=0$ and therefore $$ \\overline{\\text{mdim}_M}([0,1],(f_{n})_{n=1}^{\\infty},|\\cdot |)^{\\ast}=0< \\overline{\\text{mdim}_M}([0,1],\\phi,|\\cdot |)=1.$$ \n\\end{example}\n\n  \n\n \\begin{example}  \nThe sequence\n$$\ng_n(x)= \\begin{cases}\n    \\phi(x), &  \\text{ if }x\\in[0,a_{n+1}], \\\\\n    x, & \\hbox{ if }x\\in [a_{n+1},1].\n      \\end{cases}\n$$\n  converges uniformly to $\\phi$ as $n\\rightarrow \\infty$, where $ \\phi$ is the map  in Example \\ref{exfagner}. Note that $g_{1}^{(n+k)}|J_{n}=\\phi^{k}|_{J_{n}},$ for $n\\geq 1,k\\geq1$ (see Example \\ref{exfagner}).  Hence $$\\text{sep}(2n+k,(g_{i})_{i=1}^{\\infty}, \\varepsilon_{n})\\geq (3^{m_{n}}\/2)^{k},\\quad\\text{ and then }\\quad \\text{sep}((g_{i})_{i=1}^{\\infty},\\varepsilon_{n}) \\geq \\log (3^{m_{n}}\/2).$$ Therefore $\n\\overline{\\text{mdim}_M}([0,1],(g_{i})_{i=1}^{\\infty},| \\cdot |)\\geq 1.$ By \\eqref{tergss} we obtain that   $ \\overline{\\text{mdim}_M}([0,1],(g_{i})_{i=1}^{\\infty},|\\cdot |)= 1$.   Note that $ \\overline{\\text{mdim}_M}([0,1],g_{i},|\\cdot |)= 0$ for any $i\\geq1$. \\end{example}\n\n   \n\n\n\n\n\n\n  \n  \n\n\\section{Uniform equiconjugacy and metric mean dimension}\\label{section6}\n\n   We say that  the systems  $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ on $(X,d)$ and $\\textit{\\textbf{g}}=(g_{n})_{n=1}^{\\infty}$ on $(Y,d^{\\prime})$\nare \\textit{uniformly equiconjugate} if there exists a equicontinuous sequence of  homeomorphisms $h_n: X\\to Y$ so that\n$h_{n+1}\\circ f_n=g_n\\circ h_n$, for all $n\\in \\mathbb N$,  that is, the following diagram\n\\[ \\begin{CD}\n    X @>f_1>> X@>f_2>>\\dots @>f_n>>X\\\\\n    @VVh_{1} V      @VV h_{2} V      @.     @VVh_{n+1}V\\\\\n   Y @>g_1>> Y@>g_2 >>\\dots @>g_n>>Y\n  \\end{CD}\n\\]\nis commutative for  all $n\\in \\mathbb N$. \nIn the case where $h_n=h$, for all $n\\in\\mathbb N$, we say that\n$ \\textit{\\textbf{f}}$ and $\\textit{\\textbf{g}}$ are \\textit{uniformly conjugate}.\n\n\\medskip\n\n Note that the notion of uniform equiconjugacy does not depend on the metric on $X$ and $Y$. Indeed,  if $d^{\\ast}$   and $d^{\\star}$ are another  metrics on $X$ and $Y$, respectively, then $(X,\\textit{\\textbf{f}}, d)$ and $(X,\\textit{\\textbf{f}}, d^{\\ast})$ are uniformly equiconjugate by the sequence $(I_{X})_{n=1}^{\\infty}$   and  $(Y,\\textit{\\textbf{g}}, d')$ and $(Y,\\textit{\\textbf{g}}, d^{\\star})$ are uniformly equiconjugate by the sequence $(I_{Y})_{n=1}^{\\infty}$. Hence, if  $(X,\\textit{\\textbf{f}}, d)$ and $(Y,\\textit{\\textbf{g}}, d')$ are uniformly equiconjugate by the sequence $(h_{n})_{n=1}^{\\infty}$, then $(X,\\textit{\\textbf{f}}, d^{\\ast})$ and $(Y,\\textit{\\textbf{g}}, d^{\\star})$ are uniformly equiconjugate by the sequence $(I_{Y}\\circ h_{n}\\circ I_{X})_{n=1}^{\\infty}$.\n\n  \n\n\n\\begin{theorem}\\label{edee344}\n Let $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ and $\\textit{\\textbf{g}}=(g_{n})_{n=1}^{\\infty}$ be two non-autonomous dynamical systems defined on the  metric spaces $(X,d)$ and $(Y,d^{\\prime})$ respectively.\n\\begin{enumerate}[(i)]\n\\item  If $\\textit{\\textbf{f}}$ and $\\textit{\\textbf{g}}$ are uniformly conjugate then\n\\begin{align*}\n\\emph{mdim}(X,\\textit{\\textbf{f}}\\,)&=\\emph{mdim}(X,\\textit{\\textbf{g}}).\n\\end{align*}\n\\item If $(X,\\textit{\\textbf{f}}\\,)$ and $(Y,\\textit{\\textbf{g}})$ are uniformly equiconjugate by a sequence of homeomorphisms $(h_n)_{n=1}^{\\infty}$ that satisfies $\\inf_n \\{d(h^{-1}_n(y_1),h_n^{-1}(y_2))\\}>0$  for any   $y_1,y_2\\in Y$, then (see \\eqref{infmean})\n\\begin{equation*}\n {\\emph{mdim}_M}(X,\\textit{\\textbf{f}}\\,)\\geq  {\\emph{mdim}_M}(Y,\\textit{\\textbf{g}}).\n\\end{equation*}\n\\item  If $(X,\\textit{\\textbf{f}}\\,)$ and $(Y,\\textit{\\textbf{g}})$ are uniformly equiconjugate by a sequence of homeomorphisms $(h_n)_{n=1}^{\\infty}$ that satisfies $\\inf_n \\{ d^{\\prime}(h_n(x_1),h_n(x_2))\\}>0$  for any $x_1,x_2\\in X$, then\n\\begin{equation*}\n {\\emph{mdim}_M}(X,\\textit{\\textbf{f}}\\,)\\leq  {\\emph{mdim}_M}(Y,\\textit{\\textbf{g}}).\n\\end{equation*}\n\\item  If $(X,\\textit{\\textbf{f}}\\,)$ and $(Y,\\textit{\\textbf{g}})$ are uniformly equiconjugate by a sequence of homeomorphisms $(h_n)_{n=1}^{\\infty}$ that satisfies $\\inf_n \\{d(h^{-1}_n(y_1),h_n^{-1}(y_2)), d^{\\prime}(h_n(x_1),h_n(x_2))\\}>0$  for any $y_{1},y_{2}\\in Y$ and  $x_1,x_2\\in X$, then\n\\begin{equation*}\n {\\emph{mdim}_M}(X,\\textit{\\textbf{f}}\\,)=  {\\emph{mdim}_M}(Y,\\textit{\\textbf{g}}).\n\\end{equation*}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n (i) Let $h: X\\to Y$ be a homeomorphism which conjugates  $\\textit{\\textbf{f}}$ and $\\textit{\\textbf{g}}$, i.e.,\n$h\\circ f_1^{(n)}=g_1^{(n)}\\circ h$ for all $n\\in \\mathbb N$. For an open cover $\\alpha$\nof $X$, consider $\\beta=h(\\alpha)$, which is an  open cover of $Y$. Now we notice that\n\\begin{align*}\n  \\beta_0^{n-1} & =h(\\alpha) \\vee g_1^{-1 }(h(\\alpha))\\vee\\dots\\vee (g_1^{(n-1)})^{-1}(h(\\alpha)) \\\\\n                & =h(\\alpha)\\vee (h\\circ f_1^{-1}\\circ h^{-1})(h(\\alpha))\\vee \\dots \\vee (h\\circ (f_1^{(n-1)})^{-1}\\circ h^{-1})(h(\\alpha))\\\\\n                & =h(\\alpha_0^{n-1}).\n\\end{align*}\nIt implies that $\\mathcal{D}(h(\\alpha_0^{n-1}))=\\mathcal{D}(\\alpha_0^{n-1})$. Since, for any open cover $\\beta$ of $Y$\nis of the form $h(\\alpha)$, for some open cover $\\alpha$ of $X$,\n\\[\\text{mdim}(X,\\textit{\\textbf{f}}\\,)=\\sup_{\\alpha}\\lim_{n\\to\\infty}\\frac{\\mathcal{D}(\\alpha_0^{n-1})}{n}=\\sup_{\\beta}\\lim_{n\\to\\infty}\\frac{\\mathcal{D}(\\beta_0^{n-1})}{n}=\\text{mdim}(Y,\\textit{\\textbf{g}}).\n\\]\n\n\\noindent (ii) \nLet $(h_n)_{n=1}^{\\infty}$ be the sequence of equicontinuous homeomorphisms that equiconjugates $\\textit{\\textbf{f}}$ and $\\textit{\\textbf{g}}$. So,\n$$f_n\\circ\\dots\\circ f_1=h_{n+1}^{-1}\\circ g_n\\circ\\dots\\circ g_1\\circ h_1.$$\nBy assumption we have\n\\begin{equation*}\n \\inf_n \\{d(h^{-1}_n(y_1),h_n^{-1}(y_2))\\}>0, \\text{ for any } y_1\\not=y_2\\in Y.\n\\end{equation*}\n Hence, we can define on $Y$   the metric \n\\begin{align*}\nd^\\star(y_1,y_2):=\\inf_n\\{ d(h_{n}^{-1}(y_1),h_{n}^{-1}(y_2))\\}.\n\\end{align*}\nIn particular, if $S\\subset X$ is a $(m,\\textit{\\textbf{f}} ,\\varepsilon)$-spanning set of $X$ in the metric $d$ and $x_1,x_2\\in S$, then\n\\begin{align*}\nd_m^\\star(h_1(x_1),h_1(x_2))&=\\max\\{d^{\\star}(h_1(x_1),h_1(x_2)),\\dots,d^{\\star}\n(g_1^{m-1}(h_1(x_1)),g_1^{m-1}(h_1(x_2)))\\} \\\\\n     &\\leq \\max\\{d(x_1,x_2), d(h_2^{-1}(g_1(h_1(x_1))),h_2^{-1}(g_1(h_1(x_2)))),\\\\\n            &\\quad \\quad \\dots,d(h_{m+1}^{-1}(g_1^{m-1}(h_1(x_1))),h_{m+1}^{-1}(g_1^{m-1}(h_1(x_2))))\\}\\\\\n                            &=d_m(x_1,x_2)\\leq \\varepsilon.\n\\end{align*}\nIt follows that $h_1(S)$ is an $(m, \\textit{\\textbf{g}} ,\\varepsilon)$-spanning set of $Y$ in the metric $d^{\\star}$. So we obtain that\n$$ \n {\\text{mdim}_M}(X,\\textit{\\textbf{f}}, d)\\geq   {\\text{mdim}_M}(Y,\\textit{\\textbf{g}}, d^{\\star}) , \n$$ and therefore $\n {\\text{mdim}_M}(X,\\textit{\\textbf{f}}\\,)\\geq   {\\text{mdim}_M}(Y,\\textit{\\textbf{g}}). \n$\n\nBy an analogous argument we can prove (iii).  Item (iv) follows from (ii) and (iii). \n\\end{proof}\n\n Clearly the theorem implies that if  $\\phi:X\\rightarrow X$ and $\\psi: X\\rightarrow X$ are topologically conjugate continuous maps, then      $$\n {\\text{mdim}_M}(X,\\phi)=  {\\text{mdim}_M}(X,\\psi), \n$$ which is a well-known fact.\n\n\n\\medskip\n\nThe next corollaries  follow from  Theorem \\ref{edee344}.\n\n\\begin{corollary}\\label{newe} If $f_{1},\\dots,f_{i},g_{1},\\dots,g_{i}$ are homeomorphisms, $ \\textit{\\textbf{f}}=(f_{1},  \\dots, f_{i}, f_{i+1}, f_{i+2}, \\dots)$ and $ \\textit{\\textbf{g}}=(g_{1},\\dots, g_{i}, f_{i+1}, f_{i+2}, \\dots)$, then \\[ {\\emph{mdim}_M}(X,\\textit{\\textbf{f}}\\,)={\\emph{mdim}_M}(Y,\\textit{\\textbf{g}}). \\]\n\\end{corollary}\n\\begin{proof} Note that the following diagram is commutative\n\\[\n \\begin{CD}\n    X @>f_1>> X@>f_i>>\\dots X @>f_{i}>> X @>f_{i+1}>>X @>f_{i+2}>>X \\\\\n    @V{h_{1}}VV      @VV{h_{2}}V     @VV{h_{i}}V  @VV{Id_{X}}V    @VV{Id_{X}}V     @VV {Id_{X}}V\\\\\n   X @>g_1>> X@>g_i>>\\dots  X @>g_{i}>>X @>f_{i+1}>>X @>f_{i+2}>>X\n  \\end{CD}\n\\]\nwhere $I_{X}$ is the identity of $X$ and $h_{i}=g_{i}^{-1}\\circ f_{i}$, $h_{i-1}=g_{i-1}^{-1}\\circ h_{i}\\circ f_{i-1}$, \\dots, $h_{1}=g_{1}^{-1}h_{2}f_{1}$.  Furthermore, $(h_{1}, h_{2}, \\dots, h_{i}, I_{X}, I_{X} ,\\dots )$ is an equicontinuous sequence of homeomorphisms. Therefore, $\\textit{\\textbf{f}}$ and $\\textit{\\textbf{g}}$ are uniformly equiconjugate. The corollary follows from Theorem \\ref{edee344}, since the infimum $\\inf_n \\{d(h^{-1}_n(y_1),h_n^{-1}(y_2)),d(h_n(x_1),h_n(x_2))\\}>0$ is taken over a finite set.\n\\end{proof}\n\n \n\nNext corollary    means that if $\\textit{\\textbf{f}}$ is a sequence of homeomorphisms then  the metric mean dimension  is independent on the firsts elements in the sequence $\\textit{\\textbf{f}}.$\n\n\n\\begin{corollary}\\label{corolarioigualdad}  Let $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ be a non-autonomous dynamical system consisting of homeomorphisms. For any $i,  j\\in \\mathbb{N}$ we have  $$ {\\emph{mdim}_M}(X,\\sigma^i(\\textit{\\textbf{f}}\\,))=   {\\emph{mdim}_M}(X,\\sigma^j(\\textit{\\textbf{f}}\\,)).$$\n\\end{corollary}\n\\begin{proof} It is sufficient to prove that $ {\\text{mdim}_M}(X,\\sigma^i(\\textit{\\textbf{f}}\\,))=   {\\text{mdim}_M}(X,\\textit{\\textbf{f}}\\,)$ for all $i\\in\\mathbb{N}$. Fix $i\\in\\mathbb{N}$.  Take     $\\textit{\\textbf{g}}=(g_{n})_{n\\in \\mathbb{N}}$, where, for each $n\\leq i$,   $g_{n}=I$ is the identity on $X$  and $g_{n}=f_{n}$ for $n>i$.   It follows from Corollary \\ref{newe} that $$ {\\text{mdim}_M}(X, \\textit{\\textbf{f}}\\,)=   {\\text{mdim}_M}(X, \\textit{\\textbf{g}}).$$ For each $x,y\\in X$ and $n> i$  we have \\begin{align*}\\max \\{d(x,y),\\dots , d(g_{1}^{(i-1)}(x), &g_{1}^{(i-1)}(y)), \\dots, d(g_{1}^{(n-1)}(x), g_{1}^{(n-1)}(y)) \\}\\\\\n&= \\max \\{d(x,y), d(g_{i}(x), g_{i}(y)), \\dots, d(g_{i}^{(n-i)}(x), g_{i}^{(n-i)}(y)) \\}\n\\\\\n&= \\max \\{d(x,y), d(f_{i}(x), f_{i}(y)), \\dots, d(f_{i}^{(n-i)}(x), f_{i}^{(n-i)}(y)) \\}.\n\\end{align*}\n Hence $$ {\\text{mdim}_M}(X,\\textit{\\textbf{f}}\\,)= {\\text{mdim}_M}(X,\\textit{\\textbf{g}})=   {\\text{mdim}_M}(X,\\sigma^i(\\textit{\\textbf{f}}\\,)),$$ which proves the corollary.\n\\end{proof}\n\n\n\n \n \nNext corollary follows from  Corollary \\ref{corolarioigualdad} and Proposition \\ref{propo211} (see the proof of Corollary \\ref{desdede}). \n\\begin{corollary}  For any homeomorphisms  $f$ and $g$ defined on $X$, we have  $$ {\\emph{mdim}_M}(X,f\\circ g)=   {\\emph{mdim}_M}(X,g\\circ f).$$ \n\\end{corollary}\n  \n\n  \n\n\n\n\n\n\n\\section{On the  continuity of the metric mean dimension}\\label{section7}\n\nIn this section we will show some   results related to the continuity of the metric mean dimension of sequences of diffeomorphisms defined on a manifold.  For any $r\\geq 0,$ set  \n\\[ \\mathcal{C}^{r}(X)=\\{ (f_{n})_{n=1}^{\\infty}: f_{n}:X\\rightarrow X \\text{ is a }C^{r}\\text{-map}\\}=\\prod_{i=1} ^{+\\infty}  \\text{C}^{r}(X) ,\\]\nwhere $\\text{C}^{r}(X)=\\{\\phi:X\\rightarrow X: \\phi\\text{ is a }C^{r}\\text{-map}\\}\\footnote{If $r\\geq 1$ we assume that $X$ is a Riemannian manifold}.$ \nHence $ \\mathcal{C}^{r}(X)$ can be endowed with the \\textit{product topology}, which   is generated by the sets\n\\[ \\mathcal{U}=\\prod_{i=1} ^{j}  \\text{C}^{r}(X)\\times \\prod_{i=j+1}^{j+m}U_{i} \\times \\prod_{i>j+m} ^{+\\infty}  \\text{C}^{r}(X), \\]\nwhere $U_{i}$ is an open subset of $\\text{C}^{r}(X)$, for $j+1\\leq i\\leq j+m,$ for some  $j,m\\in \\mathbb{N}$.\nThe space $\\mathcal{C}^{r}(X)$ with the product topology will be denoted by $(\\mathcal{C}^{r}(X),\\tau_{prod}).$ We can consider the map \n\\begin{align*}\n    \\underline{\\text{mdim}_{M}}:(\\mathcal{C}^{r}(X),\\tau_{prod})&\\rightarrow \\mathbb{R}\\cup \\{+\\infty\\}\\\\\n    \\textit{\\textbf{f}} &\\to  \\underline{\\text{mdim}_{M}}(\\textit{\\textbf{f}},X). \n\\end{align*}\n\nClearly, if $ \\underline{\\text{mdim}_{M}}$ is a constant map, then is continuous.\n\n\\begin{proposition}  If $\\underline{\\emph{mdim}_{M}}:(\\mathcal{C}^{r}(X),\\tau_{prod})\\rightarrow \\mathbb{R}\\cup \\{+\\infty\\}$ is not constant then is discontinuous at any $\\textit{\\textbf{f}}\\in \\mathcal{C}^{r}(X)$.\n\\end{proposition}\n\\begin{proof} Fix $ \\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}\\in \\mathcal{C}^{r}(X)$.   Since $\\underline{\\text{mdim}_{M}}$  is not constant, there exists  $ \\textit{\\textbf{g}}=(g_{n})_{n=1}^{\\infty}\\in   \\mathcal{C}^{r}(X) $ such that\n$\\underline{\\text{mdim}_{M}}(X,\\textit{\\textbf{g}})\\neq \\underline{\\text{mdim}_{M}}(X,\\textit{\\textbf{f}}\\,).$ Let $\\mathcal{V}\\in \\tau_{prod}$ be any open neighborhood of $\\textit{\\textbf{f}}$. For some $k\\in \\mathbb{N}$, the sequence  $\\textit{\\textbf{j}}=(j_{n})_{n=1}^{\\infty}$, defined by\n\\begin{equation*}j_{n}=\n\\begin{cases}\n  f_{n} & \\mbox{if }   n=1,\\dots, k \\\\\n  g_{n} & \\mbox{if } n>k , \\\\\n        \\end{cases}\n\\end{equation*}\nbelongs to $\\mathcal{V}$, by definition of $\\tau_{prod}$. It is follow from Corollary \\ref{newe} that $ \\underline{\\text{mdim}_{M}}(X,\\textit{\\textbf{j}})= \\underline{\\text{mdim}_{M}}(X,\\textit{\\textbf{g}}).$\nwhich proves the proposition.\n\\end{proof}\n\n\n  \n\nLet $d^{1}(\\cdot,\\cdot)$ be a $C^{1}$-metric on $\\text{C}^{1}(X)$. Suppose that $\\sup_{n\\in\\mathbb{N}}\\Vert D f_{n}\\Vert <\\infty$. For any $K>0$, if $d^{1}(g_{n},f_{n})<K$, then $\\sup_{n\\in\\mathbb{N}}\\Vert D g_{n}\\Vert <\\infty$ and therefore $  \\underline{\\text{mdim}_M}(X,\\textit{\\textbf{g}},d)=0.$ On the other hand, if $\\sup_{n\\in\\mathbb{N}}\\Vert D f_{n}\\Vert =\\infty$, then $  \\underline{\\text{mdim}_M}(X,\\textit{\\textbf{f}},d)$ is not necessarily zero. \n\n\n\\medskip\n\nIn \\cite{JeoCTE}, Section 6,  is proved that: \n \\begin{proposition}\\label{escolhari} If  $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty} $ is a sequence of $C^{1}$-diffeomorphisms, there exists a sequence of positive numbers $(\\delta_{n})_{n=1}^{\\infty}$   such that every sequence of diffeomorphisms  $\\textit{\\textbf{g}}=(g_{n})_{n=1}^{\\infty}$ with  \\(d^{1}(f_{n},g_{n})<\\delta_{n}\\) for each $n\\geq 1,$\n is uniformly equiconjugate to $\\textit{\\textbf{f}}$ by a sequence $(h_{n})_{n=1}^{\\infty}$ such that $h_{n}\\rightarrow I_{X}$ as $n\\rightarrow \\infty$.\n \\end{proposition}\n  \n Note that, if $h_{n}\\rightarrow I_{X}$ as $n\\rightarrow \\infty$, then for any $x_{1}\\neq x_{2}\\in X$ and $y_{1}\\neq y_{2}\\in Y$ we have $\\inf_n \\{d(h^{-1}_{n}(y_1),h_{n}^{-1}(y_2)),d(h_{n}(x_1),h_{n}(x_2))\\}>0$. Hence, it follows from Theorems \\ref{edee344} and Proposition  \\ref{escolhari} that\n\\begin{corollary}\\label{esded} Given  a sequence of diffeomorphisms  $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$, there exists a sequence of positive numbers $(\\delta_{n})_{n=1}^{\\infty}$   such that if $\\textit{\\textbf{g}}=(g_{n})_{n=1}^{\\infty} $ is a sequence of diffeomorphisms such that   \\(d^{1} (f_{n},g_{n})<\\delta_{n}\\) for each $n\\geq 1,$  then \\begin{equation*}\n\\underline{\\emph{mdim}_M}(X,\\textit{\\textbf{g}})=\\underline{\\emph{mdim}_M}(X,\\textit{\\textbf{f}}\\,).\n\\end{equation*}\n\\end{corollary}\n\n\nRoughly, Corollary \\ref{esded} means that if $d ^{1}(f_{n},g_{n}) $ converges very quickly to zero as $n\\rightarrow \\infty$, then  \\begin{equation*}\n\\underline{\\text{mdim}_M}(X,\\textit{\\textbf{f}}\\, )=\\underline{\\text{mdim}_M}(X,\\textit{\\textbf{g}}).\n\\end{equation*}\n\n  For each sequence of diffeomorphisms  $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ and a sequence of positive numbers   $ \\varepsilon=(\\varepsilon_{n})_{n=1}^{\\infty}$, a  \\textit{strong basic neighborhood} of $\\textit{\\textbf{f}}$ is the set \\[ B^{r}(\\textit{\\textbf{f}},\\varepsilon)=  \\left\\{\\textit{\\textbf{g}}=(g_{n})_{n=1}^{\\infty}:  g_{n} \\text{ is a }C^{r}\\text{-diffeomorphism and }  d  (f_{n},g_{n}) <\\varepsilon_{n},\\text{ for all  }  n\\in\\mathbb{N}\\right\\}. \\]\n The  \\textit{strong topology} (or \\textit{Whitney topology}) on $\\mathcal{C}^{r}(X)$ is generated by the strong basic neighborhoods  of each $\\textit{\\textbf{f}}\\in \\mathcal{C}^{r}(X)$. The space $\\mathcal{C}^{r}(X)$ with the strong topology will be denoted by $(\\mathcal{C}^{r}(X),\\tau_{str}).$\n \n \n\\begin{corollary}\\label{teoprinc} For    $r\\geq 1$, let  $\\mathcal{D}^{r}(X)\\subseteq \\mathcal{C}^{r}(X)$ be the set consisting of diffeomorphisms.   Then \n\\[\\underline{\\emph{mdim}_M}:(\\mathcal{D}^{r}(X),\\tau_{str})\\rightarrow \\mathbb{R}\\cup \\{+\\infty\\}   \\]\nis a continuous map. \n\\end{corollary}\n\\begin{proof} Let $\\textit{\\textbf{f}}\\in \\mathcal{D}^{r}(X)$. If follows from  Theorem \\ref{escolhari}  that there exists a strong basic neighborhood $B ^{r}(\\textit{\\textbf{f}}, (\\delta_{n})_{n=1}^{\\infty})$ such that  every  $\\textit{\\textbf{g}}\\in B ^{r}(\\textit{\\textbf{f}}, (\\delta_{n})_{n=1}^{\\infty})$ is   uniformly equiconjugate to $\\textit{\\textbf{f}}$. Thus, from  Proposition \\ref{edee344} we have $\\underline{\\text{mdim}_M}(X,\\textit{\\textbf{g}})=\\underline{\\text{mdim}_M}(X,\\textit{\\textbf{f}}\\,)$   for all $\\textit{\\textbf{g}}\\in B^{r}(\\textit{\\textbf{f}}, (\\delta_{n})_{n=1}^{\\infty})$, which proves the corollary.\n\\end{proof}\n\n  \n   \n A real valued function $\\varphi : X \\rightarrow \\mathbb{R}\\cup \\{\\infty\\}$ is called \\textit{lower} (respectively \\textit{upper})  \\textit{semi-continuous on a point} $x\\in X$ if $$\\liminf_{y\\rightarrow x}\\varphi (y)\\geq \\varphi (x)\\quad  (\\text{repectively } \\limsup_{y\\rightarrow x}\\varphi (y)\\leq \\varphi (x) ).    $$    $\\varphi  $ is called \\textit{lower} (respectively \\textit{upper})  \\textit{semi-continuous} if is  lower (respectively  {upper})  {semi-continuous on any point} of $ X$. \n \n \n  \n \n\\begin{remark} From now on, we will consider   $\\tilde{X}=[0,1]$ or $\\mathbb{S}^1$. \\end{remark}\n\nMisiurewicz in \\cite{Misiurewicz}, Corollary 1, proved that $h_{top}: C^{0}([0,1])\\rightarrow \\mathbb{R}\\cup \\{\\infty\\}$ is lower semi-continuous. For the case of the metric mean dimension we have:  \n \n \\begin{proposition}\\label{hfjdjehr}  $\\emph{mdim}_{M}:C^{0}(\\tilde{X})\\rightarrow \\mathbb{R}$ is nor lower neither upper  semi-continuous on maps with metric mean dimension in $(0,1)$. Furthermore,   \n $\\emph{mdim}_{M}:C^{0}(\\tilde{X})\\rightarrow \\mathbb{R}$ is not  lower   semi-continuous on maps with metric mean dimension in $(0,1]$ and is not  upper   semi-continuous on maps with metric mean dimension in $[0,1)$. \n \\end{proposition}\n \\begin{proof}\n Let $\\varphi$ be a continuous map on $\\tilde{X}$.  If $\\text{mdim}_{M}(\\varphi)=1$, we can    approximate $\\varphi$ by a continuous map with zero metric mean dimension (take a sequence of $C^{1}$-maps converging to $\\varphi$). Next, suppose that $\\text{mdim}_{M}(\\varphi)=0$. Firstly, take $\\tilde{X}=[0,1]$. Fix  $\\varepsilon >0$.  \n  Let $p^{\\ast}$ be a fixed point of $\\varphi$.   Choose  $\\delta>0$ such that $d(\\varphi(x),\\varphi (p^{\\ast}))<\\varepsilon\/2$ for any $x$ with $d(x,p^{\\ast})<\\delta$.  Let $\\phi $ and $T_{2} $ be as in   Example \\ref{exfagner}, with $J_{1}=[0,p^{\\ast}]$, $J_{2}=[p^{\\ast},p^{\\ast}+\\delta\/2]$, $J_{3}=[p^{\\ast}+\\delta\/2,p^{\\ast}+\\delta] $ and $J_{4}=[p^{\\ast}+\\delta,1]$. Take the continuous map $\\psi$ on $[0,1]$ defined as $$\n\\psi(x)= \\begin{cases}\n    \\varphi(x), &  \\text{ if }x\\in J_{1}\\cup J_{4}, \\\\\n   T_{2}^{-1}\\phi T_{2}(x), &  \\text{ if }x\\in J_{2}, \\\\\n    \\psi_{1}(x), &  \\text{ if }x\\in J_{3}, \n      \\end{cases}\n$$ where $ \\psi_{1}$ is the  affine map on $J_{3}$ such that $ \\psi_{1}(p^{\\ast}+\\delta\/2)=(p^{\\ast}+\\delta\/2)$  and $ \\psi_{1}(p^{\\ast}+\\delta)=\\varphi(p^{\\ast}+\\delta).$ Note that $d(\\psi,\\varphi)<\\varepsilon.$ It follows from Proposition \\ref{invariant} that  $$\\text{mdim}_{M}(\\tilde{X},\\psi,|\\cdot |)=\\max \\{\\text{mdim}_{M}(\\tilde{X},\\psi|_{J_{1}\\cup J_{3}\\cup J_{4}},|\\cdot |)  ,\\text{mdim}_{M}(\\tilde{X},\\psi|_{J_{2}},|\\cdot |)\\}= 1,$$ \nsince $\\text{mdim}_{M}(\\tilde{X},\\psi|_{J_{1}\\cup J_{3}\\cup J_{4}},|\\cdot |)\\leq \\text{mdim}_{M}(\\tilde{X},\\varphi,|\\cdot |) =0$.\nAnalogously we can prove that any $\\varphi\\in C^{0}([0,1])$ with metric mean dimension  in $(0,1)$ can be approximated by both a continuous map with metric mean dimension equal to 1 and a continuous map with metric mean dimension equal to 0. These facts prove the proposition for $\\tilde{X}=[0,1]$.  For $\\tilde{X}=\\mathbb{S}^{1}$, we can approximate any $\\varphi\\in C^{0}(\\mathbb{S}^{1})$ by a map $\\varphi^{\\ast}$ with periodic points. We can prove analogously  that $\\varphi^{\\ast}$ can be approximate by a continuous map on $\\mathbb{S}^{1}$ with metric mean dimension equal to 0 or equal to 1, which proves the proposition for $\\tilde{X}=\\mathbb{S}^{1}$. \n \\end{proof}\n   \n \nNext, Kolyada and Snoha in \\cite{K-S}, Theorem F, showed that  $h_{top}:\\mathcal{C}([0,1])\\rightarrow \\mathbb{R}\\cup\\{\\infty\\}$ is not lower semi-continuous, endowing  $\\mathcal{C}([0,1])$   with the metric $$D((f_{n})_{n=1}^{\\infty},(g_{n})_{n=1}^{\\infty})=\\sup_{n\\in\\mathbb{N}}\\max_{x\\in [0,1]}|f_{n}(x)-g_{n}(x)|.$$ Furthermore, they proved in Theorem G that $h_{top}:\\mathcal{C}([0,1])\\rightarrow \\mathbb{R}\\cup\\{\\infty\\}$ is  lower semi-continuous on any constant sequence $(\\phi, \\phi,\\dots)\\in \\mathcal{C}(\\tilde{X})$.\nHowever, It follows from Proposition \\ref{hfjdjehr} that:\n \n \\begin{corollary}   $\\emph{mdim}_{M}:\\mathcal{C}(\\tilde{X})\\rightarrow \\mathbb{R}$ is nor lower neither upper  semi-continuous on any constant sequence $(\\phi, \\phi,\\dots)\\in \\mathcal{C}(\\tilde{X})$. Consequently, $\\emph{mdim}_{M}:\\mathcal{C}(\\tilde{X})\\rightarrow \\mathbb{R}\\cup\\{\\infty\\}$ is nor lower neither upper  semi-continuous.  \n \\end{corollary} \n \n  Take $\\textbf{\\textit{f}}=(f_{n})_{n=1}^{\\infty}$ on $\\tilde{X}$ defined by $f_{n} =\\psi^{2^{n}}$ for each $n\\in\\mathbb{N}$, where $\\psi$ is the map from Example \\ref{exfagner}. We have  $\\text{mdim}_{M}(\\tilde{X},\\textbf{\\textit{f}},|\\cdot|)=\\infty$ (see Example \\ref{hfkenrkflr}). Thus  there exist non-autonomous dynamical systems on $\\tilde{X}$ with infinite metric mean dimension. Consequently $\\text{mdim}_{M}:\\mathcal{C}(\\tilde{X})\\rightarrow \\mathbb{R}\\cup \\{\\infty\\}$ is unbounded. \n\n\\medskip\n \n \n We finish this work with the next result:\n \\begin{theorem}\\label{bnfuefnf}\n$\\emph{mdim}_{M}:\\mathcal{C}(\\tilde{X})\\rightarrow \\mathbb{R}\\cup\\{\\infty\\}$ is not lower semi-continuous on any non-autonomous dynamical system  with non-zero metric mean dimension.  \n\\end{theorem}  \n\\begin{proof}\nLet $\\textit{\\textbf{f}}=(f_{n})_{n=1}^{\\infty}$ be a non-autonomous dynamical system with positive metric mean dimension. Let $\\lambda_{m}$ be a sequence in $[0,1]$ such that $\\lambda_{m}\\rightarrow 1$ and $\\lambda_{m}\\cdots \\lambda_{1}\\rightarrow 0$ as $m\\rightarrow \\infty$. Take $\\textbf{\\textit{g}}_{m}=(\\lambda_{m+n}f_{n})_{n=1}^{\\infty}$. Thus $\\textbf{\\textit{g}}_{m}\\rightarrow \\textbf{\\textit{f}}$ as $m\\rightarrow \\infty$. However, for any $x\\in \\tilde{X}$, $({g}_{m})^{(k)}(x)\\rightarrow 0$ as $k\\rightarrow \\infty$. Consequently, the metric mean dimension of $\\textbf{\\textit{g}}_{m}$ is zero for each $m\\in \\mathbb{N}$. \n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $G=(V,E)$ be an undirected graph. All graphs considered in this paper are\nassumed to be finite and simple. The \\emph{closed neighborhood} $N_{G}\\left[\nv\\right]  $ of a vertex $v\\in V$ is the set consisting of $v$ and all its\nneighbor vertices in $G$. For any subset $W\\subseteq V$, we denote by\n$N_{G}\\left[  W\\right]  $ the \\emph{closed neighborhood} of $W$ in $G$, that is%\n\n\\[\nN_{G}\\left[  W\\right]  =\\bigcup\\limits_{v\\in W}N_{G}\\left[  v\\right]  .\n\\]\n\n\nIf the graph is clear from the context, then we write $N\\left[  v\\right]  $\nand $N\\left[  W\\right]  $ instead of $N_{G}\\left[  W\\right]  $ and\n$N_{G}\\left[  v\\right]  $, respectively. A \\emph{dominating set} of $G$ is a\nvertex subset $W\\subseteq V$ such that $N\\left[  W\\right]  =V$. Let\n$W\\subseteq V$ be a given vertex subset of the graph $G=(V,E)$. We denote by\n$\\partial(W)$ the set of all edges of $G$ that have exactly one of their end\nvertices in $W$, that is%\n\n\\[\n\\partial(W)=\\left\\{  \\left\\{  u,v\\right\\}  \\in E\\mid u\\in W,v\\in V\\setminus\nW\\right\\}  .\n\\]\n\n\nThe edges of $\\partial(W)$ link vertices of $W$ with vertices of $V\\setminus\nW$. Whether a given set $W$ is a dominating set of $G$ depends neither on\nedges lying completely inside $W$ nor on edges that have no end vertex in $W$,\nwhich gives the following statement.\n\n\\begin{proposition}\n\\label{prop_delta}Let $G=(V,E)$ be a graph, $W\\subseteq V$, and $F\\subseteq E\n$. Then $W$ is a dominating set of $\\left(  V,F\\right)  $ if and only $W$ is\ndominating in $\\left(  V,F\\cap\\partial(W)\\right)  $, i.e.%\n\n\\[\nN_{\\left(  V,F\\right)  }\\left[  W\\right]  =V\\ \\Longleftrightarrow\\ N_{\\left(\nV,F\\cap\\partial(W)\\right)  }\\left[  W\\right]  =V.\n\\]\n\n\\end{proposition}\n\n\\begin{definition}\nLet $G=(V,E)$ be an undirected graph and $d_{k}(G)$ the number of dominating\nsets of cardinality $k$ in $G$ for $k=0,...,n=\\left\\vert V\\right\\vert $. The\n\\emph{domination polynomial of }$G$ is%\n\n\\[\nD(G,x)=\\sum_{k=0}^{n}d_{k}(G)x^{k}.\n\\]\n\n\\end{definition}\n\nWe denote by $d(G)$ the number of dominating sets of $G$. Consequently, we\nfind $d(G)=D(G,1)$.\n\nThe domination polynomial of a graph has been introduced by Arocha and Llano\nin \\cite{Arocha}. More recently it has been investigated with respect to\nspecial graphs, zeros, and applications in network reliability, see\n\\cite{AAOP,AP,AP2,AAP,DT}.\n\nThe domination polynomial can also be represented as a sum over vertex subsets\nof $G$,%\n\\[\nD(G,x)=\\sum_{\\substack{U\\subseteq V \\\\N\\left[  U\\right]  =V}}x^{\\left\\vert\nU\\right\\vert }.\n\\]\n\n\nThe domination polynomial is multiplicative with respect to components, see\n\\cite{Arocha}. Let $G_{1},...,G_{k}$ be the components of a given graph $G$,\nthen%\n\\begin{equation}\nD(G,x)=%\n{\\displaystyle\\prod\\limits_{i=1}^{k}}\nD(G_{i},x). \\label{product}%\n\\end{equation}\n\n\n\\section{Spanning Subgraphs}\n\nIn this section, we provide a representation of the domination polynomial as a\nsum ranging over all bipartite spanning subgraphs of a graph.\n\n\\subsection{Connected Bipartite Graphs}\n\nAlternating sums of domination polynomials of spanning subgraphs of a given\ngraph yield a particularly simple result in case of connected bipartite graphs.\n\n\\begin{lemma}\n\\label{Lemma_bipart}Let $G=(V,E)$ be a connected bipartite graph with\nbipartition $V=Y\\cup Z$, $Y\\neq\\emptyset$, $Z\\neq\\emptyset$. Then%\n\\[\n\\sum_{F\\subseteq E}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(  V,F\\right)\n,x\\right)  =(-1)^{\\left\\vert Y\\right\\vert }x^{\\left\\vert Z\\right\\vert\n}+(-1)^{\\left\\vert Z\\right\\vert }x^{\\left\\vert Y\\right\\vert }.\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nLet $W$ be a dominating set of $G$; then we can distinguish three cases, namely%\n\n\\begin{align*}\n\\text{(a)}  &  W\\cap Y\\neq\\emptyset\\text{ and }W\\cap Z\\neq\\emptyset,\\\\\n\\text{(b)}  &  W=Y,\\\\\n\\text{(c)}  &  W=Z.\n\\end{align*}\n\n\nWe decompose the sum according to the above given cases:%\n\n\\begin{align}\n\\sum_{F\\subseteq E}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(  V,F\\right)\n,x\\right)   &  =\\sum_{F\\subseteq E}\\sum_{\\substack{W\\subseteq V \\\\N_{\\left(\nV,F\\right)  }\\left[  W\\right]  =V}}(-1)^{\\left\\vert F\\right\\vert\n}x^{\\left\\vert W\\right\\vert }\\nonumber\\\\\n&  =\\sum_{W\\subseteq V}x^{\\left\\vert W\\right\\vert }\\sum_{\\substack{F\\subseteq\nE \\\\N_{\\left(  V,F\\right)  }\\left[  W\\right]  =V}}(-1)^{\\left\\vert\nF\\right\\vert }\\nonumber\\\\\n&  =\\sum_{\\substack{W\\subseteq V \\\\W\\cap Y\\neq\\emptyset\\\\W\\cap Z\\neq\\emptyset\n}}x^{\\left\\vert W\\right\\vert }\\sum_{\\substack{F\\subseteq E \\\\N_{\\left(\nV,F\\right)  }\\left[  W\\right]  =V}}(-1)^{\\left\\vert F\\right\\vert }%\n\\tag{a}\\label{sum1}\\\\\n&  +x^{\\left\\vert Y\\right\\vert }\\sum_{\\substack{F\\subseteq E \\\\N_{\\left(\nV,F\\right)  }\\left[  Y\\right]  =V}}(-1)^{\\left\\vert F\\right\\vert }%\n\\tag{b}\\label{sum2}\\\\\n&  +x^{\\left\\vert Z\\right\\vert }\\sum_{\\substack{F\\subseteq E \\\\N_{\\left(\nV,F\\right)  }\\left[  Z\\right]  =V}}(-1)^{\\left\\vert F\\right\\vert }.\n\\tag{c}\\label{sum3}%\n\\end{align}\n\n\nWe show that the Sum (\\ref{sum1}) vanishes. According to Proposition\n\\ref{prop_delta}, a set $W$ is dominating in $(V,F)$ if and only if $W$ is a\ndominating set of $(V,F\\cap\\partial(W))$. The evaluation of the Sum\n(\\ref{sum1}) yields%\n\n\\begin{align*}\n\\sum_{\\substack{W\\subseteq V \\\\W\\cap Y\\neq\\emptyset\\\\W\\cap Z\\neq\\emptyset\n}}x^{\\left\\vert W\\right\\vert }\\sum_{\\substack{F\\subseteq E \\\\N_{\\left(\nV,F\\right)  }\\left[  W\\right]  =V}}(-1)^{\\left\\vert F\\right\\vert }  &\n=\\sum_{\\substack{W\\subseteq V \\\\W\\cap Y\\neq\\emptyset\\\\W\\cap Z\\neq\\emptyset\n}}x^{\\left\\vert W\\right\\vert }\\sum_{\\substack{F_{1}\\subseteq E\\cap\\partial(W)\n\\\\F_{2}\\subseteq E\\setminus\\partial(W) \\\\N_{\\left(  V,F_{1}\\right)  }\\left[\nW\\right]  =V}}(-1)^{\\left\\vert F_{1}\\cup F_{2}\\right\\vert }\\\\\n&  =\\sum_{\\substack{W\\subseteq V \\\\W\\cap Y\\neq\\emptyset\\\\W\\cap Z\\neq\\emptyset\n}}x^{\\left\\vert W\\right\\vert }\\sum_{\\substack{F_{1}\\subseteq E\\cap\\partial(W)\n\\\\N_{\\left(  V,F_{1}\\right)  }\\left[  W\\right]  =V}}(-1)^{\\left\\vert\nF_{1}\\right\\vert }\\sum_{F_{2}\\subseteq E\\setminus\\partial(W)}(-1)^{\\left\\vert\nF_{2}\\right\\vert }.\n\\end{align*}\n\n\nNow assume that $E\\setminus\\partial(W)=\\emptyset$. Let $y\\in Y\\cap W$ and\n$z\\in Z\\cap W$. Then there does not exist a path between $y$ and $z$ in $G$.\nThis contradicts the assumed connectedness of $G$; hence $E\\setminus\n\\partial(W)\\neq\\emptyset$, which gives%\n\\[\n\\sum_{F_{2}\\subseteq E\\setminus\\partial(W)}(-1)^{\\left\\vert F_{2}\\right\\vert\n}=(1-1)^{\\left\\vert E\\setminus\\partial(W)\\right\\vert }=0.\n\\]\nNow we turn to the Sum (\\ref{sum2}),%\n\\[\n\\sum_{\\substack{F\\subseteq E\\\\N_{\\left(  V,F\\right)  }\\left[  Y\\right]\n=V}}(-1)^{\\left\\vert F\\right\\vert }.\n\\]\nAn edge subset $F\\subseteq E$ satisfies the property \\textquotedblleft$Y$ is\ndominating in $\\left(  V,F\\right)  $\\textquotedblright\\ if and only if $F$\ncontains at least one edge from each vertex of $Z$. We denote the vertices of\n$Z$ by $v_{1},...,v_{k}$. For each $i$, $i=1,...,k$, let $E_{i}$ be the set of\nedges of $G$ that are incident to $v_{i}$. We define%\n\\[\n\\mathcal{F}=\\left\\{  A\\subseteq E\\mid\\forall i=1,...,k:\\left\\vert E_{i}\\cap\nA\\right\\vert \\geq1\\right\\}  .\n\\]\nNow the Sum (\\ref{sum2}) can be expressed as follows,%\n\\begin{align*}\n\\sum_{\\substack{F\\subseteq E\\\\N_{\\left(  V,F\\right)  }\\left[  Y\\right]\n=V}}(-1)^{\\left\\vert F\\right\\vert }  &  =\\sum_{F\\in\\mathcal{F}}%\n(-1)^{\\left\\vert F\\right\\vert }\\\\\n&  =\\sum_{\\substack{F_{1}\\cup F_{2}\\cup...\\cup F_{k}\\in\\mathcal{F}\\\\\\forall\ni=1,...,k:F_{i}\\subseteq E_{i}}}(-1)^{\\left\\vert F_{1}\\cup F_{2}\\cup...\\cup\nF_{k}\\right\\vert }\\\\\n&  =\\sum_{\\forall i=1,...,k:\\emptyset\\neq F_{i}\\subseteq E_{i}}%\n(-1)^{\\left\\vert F_{1}\\right\\vert +\\left\\vert F_{2}\\right\\vert +...+\\left\\vert\nF_{k}\\right\\vert }\\\\\n&  =\\sum_{\\substack{F_{1}\\subseteq E_{1}\\\\F_{1}\\neq\\emptyset}}(-1)^{\\left\\vert\nF_{1}\\right\\vert }\\sum_{\\substack{F_{2}\\subseteq E_{2}\\\\F_{2}\\neq\\emptyset\n}}(-1)^{\\left\\vert F_{2}\\right\\vert }\\cdot\\cdot\\cdot\\sum_{\\substack{F_{k}%\n\\subseteq E_{k}\\\\F_{k}\\neq\\emptyset}}(-1)^{\\left\\vert F_{k}\\right\\vert }\\\\\n&  =(-1)^{k}=(-1)^{\\left\\vert Z\\right\\vert },\n\\end{align*}\nwhich yields%\n\\[\nx^{\\left\\vert Y\\right\\vert }\\sum_{\\substack{F\\subseteq E\\\\N_{\\left(\nV,F\\right)  }\\left[  Y\\right]  =V}}(-1)^{\\left\\vert F\\right\\vert\n}=(-1)^{\\left\\vert Z\\right\\vert }x^{\\left\\vert Y\\right\\vert }.\n\\]\nIn the same vein, we can prove that the sum (c) satisfies%\n\\[\nx^{\\left\\vert Z\\right\\vert }\\sum_{\\substack{F\\subseteq E\\\\N_{\\left(\nV,F\\right)  }\\left[  Z\\right]  =V}}(-1)^{\\left\\vert F\\right\\vert\n}=(-1)^{\\left\\vert Y\\right\\vert }x^{\\left\\vert Z\\right\\vert }%\n\\]\nand the statement follows.\n\\end{proof}\n\n\\subsection{General Bipartite Graphs}\n\n\\begin{lemma}\n\\label{lemma_bipart2}Let $G=(V,E)$ be a bipartite graph with bipartition\n$V=Y\\cup Z$. Assume that $G$ consists of $k+l$ components such that the $k$\ncomponents $G_{1}=(V_{1},E_{1}),...,G_{k}=(V_{k},E_{k})$ have nonempty edge\nsets and the remaining $l$ components are isomorphic to $K_{1}$. Then\n\\[\n\\sum_{F\\subseteq E}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(  V,F\\right)\n,x\\right)  =x^{l}%\n{\\displaystyle\\prod\\limits_{i=1}^{k}}\n\\left[  (-1)^{\\left\\vert Y\\cap V_{i}\\right\\vert }x^{\\left\\vert Z\\cap\nV_{i}\\right\\vert }+(-1)^{\\left\\vert Z\\cap V_{i}\\right\\vert }x^{\\left\\vert\nY\\cap V_{i}\\right\\vert }\\right]  .\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nFor the one-vertex graph $K_{1}=\\left(  \\left\\{  v\\right\\}  ,\\emptyset\\right)\n$, we obtain%\n\\[\n\\sum_{F\\subseteq\\emptyset}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(\n\\left\\{  v\\right\\}  ,F\\right)  ,x\\right)  =x.\n\\]\nBy Equation (\\ref{product}), we obtain%\n\\begin{align*}\n\\sum_{F\\subseteq E}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(  V,F\\right)\n,x\\right)   &  =\\sum_{F\\subseteq E}(-1)^{\\left\\vert F\\right\\vert }x^{l}%\n{\\displaystyle\\prod\\limits_{i=1}^{k}}\nD\\left(  \\left(  V_{i},F\\cap E_{i}\\right)  ,x\\right) \\\\\n&  =x^{l}\\sum_{F\\subseteq E}%\n{\\displaystyle\\prod\\limits_{i=1}^{k}}\n(-1)^{\\left\\vert F\\cap E_{i}\\right\\vert }D\\left(  \\left(  V_{i},F\\cap\nE_{i}\\right)  ,x\\right) \\\\\n&  =x^{l}%\n{\\displaystyle\\prod\\limits_{i=1}^{k}}\n\\sum_{F\\subseteq E_{i}}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(\nV_{i},F\\right)  ,x\\right) \\\\\n&  =x^{l}%\n{\\displaystyle\\prod\\limits_{i=1}^{k}}\n\\left[  (-1)^{\\left\\vert Y\\cap V_{i}\\right\\vert }x^{\\left\\vert Z\\cap\nV_{i}\\right\\vert }+(-1)^{\\left\\vert Z\\cap V_{i}\\right\\vert }x^{\\left\\vert\nY\\cap V_{i}\\right\\vert }\\right]  ,\n\\end{align*}\nwhere the last equality is valid due to Lemma \\ref{Lemma_bipart}.\n\\end{proof}\n\nObserve that $(-1)^{\\left\\vert Y\\right\\vert }x^{\\left\\vert Z\\right\\vert\n}+(-1)^{\\left\\vert Z\\right\\vert }x^{\\left\\vert Y\\right\\vert }\\neq0$ for any\nbipartition $V=Y\\cup Z$, which shows together with Lemma \\ref{lemma_bipart2}\nthat\n\\[\n\\sum_{F\\subseteq E}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(  V,F\\right)\n,x\\right)  \\neq0\n\\]\nfor any bipartite graph $G=(V,E)$. Moreover, we have the following statement.\n\n\\begin{theorem}\n\\label{theo_main}Let $G=(V,E)$ be a graph. Then%\n\\[\n\\sum_{F\\subseteq E}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(  V,F\\right)\n,x\\right)  \\neq0\n\\]\nif and only if $G$ is bipartite.\n\\end{theorem}\n\n\\begin{proof}\nIt remains to show that the sum vanishes for non-bipartite graphs. Using\nProposition \\ref{prop_delta}, we obtain\n\\begin{align*}\n\\sum_{F\\subseteq E}(-1)^{\\left\\vert F\\right\\vert }D\\left(  \\left(  V,F\\right)\n,x\\right)   &  =\\sum_{F\\subseteq E}\\sum_{\\substack{W\\subseteq V \\\\N_{\\left(\nV,F\\right)  }\\left[  W\\right]  =V}}(-1)^{\\left\\vert F\\right\\vert\n}x^{\\left\\vert W\\right\\vert }\\\\\n&  =\\sum_{W\\subseteq V}x^{\\left\\vert W\\right\\vert }\\sum_{\\substack{F\\subseteq\nE \\\\N_{\\left(  V,F\\right)  }\\left[  W\\right]  =V}}(-1)^{\\left\\vert\nF\\right\\vert }\\\\\n&  =\\sum_{W\\subseteq V}x^{\\left\\vert W\\right\\vert }\\sum_{\\substack{F_{1}%\n\\subseteq\\partial(W) \\\\N_{\\left(  V,F_{1}\\right)  }\\left[  W\\right]  =V\n}}(-1)^{\\left\\vert F_{1}\\right\\vert }\\sum_{F_{2}\\subseteq E\\setminus\n\\partial(W)}(-1)^{\\left\\vert F_{2}\\right\\vert }.\n\\end{align*}\nSince $G$ is not a bipartite graph, the set $F_{2}$ is nonempty, which yields%\n\\[\n\\sum_{F_{2}\\subseteq E\\setminus\\partial(W)}(-1)^{\\left\\vert F_{2}\\right\\vert\n}=0\n\\]\nand hence the statement of the theorem.\n\\end{proof}\n\nThere is also a \\textquotedblleft local version\\textquotedblright\\ for one\ndirection of Theorem \\ref{theo_main}, which can be proved by the same method.\n\n\\begin{theorem}\nLet $G=(V,E)$ be a graph and $A\\subseteq E$ an edge subset such that $\\left(\nV,A\\right)  $ contains an odd cycle. Then%\n\\[\n\\sum_{F\\subseteq A}(-1)^{\\left\\vert F\\right\\vert }D\\left(  G-F,x\\right)  =0.\n\\]\n\n\\end{theorem}\n\n\\subsection{Applications of Spanning Subgraph Expansions}\n\nLet $G=(V,E)$ be a given graph. We define for any edge subset $F$ of $G$,%\n\\[\nh(F)=\\sum_{A\\subseteq F}(-1)^{\\left\\vert A\\right\\vert }D\\left(  \\left(\nV,A\\right)  ,x\\right)  .\n\\]\nM\\\"{o}bius inversion yields%\n\\[\nD\\left(  \\left(  V,F\\right)  ,x\\right)  =\\sum_{A\\subseteq F}(-1)^{\\left\\vert\nA\\right\\vert }h(A).\n\\]\nAccording to Lemma \\ref{Lemma_bipart}, Lemma \\ref{lemma_bipart2}, and Theorem\n\\ref{theo_main}, we define%\n\\begin{equation}\nh(F)=\\left\\{\n\\begin{array}\n[c]{l}%\nx^{l}%\n{\\displaystyle\\prod\\limits_{i=1}^{k}}\n(-1)^{\\left\\vert E_{i}\\right\\vert }\\left[  (-1)^{\\left\\vert Y\\cap\nV_{i}\\right\\vert }x^{\\left\\vert Z\\cap V_{i}\\right\\vert }+(-1)^{\\left\\vert\nZ\\cap V_{i}\\right\\vert }x^{\\left\\vert Y\\cap V_{i}\\right\\vert }\\right]  \\text{,\nif }\\left(  V,F\\right)  \\text{ is bipartite,}\\\\\n0\\text{ otherwise.}%\n\\end{array}\n\\right.  \\label{h-function}%\n\\end{equation}\nHere the notations are as in Lemma \\ref{lemma_bipart2}. We can now conclude\nthat the domination polynomial of a graph $G=(V,E)$ is a sum of $h$-function\nvalues of spanning bipartite subgraphs, i.e.%\n\\begin{equation}\nD\\left(  G,x\\right)  =\\sum_{\\substack{B\\subseteq E\\\\\\left(  V,B\\right)  \\text{\nis bipartite}}}h(B). \\label{h-sum}%\n\\end{equation}\nThe number of dominating sets of $G=(V,E)$ is $D(G,1)$. In order to derive\nthis number from Equation (\\ref{h-sum}), we define $h_{1}$ by substituting\n$x=1$ in $h$, that is%\n\\[\nh_{1}(F)=%\n{\\displaystyle\\prod\\limits_{i=1}^{k}}\n(-1)^{\\left\\vert E_{i}\\right\\vert }\\left[  (-1)^{\\left\\vert Y\\cap\nV_{i}\\right\\vert }+(-1)^{\\left\\vert Z\\cap V_{i}\\right\\vert }\\right]  .\n\\]\nObserve that $h_{1}(\\emptyset)=1$ and $h_{1}(F)\\equiv0$ $(\\operatorname{mod}%\n2)$ for $F\\neq\\emptyset$, which gives the following statement.\n\n\\begin{corollary}\n\\label{coro_odd}For any graph $G$, the number of dominating sets of $G$ is odd.\n\\end{corollary}\n\nFor alternative proofs of this corollary, see \\cite{Brouwer}.\n\n\\begin{remark}\nIn almost the same manner, by substituting $x=-1$ in $h$, we can prove that\n$D(G,-1)$ is odd. Moreover, from the Equations (\\ref{h-function}) and\n(\\ref{h-sum}), we obtain%\n\\[\nD(G,-1)=(-1)^{\\left\\vert V\\right\\vert }\\sum_{\\substack{F\\subseteq\nE\\\\(V,F)\\text{ is bipartite}}}(-1)^{\\left\\vert F\\right\\vert }2^{c(F)},\n\\]\nwhere $c(F)$ denotes here the number of components of $\\left(  V,F\\right)  $\nthat have at least one edge.\n\\end{remark}\n\n\\section{Vertex Induced Subgraphs}\n\nLet $G=(V,E)$ be a graph and $W\\subseteq V$. We denote by $G\\left[  W\\right]\n$ the \\emph{vertex induced subgraph} of $G$:\n\\[\nG\\left[  W\\right]  =\\left(  W,\\left\\{  \\left\\{  u,v\\right\\}  \\in E\\mid u\\in\nW\\text{ and }v\\in W\\right\\}  \\right)  .\n\\]\n\n\n\\begin{theorem}\n\\label{theo_vertex_sum}Any connected graph $G=(V,E)$ satisfies%\n\\[\n\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }D\\left(  G\\left[  W\\right]\n,x\\right)  =1+\\left(  -x\\right)  ^{\\left\\vert V\\right\\vert }.\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nBy switching the order of summation, we have%\n\\begin{align*}\n\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }D\\left(  G\\left[  W\\right]\n,x\\right)   &  =\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }%\n\\sum_{\\substack{T\\subseteq W\\\\N_{G\\left[  W\\right]  }\\left[  T\\right]\n=W}}x^{\\left\\vert T\\right\\vert }\\\\\n&  =\\sum_{T\\subseteq V}x^{\\left\\vert T\\right\\vert }\\sum_{\\substack{W\\supseteq\nT\\\\N_{G\\left[  W\\right]  }\\left[  T\\right]  =W}}(-1)^{\\left\\vert W\\right\\vert\n}\\\\\n&  =\\sum_{T\\subseteq V}x^{\\left\\vert T\\right\\vert }\\sum_{W:T\\subseteq\nW\\subseteq N_{G\\left[  W\\right]  }\\left[  T\\right]  }(-1)^{\\left\\vert\nW\\right\\vert }\\\\\n&  =\\sum_{T\\subseteq V}x^{\\left\\vert T\\right\\vert }\\sum_{W:T\\subseteq\nW\\subseteq N_{G}\\left[  T\\right]  }(-1)^{\\left\\vert W\\right\\vert }\\\\\n&  =\\sum_{T\\subseteq V}(-x)^{\\left\\vert T\\right\\vert }\\sum_{Y\\subseteq\nN_{G}\\left[  T\\right]  \\setminus T}(-1)^{\\left\\vert Y\\right\\vert }.\n\\end{align*}\nSince $G$ is connected, the set $N_{G}\\left[  T\\right]  \\setminus T$ is empty\nif and only if $T=\\emptyset$ or $T=V$. Hence we obtain%\n\\[\n\\sum_{T\\subseteq V}(-x)^{\\left\\vert T\\right\\vert }\\sum_{Y\\subseteq\nN_{G}\\left[  T\\right]  \\setminus T}(-1)^{\\left\\vert Y\\right\\vert\n}=1+(-x)^{\\left\\vert V\\right\\vert }.\\text{ }%\n\\]\n\n\\end{proof}\n\n\\begin{definition}\nLet $G=(V,E)$ be a graph with $n$ vertices. The \\emph{type} of $G$ is an\ninteger partition $\\lambda_{G}=\\left(  \\lambda_{1},...,\\lambda_{k}\\right)\n\\vdash n$ that gives the sequence of orders of the components of $G$. We write\n$i\\in\\lambda_{G}$ in order to indicate that $i$ is a part of $\\lambda_{G}$.\nThe number of parts of $\\lambda_{G}$ is denoted by $\\left\\vert \\lambda\n_{G}\\right\\vert $.\n\\end{definition}\n\nObserve that for all $W\\subseteq V$ the relation $\\left\\vert \\lambda_{G\\left[\nW\\right]  }\\right\\vert \\leq\\alpha(G)$ is satisfied, where $\\alpha(G)$ denotes\nthe independence number of $G$. Theorem \\ref{theo_vertex_sum} and Equation\n(\\ref{product}) immediately imply the following statement.\n\n\\begin{corollary}\n\\label{coro_type}For any graph $G=(V,E)$, we have%\n\\begin{equation}\n\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }D\\left(  G\\left[  W\\right]\n,x\\right)  =%\n{\\displaystyle\\prod\\limits_{i\\in\\lambda_{G}}}\n\\left(  1+(-x)^{i}\\right)  . \\label{vertex sum}%\n\\end{equation}\n\n\\end{corollary}\n\nThe application of the M\\\"{o}bius inversion to Equation (\\ref{vertex sum})\nyields%\n\\begin{align}\nD(G,x)  &  =\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }%\n{\\displaystyle\\prod\\limits_{i\\in\\lambda_{G\\left[  W\\right]  }}}\n\\left(  1+(-x)^{i}\\right) \\nonumber\\\\\n&  =\\sum_{W\\subseteq V}%\n{\\displaystyle\\prod\\limits_{i\\in\\lambda_{G\\left[  W\\right]  }}}\n\\left(  x^{i}+(-1)^{i}\\right)  . \\label{moeb2}%\n\\end{align}\n\n\n\\begin{remark}\nIf we substitute $x=1$ (or $x=-1$) in Equation (\\ref{moeb2}) then all the\nproducts are equal to $0$ $(\\operatorname{mod}2)$. There is only one\nexception, namely the empty product corresponding to $W=\\emptyset$, which is\n1. This gives an alternative proof of Corollary \\ref{coro_odd}.\n\\end{remark}\n\nWe call a graph $G$ \\emph{conformal} if all of its components are of even\norder. Let $\\mathrm{Con}(G)$ be the set of all vertex-induced conformal\nsubgraphs of $G$ and let $k(G)$ be the number of components of $G$.\n\n\\begin{theorem}\n\\label{theo_con}The number of dominating sets of a graph $G$ satisfies%\n\\[\nd(G)=\\sum_{H\\in\\mathrm{Con}(G)}2^{k(H)}.\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nThe statement follows from Equation (\\ref{moeb2}) by substituting $x=1$. In\nthis case any component of odd order leads to a zero product, such that only\nconformal vertex-induced subgraphs count. Observe that the null graph is\nconformal and has no components, which produces the only odd term of the sum,\nnamely $2^{0}=1$.\n\\end{proof}\n\nEquation (\\ref{moeb2}) offers a possibility to derive a decomposition for the\ndomination polynomial.\n\n\\begin{theorem}\n\\label{theo_rec}Let $G=(V,E)$ be a graph and $v\\in V$. Then%\n\\[\nD(G,x)=D(G-v,x)+\\sum_{\\substack{\\left\\{  v\\right\\}  \\subseteq W\\subseteq\nV\\\\G\\left[  W\\right]  \\text{ is connected}}}\\left(  x^{\\left\\vert W\\right\\vert\n}+(-1)^{\\left\\vert W\\right\\vert }\\right)  D\\left(  G-N[W],x\\right)  .\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nWe start from Equation (\\ref{moeb2}):%\n\\begin{align*}\nD(G,x)  &  =\\sum_{W\\subseteq V}%\n{\\displaystyle\\prod\\limits_{i\\in\\lambda_{G\\left[  W\\right]  }}}\n\\left(  x^{i}+(-1)^{i}\\right) \\\\\n&  =\\sum_{W\\subseteq V\\setminus\\left\\{  v\\right\\}  }%\n{\\displaystyle\\prod\\limits_{i\\in\\lambda_{G\\left[  W\\right]  }}}\n\\left(  x^{i}+(-1)^{i}\\right)  +\\sum_{\\left\\{  v\\right\\}  \\subseteq W\\subseteq\nV}%\n{\\displaystyle\\prod\\limits_{i\\in\\lambda_{G\\left[  W\\right]  }}}\n\\left(  x^{i}+(-1)^{i}\\right) \\\\\n&  =D(G-v,x)\\\\\n&  +\\sum_{\\substack{\\left\\{  v\\right\\}  \\subseteq W\\subseteq V\\\\G\\left[\nW\\right]  \\text{ is connected}}}\\left(  x^{\\left\\vert W\\right\\vert\n}+(-1)^{\\left\\vert W\\right\\vert }\\right)  \\sum_{T\\subseteq V\\setminus N[W]}%\n{\\displaystyle\\prod\\limits_{i\\in\\lambda_{G\\left[  T\\right]  }}}\n\\left(  x^{i}+(-1)^{i}\\right) \\\\\n&  =D(G-v,x)+\\sum_{\\substack{\\left\\{  v\\right\\}  \\subseteq W\\subseteq\nV\\\\G\\left[  W\\right]  \\text{ is connected}}}\\left(  x^{\\left\\vert W\\right\\vert\n}+(-1)^{\\left\\vert W\\right\\vert }\\right)  D\\left(  G-N[W],x\\right)  .\n\\end{align*}\n\n\\end{proof}\n\nThe following statement for the number of dominating sets of $G$ is an\nimmediate consequence of Theorem \\ref{theo_rec}.\n\n\\begin{corollary}\nLet $G=(V,E)$ be a graph and $v\\in V$. Then the difference $d(G)-d(G-v)$ is even.\n\\end{corollary}\n\n\\begin{proof}\nWhen we substitute $x=1$ in Theorem \\ref{theo_rec}, then we obtain%\n\\[\nd(G)=d(G-v)+2\\sum_{\\substack{_{\\substack{\\left\\{  v\\right\\}  \\subseteq\nW\\subseteq V\\\\G\\left[  W\\right]  \\text{ is connected}}}\\\\\\left\\vert\nW\\right\\vert \\text{ is even}}}d\\left(  G-N[W]\\right)  ,\n\\]\nwhich gives the desired result.\n\\end{proof}\n\n\\section{Inclusion--Exclusion}\n\nWe obtain a different representation of the domination polynomial as a sum\nranging over vertex subsets by counting all vertex subsets of $G=(V,E)$ that\ndo not dominate the whole vertex set $V$ and applying inclusion-exclusion.\n\n\\begin{theorem}\n[\\cite{DT}]\\label{theo_inc_exc}Let $G=(V,E)$ be a graph. Then\n\\begin{equation}\nD(G,x)=\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }(1+x)^{\\left\\vert\nV\\setminus N\\left[  W\\right]  \\right\\vert }. \\label{incl-excl}%\n\\end{equation}\n\n\\end{theorem}\n\n\\begin{corollary}\n\\label{coro_binomial}The domination polynomial of a graph $G=(V,E)$ with $n$\nvertices satisfies%\n\\[\nD(G,x)=\\sum_{k=0}^{n}x^{k}\\sum_{\\substack{W\\subseteq V\\\\\\left\\vert N\\left[\nW\\right]  \\right\\vert \\leq n-k}}(-1)^{\\left\\vert W\\right\\vert }\\binom\n{n-\\left\\vert N\\left[  W\\right]  \\right\\vert }{k}.\n\\]\n\n\\end{corollary}\n\n\\begin{proof}\nUsing Equation (\\ref{incl-excl}), we obtain%\n\\begin{align*}\nD(G,x)  &  =\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }%\n(1+x)^{\\left\\vert V\\setminus N\\left[  W\\right]  \\right\\vert }\\\\\n&  =\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }\\sum_{k=0}^{\\left\\vert\nV-N\\left[  W\\right]  \\right\\vert }\\binom{n-\\left\\vert N\\left[  W\\right]\n\\right\\vert }{k}x^{k}\\\\\n&  =\\sum_{k=0}^{n}x^{k}\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert\n}\\binom{n-\\left\\vert N\\left[  W\\right]  \\right\\vert }{k}\\\\\n&  =\\sum_{k=0}^{n}x^{k}\\sum_{\\substack{W\\subseteq V\\\\\\left\\vert N\\left[\nW\\right]  \\right\\vert \\leq n-k}}(-1)^{\\left\\vert W\\right\\vert }\\binom\n{n-\\left\\vert N\\left[  W\\right]  \\right\\vert }{k}.\n\\end{align*}\n\n\\end{proof}\n\n\\begin{remark}\nAn interesting consequence of Corollary \\ref{coro_binomial} is the\ncharacterization of the domination number $\\gamma(G)$ of a graph $G=(V,E)$ as\nthe smallest nonnegative integer $k$ such that the sum%\n\\[\n\\sum_{\\substack{W\\subseteq V\\\\\\left\\vert N\\left[  W\\right]  \\right\\vert \\leq\nn-k}}(-1)^{\\left\\vert W\\right\\vert }\\binom{n-\\left\\vert N\\left[  W\\right]\n\\right\\vert }{k}%\n\\]\ndoes not vanish.\n\\end{remark}\n\nWe call a vertex subset $W\\subseteq V$ of a graph $G=(V,E)$ \\emph{essential}\nin $G$ if $W$ contains the closed neighborhood $N\\left[  v\\right]  $ of at\nleast one vertex $v\\in V$. We denote by $\\mathrm{Ess}(G)$ the family of all\nessential sets of $G$.\n\n\\begin{theorem}\n\\label{theo_neighborhood}Let $G=(V,E)$ be a graph with nonempty vertex set.\nThen the domination polynomial of $G$ satisfies%\n\\[\nD(G,x)=(-1)^{\\left\\vert V\\right\\vert }\\sum_{U\\in\\mathrm{Ess}(G)}%\n(-1)^{\\left\\vert U\\right\\vert }\\left[  (1+x)^{\\left\\vert \\left\\{  u\\in U\\mid\nN\\left[  u\\right]  \\subseteq U\\right\\}  \\right\\vert }-1\\right]  .\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nAccording to Equation (\\ref{incl-excl}), we have%\n\\begin{align*}\nD(G,x)  &  =\\sum_{W\\subseteq V}(-1)^{\\left\\vert W\\right\\vert }%\n(1+x)^{\\left\\vert V\\setminus N\\left[  W\\right]  \\right\\vert }\\\\\n&  =\\sum_{U\\subseteq V}(-1)^{\\left\\vert V\\right\\vert -\\left\\vert U\\right\\vert\n}(1+x)^{\\left\\vert V\\setminus N\\left[  V\\setminus U\\right]  \\right\\vert }\\\\\n&  =\\sum_{U\\subseteq V}(-1)^{\\left\\vert V\\right\\vert -\\left\\vert U\\right\\vert\n}(1+x)^{\\left\\vert \\left\\{  u\\in U\\mid N\\left[  u\\right]  \\subseteq U\\right\\}\n\\right\\vert }.\n\\end{align*}\nIn order to see the last equality, we verify%\n\\begin{align*}\nN\\left[  V\\setminus U\\right]   &  =\\bigcup\\limits_{v\\in V\\setminus U}N\\left[\nv\\right] \\\\\n&  =(V\\setminus U)\\cup\\left\\{  u\\in U\\mid N\\left[  u\\right]  \\cap(V\\setminus\nU)\\neq\\emptyset\\right\\} \\\\\n&  =(V\\setminus U)\\cup\\left\\{  u\\in U\\mid N\\left[  u\\right]  \\nsubseteq\nU\\right\\}\n\\end{align*}\nand consequently,%\n\\begin{align*}\nV\\setminus N\\left[  V\\setminus U\\right]   &  =V\\setminus\\left[  (V\\setminus\nU)\\cup\\left\\{  u\\in U\\mid N\\left[  u\\right]  \\nsubseteq U\\right\\}  \\right] \\\\\n&  =U\\setminus\\left\\{  u\\in U\\mid N\\left[  u\\right]  \\nsubseteq U\\right\\} \\\\\n&  =\\left\\{  u\\in U\\mid N\\left[  u\\right]  \\subseteq U\\right\\}  .\n\\end{align*}\n\n\nAll polynomials of the form $(1+x)^{\\left\\vert \\left\\{  u\\in U\\mid N\\left[\nu\\right]  \\subseteq U\\right\\}  \\right\\vert }$ have the constant term 1. As\n$V\\neq\\emptyset$, the constant term in%\n\\[\n\\sum_{U\\subseteq V}(-1)^{\\left\\vert V\\right\\vert -\\left\\vert U\\right\\vert\n}(1+x)^{\\left\\vert \\left\\{  u\\in U\\mid N\\left[  u\\right]  \\subseteq U\\right\\}\n\\right\\vert }%\n\\]\nvanishes, which gives%\n\\[\nD(G,x)=\\sum_{U\\subseteq V}(-1)^{\\left\\vert V\\right\\vert -\\left\\vert\nU\\right\\vert }\\left[  (1+x)^{\\left\\vert \\left\\{  u\\in U\\mid N\\left[  u\\right]\n\\subseteq U\\right\\}  \\right\\vert }-1\\right]  .\n\\]\nIf $U$ is a non-essential set of $G$ then we have $\\left\\{  u\\in U\\mid\nN\\left[  u\\right]  \\subseteq U\\right\\}  =\\emptyset$ and hence\n$(1+x)^{\\left\\vert \\left\\{  u\\in U\\mid N\\left[  u\\right]  \\subseteq U\\right\\}\n\\right\\vert }=1$. Consequently, all non-vanishing terms correspond to\nessential sets, yielding the statement of the theorem.\n\\end{proof}\n\nAnother interesting consequence of Theorem \\ref{theo_inc_exc} is the following\nrelation between $D(G,x)$ and $D\\left(  G,\\frac{1}{x}\\right)  $.\n\n\\begin{theorem}\n\\label{theo_reciprocal}Let $G=(V,E)$ be a graph. Then%\n\\[\nD(G,x)=(1+x)^{\\left\\vert V\\right\\vert }\\sum_{W\\subseteq V}\\left(  \\frac\n{-x}{1+x}\\right)  ^{\\left\\vert W\\right\\vert }D\\left(  G\\left[  W\\right]\n,\\frac{1}{x}\\right)  .\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nWe consider the right-hand side of the equation from the theorem. Substituting\n$D\\left(  G\\left[  W\\right]  ,\\frac{1}{x}\\right)  $ according to the\ndefinition of the domination polynomial yields%\n\\begin{align*}\n&  (1+x)^{\\left\\vert V\\right\\vert }\\sum_{W\\subseteq V}\\left(  \\frac{-x}%\n{1+x}\\right)  ^{\\left\\vert W\\right\\vert }\\sum_{\\substack{T\\subseteq\nW\\\\N_{G\\left[  W\\right]  }\\left[  T\\right]  =W}}x^{-\\left\\vert T\\right\\vert\n}\\\\\n&  =(1+x)^{\\left\\vert V\\right\\vert }\\sum_{W\\subseteq V}\\left(  \\frac{-x}%\n{1+x}\\right)  ^{\\left\\vert W\\right\\vert }\\sum_{T:T\\subseteq W\\subseteq\nN_{G}\\left[  T\\right]  }x^{-\\left\\vert T\\right\\vert }.\n\\end{align*}\nSwitching the order of summation gives%\n\\[\n(1+x)^{\\left\\vert V\\right\\vert }\\sum_{T\\subseteq V}x^{-\\left\\vert T\\right\\vert\n}\\sum_{W:T\\subseteq W\\subseteq N_{G}\\left[  T\\right]  }\\left(  \\frac{-x}%\n{1+x}\\right)  ^{\\left\\vert W\\right\\vert }.\n\\]\nNow we define $U=W\\setminus T$ and substitute $W=U\\cup T$, yielding%\n\\begin{align*}\n&  (1+x)^{\\left\\vert V\\right\\vert }\\sum_{T\\subseteq V}x^{-\\left\\vert\nT\\right\\vert }\\sum_{U\\subseteq N_{G}\\left[  T\\right]  \\setminus T}\\left(\n\\frac{-x}{1+x}\\right)  ^{\\left\\vert U\\right\\vert +\\left\\vert T\\right\\vert }\\\\\n&  =(1+x)^{\\left\\vert V\\right\\vert }\\sum_{T\\subseteq V}\\left(  \\frac{-1}%\n{1+x}\\right)  ^{\\left\\vert T\\right\\vert }\\sum_{U\\subseteq N_{G}\\left[\nT\\right]  \\setminus T}\\left(  \\frac{-x}{1+x}\\right)  ^{\\left\\vert U\\right\\vert\n},\n\\end{align*}\nwhich simplifies via the binomial theorem to%\n\\begin{align*}\n&  (1+x)^{\\left\\vert V\\right\\vert }\\sum_{T\\subseteq V}\\left(  \\frac{-1}%\n{1+x}\\right)  ^{\\left\\vert T\\right\\vert }\\left(  1-\\frac{x}{1+x}\\right)\n^{\\left\\vert N_{G}\\left[  T\\right]  \\right\\vert -\\left\\vert T\\right\\vert }\\\\\n&  =(1+x)^{\\left\\vert V\\right\\vert }\\sum_{T\\subseteq V}\\left(  -1\\right)\n^{\\left\\vert T\\right\\vert }\\left(  1+x\\right)  ^{-\\left\\vert N_{G}\\left[\nT\\right]  \\right\\vert }.\n\\end{align*}\nThe statement follows now by Theorem \\ref{theo_inc_exc}.\n\\end{proof}\n\nThe following statement can be shown by substituting $x=1$ in Theorem\n\\ref{theo_reciprocal}.\n\n\\begin{corollary}\nLet $G=(V,E)$ be a graph. The numbers of dominating sets of vertex-induced\nproper subgraphs of $G$ satisfy%\n\\[\n\\sum_{W\\subset V}(-1)^{\\left\\vert W\\right\\vert }\\frac{d(G[W])}{2^{\\left\\vert\nW\\right\\vert }}=0.\n\\]\n\n\\end{corollary}\n\n\\section{Conclusions and Open Problems}\n\nThe domination polynomial of a graph can be expressed as a sum of quite simple\npolynomials of vertex-induced or spanning subgraphs. In case of spanning\nsubgraphs, we can show that the domination polynomial depends only on\nbipartite spanning subgraphs.\n\nThere remain interesting open problems for further research in this field. The\nfirst one concerns the number of dominating sets of a graph given by Theorem\n\\ref{theo_con}.\n\n\\begin{problem}\nThe simple formula%\n\\[\nd(G)=\\sum_{H\\in\\mathrm{Con}(G)}2^{k(H)}%\n\\]\nsuggests that there is a bijection between subsets of components of conformal\ngraphs and dominating sets of $G$. Is there a bijective proof for Theorem\n\\ref{theo_con}? What is the best way to enumerate the set $\\mathrm{Con}(G)$?\n\\end{problem}\n\nIn Corollary \\ref{coro_type}, we showed that the type of a subgraph yields the\nessential information for a representation of $D(G,x)$ as a sum over\nvertex-induced subgraphs. Here it seems interesting to investigate whether we\nneed all vertex-induced subgraphs in order to derive the domination polynomial.\n\n\\begin{problem}\nComponents of $G\\left[  W\\right]  $ that have odd order lead to cancellation\nof terms of the sum in Equation (\\ref{moeb2}),%\n\\[\nD(G,x)=\\sum_{W\\subseteq V}%\n{\\displaystyle\\prod\\limits_{i\\in\\lambda_{G\\left[  W\\right]  }}}\n\\left(  x^{i}+(-1)^{i}\\right)  .\n\\]\nIs there a way to identify those cancelling terms?\n\\end{problem}\n\n\\begin{problem}\nIn Theorem \\ref{theo_neighborhood}, we showed that the restriction to\nessential sets is sufficient in order to compute the domination polynomial of\na graph. Can we reduce the number of terms needed to derive $D(G,x)$ further?\n\\end{problem}\n\nFurther topics of interest for future research include the investigation of\nspecial graph classes with respect to the given representations of the\ndomination polynomial and the application of these representations to special\ngraph classes. Since bipartite graphs play an important role for the\nrepresentation of the domination polynomial, we conjecture that also matchings\nhave a close relation to dominating sets. However, until now all attempts to\nfind a sum representation of $D(G,x)$ based on matchings of $G$ failed.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years, extensive research efforts have been devoted to studying cooperative networks where agents interact with each other over a topology repeatedly, by sharing information such as measurements, estimates, beliefs, or opinions. Such networks involve various levels of coordinated behavior among agents in order to solve important tasks in an efficient and distributed manner such as target tracking, object detection, resource allocation, learning, inference, and estimation. Collaboration among the agents via repeated information sharing is critical for the enhanced performance and robustness of the distributed solution, as already demonstrated in various insightful studies on social learning in multi-agent networks \\cite{KrishnamurthyA}-\\cite{Jadbabaie}, belief consensus in social networks \\cite{Chamley}\\cite{Acemouglu}, distributed optimization in resource allocation problems \\cite{Tsitsiklis}-\\cite{Dimakis} and in the diffusion of information for adaptation and learning purposes \\cite{Chen}-\\cite{Sayed}. However, in many scenarios, participating in the cooperative process entails costs to the agents, such as the cost of producing, transmitting, and sharing information with their neighbors. In these situations, the cost of sharing information may outweigh the benefit of cooperation and agents may not see an immediate benefit to being cooperative. For networks where agents are strategic, meaning that they aim to maximize their own utilities by strategically choosing their actions, the agents will choose to participate in the collaborative process only if they believe this action is beneficial to their current and long-term interests. Absent incentives for collaboration, these networks will work inefficiently or can even collapse \\cite{Lucky}. A distinct feature of the network under consideration is that agents' incentives can be coupled in a possibly extremely complex way due to the underlying topology. Thus, a key challenge to ensure the survivability and efficient operation of networks in the presence of selfish agents is the design of incentive schemes that adapt to the network topology and encourage the agents' cooperation in accordance with the network objective.\n\nWe propose to resolve the above incentive problem by exploiting the repeated interactions among agents to enable social reciprocation, by deploying a \\textit{distributed} rating protocol. Such rating protocols are designed and implemented in a distributed manner and are tailored to the underlying topologies. The rating protocol, via the (non-strategic) Social Network Interface (SNI)\\footnote{ For example, the SNIs are tamper-proof software\/hardware modules that can communicate with other SNIs in the neighborhood. However, they do not communicate with a central entity and hence, they are also distributed.} with which each agent is equipped, recommends (online and in a distributed way) to every agent how much information they should share with their neighbors depending on each neighbor's current rating according to the network topology. We refer to this recommendation as the \\textit{recommended strategy}. Importantly, the protocol has to be designed in such a way that this recommendation is incentive-compatible, meaning that agents have incentives to follow it. (We will later define a more formal version of ``incentive-compatibility''.)  In each period, agents have the freedom to decide how much information they should share with each of their neighbors. Their decision may comply or not with the strategy recommended (i.e. agents may follow or deviate from this strategy).  The agent's rating is then increased\/decreased by the SNI based on its current rating, and whether it has followed\/deviated from the recommended strategy. We refer to this as the \\textit{rating update} rule. High-rated agents will be rewarded -- the protocol recommends more information sharing by their neighbors and hence they receive more benefit in the future; low-rated agents will be punished -- the protocol recommends less information sharing by their neighbors and hence, they receive less benefit in the future.\n\nNext, we highlight two distinct features of the networks under consideration and the resulting key challenges for designing rating protocols for agents to cooperate. The first feature is that agents interact over an underlying topology. This is in stark contrast with existing works in repeated games relying on social reciprocation which assume that the agents are randomly matched \\cite{Kandori}\\cite{Zhang}\\cite{Xu}. In this paper, agents' incentives are coupled in a complex manner since their utilities depend on the behavior of the other agents with which they are interconnected. Since agents have different neighbors, their incentives can also be very diverse. A recommended strategy and rating update rule may provide sufficient incentives for some agents to follow but may fail in incentivizing others. In the worst cases, even a single agent deviating from the recommended strategy may cause a ``chain effect'' where eventually all agents deviate, leading to the collapse of the network. Hence, the rating protocol must be designed to adapt to the specific network topology.\n\nThe second feature is that the networks under consideration are distributed and hence, they are informationally decentralized, in the sense that (i) communication can only occur between neighboring agents (and SNIs) and (ii) there is no central planner that can monitor the entire network and communicate to the individual agents information about each agent's behavior (e.g. its compliance with the recommended strategy in the past, its rating etc.). Decentralization prevents rating protocols proposed in prior works \\cite{Zhang}\\cite{Xu} from being applicable in the considered scenarios since they are designed and implemented in a centralized manner. Therefore, a new distributed rating protocol needs to be developed which can operate successfully in an informationally-decentralized network.\n\nThe remaining part of this paper is organized as follows. In Section II, we review related works and existing solutions, and highlight the key differences to this work. Section III outlines the system model and formulates the protocol design problem. The structure of the rating protocol is unraveled in Section IV. In Section V, we design the optimal rating protocol to maximize the social welfare. The performance of the optimal design is then analyzed in Section VI. Section VII studies the rating protocol design in a class of time-varying topologies. Section VIII provides numerical results to highlight the features of the proposal. Finally, we conclude this paper in Section XI.\n\n\n\\section{Related Works}\n\n\\begin{table*}[t]\n\\centerline{\\includegraphics[scale = 1]{Table1.pdf}}\n\\caption{Comparison with existing works. }\n\\label{com_existing}\n\\vspace{-10pt}\n\\end{table*}\n\n\nCollaboration among the agents via repeated information sharing is critical for the enhanced performance and robustness of various types of social networks \\cite{KrishnamurthyA}-\\cite{Sayed}. The main focus of this literature is on determining the resulting network performance if agents repeatedly share and process information in various ways. However, absent incentives and in the presence of selfish agents, these networks will work inefficiently or can even collapse \\cite{Lucky}. Thus, the main focus of the current paper is how to incentivize strategic agents to share such information such that this type of social networks can operate efficiently.\n\nA variety of incentive schemes has been proposed to encourage cooperation among agents (see e.g. \\cite{Park} for a review of different game theoretic solutions). Two popular incentive schemes are pricing and differential service. Pricing schemes \\cite{Bergemann}\\cite{MacKie-Mason} use payments to reward and punish individuals for their behavior. However, they often require complex accounting and monitoring infrastructure, which introduce substantial communication and computation overhead. Differential service schemes, on the other hand, reward and punish individuals by providing differential services depending on their behavior. Differential services can be provided by the network operator. However, in many distributed information sharing networks, such a centralized network operator does not exist. Alternatively, differential services can also be provided by the other agents participating in the network since agents in the considered applications derive their utilities from their interactions with other agents \\cite{Axelrod}-\\cite{Jackson}\\cite{Zhang}\\cite{Xu}. Such incentive schemes are based on the principle of reciprocity and can be classified into direct (personal) reciprocation and social reciprocation. In direct (personal) reciprocation schemes (e.g. the widely adopted Tit-for-Tat strategy \\cite{Axelrod}-\\cite{Milan}), the behavior of an individual agent toward another is based on its personal experience with that agent. However, they only work when two interacting agents have common interests. In social reciprocation schemes \\cite{Song}-\\cite{Jackson}\\cite{Zhang}\\cite{Xu}, individual agents obtain some (public) information about other individuals (e.g. their ratings) and decide their behavior toward other agents based on this information.\n\nIncentive mechanisms based on social reciprocation are often studied using the familiar framework of repeated games. In \\cite{Song}, the information sharing game is studied in a narrower context of cooperative spectrum sensing and various simple strategies are investigated. Agents are assumed to be able to communicate and share information with all other agents, effectively forming a clique topology where the agents' knowledge of the network is complete and symmetric. However, such an assumption rarely holds in distributed networks where, instead, agents may interact over arbitrary topologies and have incomplete and asymmetric knowledge of the entire network. In such scenarios, simple strategies proposed in \\cite{Song} will fail to work and the incentives design becomes significantly more challenging.\n\nContagion strategies on networks \\cite{Kandori}-\\cite{Jackson} are proposed as a simple strategy to provide incentives for agents to cooperate. However, such strategies do not perform well if monitoring is imperfect since any single error can lead to a network collapse. Even if certain forms of forgiveness are introduced, contagion strategies are shown to be effective only in very specific topologies \\cite{Ali}\\cite{Jackson}. It is still extremely difficult, if not impossible, to design efficient forgiving schemes in distributed networks with arbitrary topologies since agents will have difficulty in conditioning their actions on history, e.g. whether they are in the contagion phase or the forgiving phase, due to the asymmetric and incomplete knowledge.\n\nRating\/reputation mechanisms are proposed as another promising solution to implement social reciprocation. Much of the existing work on reputation mechanism is concerned with practical implementation details such as effective information gathering techniques \\cite{Kamvar} or determining the impact of reputation on a seller's prices and sales \\cite{Ba}\\cite{Resnick}. The few works providing theoretical results on rating protocol design consider either one (or a few) long-lived agent(s) interacting with many short-lived agents \\cite{Dellarocas}-\\cite{Zacharia} or anonymous, homogeneous and unconnected agents selected to interact with each other using random matching \\cite{Kandori}\\cite{Zhang}\\cite{Xu}. Importantly, few of the prior works consider the design of such rating protocols for networks where agents interact over an underlying topology which leads to extremely complexly-coupled interactions among agents. Moreover, the distributed nature of the considered information sharing networks imposes unique challenges for the rating protocol design and implementation which are not addressed in prior works \\cite{Zhang}\\cite{Xu}.\n\nIn Table 1, we compare the current paper with existing works on social learning and incentive schemes based on direct reciprocation and social reciprocation.\n\n\n\n\\section{System Model}\nWe consider a network of $N$ agents, indexed by $\\{1,2,...,N\\} = {\\mathcal N}$. Agents are connected subject to an underlying topology $G=\\{ g_{ij} \\} _{i,j\\in {\\rm {\\mathcal N}}} $ with $g_{ij} =g_{ji} =1$ (here we consider undirected connection) representing agent $i$ and $j$ being connected (e.g. there is a communication channel between them) and $g_{ij} =g_{ji} =0$ otherwise. Moreover, we set $g_{ii} =0$. We say that agent $i$ and agent $j$ are neighbors if they are connected. For now we assume a fixed topology $G$ but certain types of time-varying topologies are allowed in our framework and this will be discussed in detail in Section VII.\n\nTime is divided into discrete periods. In each time period, each agent $i$ decides an information sharing action with respect to each of its neighbors $j$, denoted by $a_{ij} \\in [0,1]$. For example, $a_{ij} $ can represent the information sharing effort by agent $i$ with agent $j$. We collect the actions of agent $i$ with respect to all its neighbors in the notation ${\\boldsymbol{a}}_{i} =\\{ a_{ij} \\} _{j:g_{ij} =1} $. Denote ${\\boldsymbol{a}}=({\\boldsymbol{a}}_{1} ,...,{\\boldsymbol{a}}_{N} )$ as the action profile of all agents and ${\\boldsymbol{a}}_{-i} =({\\boldsymbol{a}}_{1} ,...,{\\boldsymbol{a}}_{i-1} ,{\\boldsymbol{a}}_{i+1} ,...,{\\boldsymbol{a}}_{N} )$ as the action profile of agents except $i$. Let ${\\rm {\\mathcal A}}_{i} =[0,1]^{m_{i} } $ be the action space of agent $i$ where $m_{i} =\\sum _{j}g_{ij}  $. Let ${\\rm {\\mathcal A}}=\\times _{i\\in {\\rm {\\mathcal N}}} {\\rm {\\mathcal A}}_{i} $ be the action space of all agents.\n\nAgents obtain benefits from neighbors' sharing actions. We denote the actions of agent $i$'s neighbors with respect to agent $i$ by $\\hat{{\\boldsymbol{a}}}_{i} {\\rm =}\\{ a_{ji} \\} _{j:g_{ij} =1} $ and let $b_{i} (\\hat{{\\boldsymbol{a}}}_{i} )$ be the benefit that agent $i$ obtains from its neighbors \\footnote{ In principle, an agent can obtain benefits from the information sharing over indirect links relayed by its neighbor. In this case, the action will also include the relaying action. }. Sharing information is costly and the cost $c_{i} ({\\boldsymbol{a}}_{i} )$ depends on an agent $i$'s  own actions ${\\boldsymbol{a}}_{i}$. Hence, given the action profile ${\\boldsymbol{a}}$ of all agents, the utility of agent $i$ is\n\\begin{equation} \\label{ZEqnNum432082}\nu_{i} ({\\boldsymbol{a}})=b_{i} (\\hat{{\\boldsymbol{a}}}_{i} )-c_{i} ({\\boldsymbol{a}}_{i} )\n\\end{equation}\n\nWe impose some constraints on the benefit and cost functions.\n\n\\textit{Assumption}: (1) For each $i$,  the benefit $b_{i} (\\hat{{\\boldsymbol{a}}}_{i} )$ is non-decreasing in each $a_{ij} ,\\forall j:g_{ij} =1$ and is concave in $\\hat{{\\boldsymbol{a}}}_{i} $ (in other words, jointly concave in $a_{ji} ,\\forall j:g_{ij} =1$). (2) For each $i$, the cost is linear in its sum actions, i.e. $c_{i} ({\\boldsymbol{a}}_{i} )=\\|{\\boldsymbol{a}}_{i}\\|_1 =\\sum _{j:g_{ij} } a_{ij} $.\n\nThe above assumption states that (1) agents receive decreasing marginal benefits of information acquisition, which captures the fact that agents become more or less ``satiated'' when they possess sufficient information, in the sense that additional information would only generate little additional payoff; (2) the cost incurred by an agent is equal (or proportional) to the sum effort of collaboration with all its neighbors.\n\n\n\\subsection{Example: Cooperative Estimation}\nWe illustrate the generality of our formalism by showing how well-studied cooperative estimation problems \\cite{Mishra}\\cite{Unnikrishnan} can be cast into it. Consider that each agent observes in each period a noisy version of a time-varying underlying system parameter $s(t)$ of interest. Denote the observation of agent $i$ by $o_{i} (t)$. We assume that $o_{i} (t)=s(t)+ \\epsilon_i(t)$, where the observation error $\\epsilon_{i} (t)$ is i.i.d. Gaussian across agents and time with mean zero and variance $r^{2} $. Agents can exchange observations with their neighbors to obtain better estimations of the system parameter. Let $a_{ij} (t)$ be the transmission power spent by agent $i$. The higher the transmission power the larger probability that agent $j$ receives this additional observation from agent $i$. Agents can use various combination rules \\cite{Chen} to obtain the final estimations. The expected mean square error (MSE) of agent $i$'s final estimation will depend on the actions of its neighbors, denoted by $MSE_{i} (\\hat{{\\boldsymbol{a}}}_{i} (t))$. If we define the MSE improvement as the benefit of agents, i.e. $b_{i} (\\hat{{\\boldsymbol{a}}}_{i} (t))=r^{2} -MSE(\\hat{{\\boldsymbol{a}}}_{i} (t))$, then the utility of agent $i$ in period $t$ given the received benefit and its incurred cost is $u_{i} ({\\boldsymbol{a}}(t))=r^{2} -MSE_{i} (\\hat{{\\boldsymbol{a}}}_{i} (t))-{\\boldsymbol{a}}_{i} (t)$.\n\n\n\n\\subsection{Obedient Agents -- Benchmark}\nEven though this paper focuses on strategic agents in information sharing networks, it is useful to first study how obedient agents (i.e. non-strategic agents who follow any prescribed strategy) interact in order to obtain a better understanding of the interactions and the achievable performance. The objective of the protocol designer in this benchmark case is to maximize the social welfare of the network, which is defined as the time-average sum utility of all agents, i.e.\n\\begin{equation} \\label{ZEqnNum110798}\nV=\\mathop{\\lim }\\limits_{T\\to \\infty } \\frac{1}{T} \\sum _{t=0}^{\\infty }\\sum _{i}u_{i}   ({\\boldsymbol{a}}(t))\n\\end{equation}\nwhere ${\\boldsymbol{a}}(t)$ is the action profile in period $t$. If agents are obedient, then the system designer can assign socially optimal actions, denoted by ${\\boldsymbol{a}}^{opt} (t),\\forall t$, to agents and then agents will simply take the actions prescribed by the system designer. Determining the socially optimal actions involves solving the following utility maximization problem \\cite{Palomar}:\n\\begin{equation} \\label{ZEqnNum918213}\n\\begin{array}{l} {\\mathop{{\\rm maximize}}\\limits_{a} {\\rm \\; \\; \\; \\; \\; \\; \\; \\; }V} \\\\ {{\\rm subject\\; to\\; \\; \\; \\; \\; \\; \\; \\; }a_{ij} (t)\\in [0,1],\\forall i,j:g_{ij} =1,\\forall t} \\end{array}\n\\end{equation}\nThis problem can be easily solved and any action profile ${\\boldsymbol{a}}^{opt}$ that satisfies\n\\begin{equation} \\label{ZEqnNum389088}\n\\hat{{\\boldsymbol{a}}}_{i}^{opt} (t)\\in \\arg \\max _{\\hat{{\\boldsymbol{a}}}} b_{i} (\\hat{{\\boldsymbol{a}}}_{i} (t))-\\hat{{\\boldsymbol{a}}}_{i} (t)\n\\end{equation}\nis its solution. We denote the optimal social welfare by $V^{opt} $.\n\nIn a distributed network, there is no central planner that knows everything about the network (including the network size, topology and individual agents' utility functions) and can communicate to all agents. However, the structure of problem \\eqref{ZEqnNum918213} lends itself to a fully decentralized implementation \\cite{Rockafellar}: each SNI can compute the optimal actions for its neighbors by solving \\eqref{ZEqnNum389088} and sending the solution to their neighboring SNIs. In this way, if all agents take the actions solved by the SNIs, the social welfare is maximized.\n\nIt is helpful to give an illustrative example of the optimal information sharing actions for agents connected using different topologies. We will revisit this example when we study strategic agents and show how incentives design and information sharing strategies are affected by the topologies.\n\n\\textit{Example}: (Ring and Star topologies) We consider a set of 4 agents performing cooperative estimation (as in Section III. A) over two fixed topologies -- a ring and a star. A possible approximation of the utility function of each agent $i$ when the uniform combination rule is used is $u_{i} ({\\boldsymbol{a}}(t))=r^{2} -\\frac{r^{2} }{1+\\sum _{j:g_{ij} } a_{ji} } -\\sum _{j:g_{ij} } a_{ij} $. We assume that the noise variance $r^{2} =4$. Figure \\ref{ring-star1} illustrates the optimal actions in different topologies by solving \\eqref{ZEqnNum918213}. In both topologies, the optimal social welfare is $V^{opt} =4$.\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale = 0.7]{ring_star1.pdf}}\n\\caption{Optimal strategies for obedient agents interacting over a ring and a star.}\\label{ring-star1}\n\\end{figure}\n\n\\subsection{Strategic Agents}\n\nThe information sharing problem becomes much more difficult in the presence of strategic agents: strategic agents may not want to take the prescribed actions because they do not maximize their own utilities. We formally define the network information sharing games below.\n\n\\textit{Definition 1}: A (one-shot) \\textit{network information sharing game} is a tuple ${\\rm {\\mathcal G}}=\\left\\langle {\\rm {\\mathcal N}},{\\rm {\\mathcal A}},\\{ u_{i} (\\cdot )\\} _{i\\in {\\rm {\\mathcal N}}} ;G\\right\\rangle $ where ${\\rm {\\mathcal N}}$ is the set of players, ${\\rm {\\mathcal A}}$ is the action space of all players, $u_{i} (\\cdot )$ is the utility function of player $i$ (defined by \\eqref{ZEqnNum432082}) and $G$ is the underlying topology.\n\n\\begin{theorem}\nThere exists a unique Nash equilibrium (NE) ${\\boldsymbol{a}}^{NE} =0$ in the network information sharing game in any period.\n\\end{theorem}\n\\begin{proof}\nConsider the utility of an agent $i$ in \\eqref{ZEqnNum432082}, the dominant\nstrategy of agent $i$ is ${\\boldsymbol{a}}_{i} =0$ regardless of other agents' actions ${\\boldsymbol{a}}_{-i} $. Therefore, the only NE is ${\\boldsymbol{a}}_{i} =0,\\forall i$.\n\\end{proof}\n\n\n\nWe now proceed to show how to build incentives for agents to share information with each other by exploiting their repeated interactions. In the repeated game, the (one-shot) network information sharing game is played in every period $t=0,1,2,...$. Let $y_{i}^{t} \\in Y$ be the public monitoring signal related to agent $i$'s actions ${\\boldsymbol{a}}_{i} (t)$ at time $t$. A public history of length $t$ is a sequence of public signals $(y^{0} ,y^{1} ,...,y^{t-1} )\\in Y^{t} $. We note that in the considered network setting, public signals are ``locally public'' in the sense that agents only observe the public signals within their own neighborhood but not all public signals. For example, a public signal $y_{i}^{t} $ can indicate whether or not agent $i$ followed the strategy at time $t$ and only the neighbors of agent $i$ observe it.  We write ${\\rm {\\mathcal H}}(t)$ for the set of public histories of length $t$, ${\\rm {\\mathcal H}}^{T} =\\bigcup _{t=0}^{T} {\\rm {\\mathcal H}}(t)$ for the set of public histories of length at most $T$ and ${\\rm {\\mathcal H}}=\\bigcup _{t=0}^{\\infty } {\\rm {\\mathcal H}}(t)$ for the set of all public histories of all finite lengths. A public strategy of agent $i$ is a mapping from public histories (in fact, only those public signals $\\{ y_{j}^{t} {\\rm \\} }_{j:g_{ij} =1} $ that agent $i$ can observe) to $i$'s pure actions ${\\it \\bm\\sigma }_{i} :{\\rm {\\mathcal H}}\\to {\\rm {\\mathcal A}}_{i} $. We write ${\\bm\\sigma }$ as the collection of public strategies for all agents. Let $\\delta \\in (0,1]$ be the discount factor of agents. Since interactions are on-going, agents care about their long-term utilities. The long-term utility  for an agent $i$ is defined as follows:\n\\begin{equation} \\label{5)}\nU_{i} (t)=u_{i} ({\\boldsymbol{a}}(t))+\\delta u_{i} ({\\boldsymbol{a}}(t+1))+\\delta ^{2} u_{i} ({\\boldsymbol{a}}(t+2))+...\n\\end{equation}\nA public strategy profile ${\\bm\\sigma }$ induces a probability distribution over public histories and hence over\\textit{ ex ante }utilities. We abuse notation and write $U_{i} ({\\bm \\sigma };h)$ for the expected long-run average \\textit{ex ante} utility of agent $i$ when agents follow the strategy profile ${\\bm \\sigma }$ after the public history $h\\in {\\rm {\\mathcal H}}$.\n\n\\textit{Definition 1}: (Perfect Public Equilibrium) A strategy profile ${\\bm \\sigma }$ is a perfect public equilibrium  if $\\forall h\\in {\\mathcal H}$,$\\forall i$, $U_{i} ({\\it \\sigma }_{i} ,{\\it \\sigma }_{-i} ;h)\\ge U_{i} ({\\it \\sigma }_{i} ',{\\it \\sigma }_{-i} ;h),\\forall {\\it \\sigma }_{i} '\\ne {\\it \\sigma }_{i} $.\n\nIn the above formulation, we restrict agents to use public strategies and assume that agents make no use of any information other than provided by the (local) public signal (See Figure \\ref{localpublicsignal}); in particular, agents make no use of their private history (i.e. the history sequence of its own actions ${\\boldsymbol{a}}_{i} (t)$, its own utilities $u_{i} (t)$ and its neighbors' action toward it $\\hat{{\\boldsymbol{a}}}_{i} (t)$). This assumption admits a number of possible interpretations \\cite{Mailath}, each of which is appropriate in some circumstances. In the considered scenarios where agents interact over a topology, the most important reason why we consider the design of public strategies and PPE is due to agents' partial observations and asymmetric knowledge of the network. In particular, since agent  $i$ only observes its own neighborhood subject to the underlying topology, it cannot distinguish based solely on its private history between the case in which its neighbor is deviating from the recommended strategy and the case in which its neighbor is following the recommended strategy and punishing its own neighbors' deviation actions. Using (local) public histories is more practical in the considered scenarios since it allows agents to have common knowledge within each neighborhood. The proposed rating protocols go one step further in reducing the implementation complexity by associating each agent with a rating that summarizes the public history of that agent. In this way, the space of public histories is reduced to a finite set and hence, much simpler strategies can be constructed which can still achieve the optimal social welfare.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale = 0.8]{localpublicsignal.pdf}}\n\\caption{Illustration of local public signals. (Agents observe only public signals generated by the SNIs in their neighborhood)}\\label{localpublicsignal}\n\\vspace{-5pt}\n\\end{figure}\n\n\\section{Proposed Rating Protocols}\nIn this section, we describe the proposed distributed rating protocol and its operation in a distributed network. As mentioned in the Introduction, each agent is equipped with an SNI. These SNIs are non-strategic software\/hardware components available to the agents and will assist in the distributed design and implementation of the rating protocol. Importantly though, note that the agents \\textit{are strategic} in choosing the information sharing actions (i.e. they will selfishly decide whether or not to follow the strategy recommended by the SNIs) such that their own utilities are maximized.\n\n\n\\subsection{Considered Rating Protocol}\nA rating protocol, which is designed and implemented by the SNIs, consists of three components -- a set of ratings, a set of recommended strategies to agents, and a rating update rule.\n\n\n\\begin{enumerate}\n\\item  We consider a set of $K$ ordered ratings $\\Theta =\\{ 1,2,...,K\\} $ with $1$ being the lowest and $K$ being the highest rating. Denote agent $i$'s rating in period $t$ by $\\theta _{i} (t)\\in \\Theta $ and agent $i$'s neighbors' ratings by $\\hat{{\\bm \\theta }}_{i} =\\{ \\theta _{j} \\} _{j:g_{ij} =1} $. $K$ serves as an upper bound of the rating set size and is predetermined before the system operates.\n\n\\item  The SNIs determine the recommended (public) strategy in a distributed manner and recommend actions to their own agent depending on neighbors' ratings $\\bm\\sigma :{\\rm {\\mathcal N}}\\times {\\rm {\\mathcal N}}\\times \\Theta \\to [0,1]$, where $\\sigma _{ij} (\\theta _{j} )$ represents the recommended sharing action of agent $i$ with respect to agent $j$ if agent $j$'s rating is $\\theta _{j} $. Since it is reasonable that high-rated agents should be rewarded while low-rated agents should be punished, the recommended strategy should satisfy that $\\sigma _{ij} (\\theta )\\le \\sigma _{ij} (\\theta ')$ if $\\theta <\\theta '$. We collect the strategies of agent $i$ to all its neighbors in ${\\bm \\sigma }_{i} (\\hat{{\\bm \\theta }}_{i} )=\\{ \\sigma _{ij} (\\theta _{j} )\\} _{j:g_{ij} =1} $ and the strategies of agent $i$'s neighbors to itself in $\\hat{{\\bm \\sigma }}_{i} (\\theta _{i} )=\\{ \\sigma _{ji} (\\theta _{i} )\\} _{j:g_{ij} =1} $.\n\n\\item  Depending on whether an agent $i$ followed or not the recommended strategy, the SNI of agent $i$ updates agent $i$'s rating at the end of each period.  Let $y_{i} \\in Y=[0,1]$  be the monitoring signal with respect to agent $i$. Specifically, $y_{i} =1$ if ${\\boldsymbol{a}}_{i} =\\bm\\sigma _{i} $ and $y_{i} =0$ if ${\\boldsymbol{a}}_{i} \\ne \\bm\\sigma _{i} $. The rating update rule is therefore a mapping $\\tau :{\\rm {\\mathcal N}}\\times \\Theta \\times Y\\to \\Delta (\\Theta )$, where $\\tau _{i} (\\theta _{i}^{+} ;\\theta _{i} ,y_{i} )$ is the probability that the updated rating is $\\theta _{i}^{+} $ if agent $i$'s current rating is $\\theta _{i} $ and the public signal is $y_{i} $. In particular, we consider the following parameterized rating update rule (see also Figure \\ref{ratingupdate}), for agent $i$, if $\\theta _{i} =k$,\n\\begin{equation} \\label{6)}\n\\tau _{i} (\\theta _{i}^{+} ;\\theta _{i} ,y){\\rm =}\n\\left\\{\\begin{array}{l}\n{\\alpha _{i,k} ,{\\rm \\; \\; \\; \\;if\\; \\; }\\theta _{i}^{+} =\\max \\{ 1,k-1\\} ,y_{i} =0} \\\\ {1-\\alpha _{i,k} ,{\\rm \\; \\; \\; \\; if\\; \\; }\\theta _{i}^{+} =k,y_{i} =0} \\\\ {\\beta _{i,k} ,{\\rm \\; \\; \\; \\;  if\\; \\; }\\theta _{i}^{+} =\\min \\{ K,k+1\\} ,y_{i} =1} \\\\ {1-\\beta _{i,k} ,{\\rm \\; \\; \\; \\; if\\; \\; }\\theta _{i}^{+} =k,y_{i} =1}\n\\end{array}\\right.\n\\end{equation}\n\n    In words, compliant agents are rewarded to receive a higher rating with some probability while deviating agents are punished to receive a lower rating with some (other) probability. These probabilities $\\alpha _{i,k} ,\\beta _{i,k} $ are chosen from $[0,1]$. Note that when $\\alpha _{i,k} =0$, the rating label set of agent $i$ effectively reduces to a subset $\\{ k,k+1,...,K\\} $ since its rating will never drop below $k$ (if its initial rating is higher than $k$). Note also that agents remain at the highest rating $\\theta =K$ if they always follow the recommended strategy regardless of the choice of $\\beta _{i,K} $.\n\\end{enumerate}\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale = 0.9]{ratingupdate.pdf}}\n\\caption{Rating update rule.}\\label{ratingupdate}\n\\vspace{-15pt}\n\\end{figure}\n\n\nMonitoring may not be perfect in implementation and hence it is possible that even if $a_{i} =\\sigma _{i} $, it can still be $y_{i} =0$ (and if $a_{i} \\ne \\sigma _{i} $, $y_{i} =1$). If monitoring is perfect, then the strongest punishment (i.e. the agent receives the lowest rating forever once a deviation is detected) will provide the strongest incentives for agents to cooperate. However, in the imperfect monitoring environment, such punishment will lead to the network collapse where no agents share information with others. Hence, when designing the rating update rule, the monitoring errors should also be taken into account.\n\n\n\nTo sum up, the rating protocol is uniquely determined by the recommended (public) strategies ${\\bm \\sigma }_{i} (\\hat{{\\bm \\theta }}_{i} ),\\forall i,\\forall \\hat{{\\bm \\theta }}_{i}$ and the rating update probabilities $\\alpha _{i,k} ,\\beta _{i,k}, \\forall i,\\forall k$. We denote the rating protocol by $\\pi =(\\Theta ,{\\bm \\sigma },{\\bm \\alpha },{\\bm \\beta })$. Different rating protocols lead to different social welfare. Denote the achievable social welfare by adopting the rating protocol by $V(\\pi )$. The rating protocol design problem thus is\n\\begin{equation}\n\\begin{array}{l}\n{\\mathop{{\\rm maximize}}\\limits_{\\pi =(\\Theta ,\\bm\\sigma, \\bm\\alpha, \\bm\\beta )} {\\rm \\; \\; \\; \\; \\; \\; \\; \\; }V(\\pi )} \\\\ {{\\rm \\;\\;subject\\; to\\; \\; \\; \\; \\; \\; \\; \\; \\;\\;}\\bm\\sigma {\\rm \\; constitutes\\; a\\; PPE}} \\end{array}\\label{ZEqnNum479148}\n\\end{equation}\n\n\\subsection{Operation of the Rating Protocol}\nThe operation of the rating protocol comprises two phases: the design phase and the implementation phase. In the design phase, the SNIs determine in a distributed way the recommended strategy and rating update rules according to the network topology, and the agents do nothing except being informed of the instantiated rating protocol. In the implementation phase (run-time), the agents (freely and selfishly) choose their actions in each information sharing period in order to maximize their own utilities (i.e. they can freely decide whether to follow or not the recommended strategies). Depending on whether the agents are following or deviating from the recommended strategy, each SNI executes the rating update of its agent and sends the new ratings of its agent to the neighboring SNIs. Note that if the rating protocol constitutes a PPE, then the agents will follow the recommended strategy in any period. When the network topology is static, the rating protocol goes through the design phase only once, when the network becomes operational, and then enters the implementation phase. When the network topology is dynamic, the rating protocol re-enters the design phase periodically, to adapt to the varying topology. However, both the design and implementation have to be carried out in a distributed way in the informationally decentralized environment. Table 2 summarizes the available information and actions of the agents and SNIs in both the design and implementation phases.\n\n\n\\begin{table*}[t]\n\\centerline{\\includegraphics[scale = 0.9]{Table2.pdf}}\n\\caption{Operation of the rating protocol. }\n\\label{com_existing}\n\\vspace{-20pt}\n\\end{table*}\n\n\n\\section{Distributed Optimal Rating Protocol Design}\nIf a rating protocol constitutes a PPE, then all agents will find it in their self-interest to follow the recommended strategies. If the rating update rule updates compliant agents' to a higher rating with positive probabilities, then eventually all agents will have the highest ratings forever (assuming no update errors). Therefore, the social welfare, which is the time-average sum utilities, is asymptotically the same as the sum utilities of all agents when they have the highest ratings and follow the recommended strategy, i.e.\n\\begin{equation} \\label{ZEqnNum559005}\nV=\\sum _{i}(b_{i} (\\hat{{\\bm \\sigma }}_{i} (K))-{\\bm \\sigma }_{i} ({\\boldsymbol{K}}))\n\\end{equation}\nThis means that the recommended strategies for the highest ratings determine the social welfare that can be achieved by the rating protocol. If these strategies can be arbitrarily chosen, then we can solve a similar problem as \\eqref{ZEqnNum918213} for the obedient agent case. However, in the presence of self-interested agents, these strategies, together with the other components of a rating protocol, need to satisfy the equilibrium constraint such that self-interested agents have incentives to follow the recommended strategies. In Theorem 2, we identify a sufficient and necessary condition on ${\\bm \\sigma }({\\boldsymbol{K}})$ (i.e. the recommended strategies when agents have the highest ratings) such that an equilibrium rating protocol can be constructed. With this, the SNIs are able to determine the optimal rating protocol in a distributed way in order to maximize the social welfare. We denote the social welfare that can be achieved by the optimal rating protocol as $V^{*} $ and use \\textit{the price of anarchy} (PoA)\\footnote{ We can also use the price of stability (PoS) as the performance measure. However, since there is a unique equilibrium given the specific rating protocol, these two measures are equivalent. }, defined as $PoA=V^{opt} \/V^{*} $, as the performance measure of the rating protocol.\n\n\n\n\n\\subsection{Sufficient and Necessary Condition}\nTo see whether a rating protocol can constitute a PPE, it suffices to check whether agents can improve their long-term utilities by one-shot unilateral deviation from the recommended strategy after any history (according to the one-shot deviation principle in repeated game theory \\cite{Mailath}). Since in the rating protocol, the history is summarized by the ratings, this reduces to checking the long-term utility in any state (any rating profile ${\\bm \\theta }$ of agents). Agent $i$'s long-term utility when agents choose the action profile ${\\boldsymbol{a}}$ is\n\\begin{equation} \\label{9)}\nU_{i} ({\\bm \\theta },{\\boldsymbol{a}})=u_{i} ({\\bm \\theta },{\\boldsymbol{a}})+\\delta \\sum _{\\bm\\theta '} p({\\bm \\theta }'|{\\bm \\theta },{\\boldsymbol{a}})U_{i}^{*} ({\\bm \\theta }'),\n\\end{equation}\nwhere $p({\\bm \\theta }'|{\\bm \\theta },{\\boldsymbol{a}})$ is the rating profile transition probability which can be fully determined by the rating update rule based on agents' actions and $U_{i}^{*} ({\\bm \\theta }')$ is the optimal value of agent $i$ at the rating profile ${\\bm \\theta }'$, i.e. $U_{i}^{*} ({\\bm \\theta }') = \\max\\limits_{{\\boldsymbol{a}}_{i} } U_{i} ({\\bm \\theta },{\\boldsymbol{a}})$. PPE requires that the recommended actions for any rating profile are the optimal actions that maximize agents' long-term utilities. Before we proceed to the proof of Theorem 2, we prove the following Lemma, whose proof is deferred to online appendix \\cite{onlineappendix} due to space limitation.\n\n\\smallskip\n{\\bf Lemma} (1) $\\forall {\\bm \\theta }$, the optimal action of agent $i$ is either ${\\boldsymbol{a}}_{i}^{*} ({\\bm \\theta })={\\bf 0}$ or ${\\boldsymbol{a}}_{i}^{*} ({\\bm \\theta })={\\bm \\sigma }_{i} (\\hat{{\\bm \\theta }}_{i} )$.\n\n(2) $\\forall \\theta _{i} $, if for $\\hat{{\\bm \\theta }}_{i} ={\\boldsymbol{K}}$, ${\\boldsymbol{a}}_{i}^{*} ({\\bm \\theta })={\\bm \\sigma }_{i} (\\hat{{\\bm \\theta }}_{i} )$, then for any other $\\hat{{\\bm \\theta }}_{i} $, ${\\boldsymbol{a}}_{i}^{*} ({\\bm \\theta })={\\bm \\sigma }_{i} (\\hat{{\\bm \\theta }}_{i} )$.\n\n(3) Let $\\hat{{\\bm \\theta }}_{i} ={\\boldsymbol{K}}$, suppose $\\forall \\theta _{i} $, ${\\boldsymbol{a}}_{i}^{*} ({\\bm \\theta })={\\bm \\sigma }_{i} (\\hat{{\\bm \\theta }}_{i} )$, then $\\theta _{i} <\\theta '_{i} {\\rm \\; \\; \\; }\\Leftrightarrow {\\rm \\; \\; }U^*_{i} (\\theta _{i} ,\\hat{{\\bm \\theta }}_{i} )\\le U^*_{i} (\\theta '_{i} ,\\hat{{\\bm \\theta }}_{i} )$\n\\smallskip\n\nLemma (1) characterizes the set of possible optimal actions. That is, self-interested agents choose to either share nothing with their agents or share the recommended amount of information with their neighbors.  Lemma (2) states that if an agent has incentives to follow the recommended strategy when all its neighbors have the highest ratings, then it will also have incentives to follow the recommended strategy in all other cases. Lemma (3) shows that the optimal long-term utility of an agent is monotonic in its ratings when all its neighbors have the highest rating -- the higher the rating the larger the long-term utility the agent obtains. With these results in hand, we are ready to present and prove Theorem 2.\n\n\\begin{theorem}\nGiven the rating protocol structure and the network structure (topology and individual utility functions), there exists at least one PPE (of the rating protocol) if and only if $\\delta b_{i} (\\hat{{\\bm \\sigma }}_{i} (K))\\ge c_i({\\bm \\sigma }_{i} ({\\boldsymbol{K}})),\\forall i$.\n\\end{theorem}\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\n\n\\subsection{Computing the Recommended Strategy}\nTheorem 2 provides a sufficient and necessary condition for the existence of a PPE with respect to the recommended strategies when agents have the highest ratings. From \\eqref{ZEqnNum559005} we already know that these strategies fully determine the social welfare that can be achieved by the rating protocol. Therefore, the optimal values of ${\\bm \\sigma }({\\boldsymbol{K}})$ can be determined by solving the following \\textit{optimal recommended strategy design} problem:\n\\begin{equation} \\label{ZEqnNum960030}\n\\begin{array}{l} {\\mathop{{\\rm maximize}}\\limits_{{\\bm \\sigma }} {\\rm \\; \\; \\; \\; \\; \\; \\; \\; }\\sum _{i}(b_{i} (\\hat{{\\bm \\sigma }}_{i} (K))-c_i({\\bm \\sigma }_{i} ({\\boldsymbol{K}}))) } \\\\ {{\\rm subject\\; to\\; \\; \\; \\; \\; \\; \\; \\; }c_i({\\bm \\sigma }_{i} ({\\boldsymbol{K}}))\\le \\delta b_{i} (\\hat{{\\bm \\sigma }}_{i} (K)),\\forall i} \\end{array}\n\\end{equation}\nwhere the constraint ensures that an equilibrium rating protocol can be constructed. Note that this problem implicitly depends on the network topology since both $\\hat{{\\bm \\sigma }}_{i} (K)$ and ${\\bm \\sigma }_{i} ({\\boldsymbol{K}}),\\forall i$ are topology-dependent (since for each agent $i$, the strategy is only with respect to its neighbors). In this subsection, we will write ${\\bm \\sigma }_{i} ({\\boldsymbol{K}})$ as ${\\bm \\sigma }_{i} $ and $\\hat{{\\bm \\sigma }}_{i} (K)$ as $\\hat{{\\bm \\sigma }}_{i} $ to keep the notation simple.\n\nNow, we propose a distributed algorithm to compute these recommended strategies using dual decomposition and Lagrangian relaxation. The Optimal Recommended Strategy Design problem \\eqref{ZEqnNum960030} is decomposed into $N$ sub-problems each of which is solved locally by the SNIs. Note that unlike the case with obedient agents, these sub-problems have coupled constraints. Therefore, SNIs will need to go through an iterative process to exchange messages (the Lagrangians) with their neighboring SNIs such that their local solutions converge to the global optimal solution. We perform dual decomposition on \\eqref{ZEqnNum960030} and relax the constraints as follows\n\\begin{equation} \\label{ZEqnNum321261}\n\\mathop{{\\rm maximize}}\\limits_{{\\bm \\sigma }} {\\rm \\;  \\; \\; \\; }\\sum _{i}(b_{i} (\\hat{{\\bm \\sigma }}_{i} )-\\|{\\bm \\sigma }_{i}\\| ) -\\sum _{i}\\lambda _{i}  (\\|{\\bm \\sigma }_{i}\\| -\\delta b_{i} (\\hat{{\\bm \\sigma }}_{i} ))\n\\end{equation}\nwhere $\\lambda _{i} \\ge 0,\\forall i$ are the Lagrangian multiplexers. The optimization thus separates into two levels of optimization. At the lower level, we have the sub-problems (one for each agent), $\\forall i$\n\\begin{equation} \\label{ZEqnNum852632}\n\\mathop{{\\rm maximize}}\\limits_{\\hat{{\\bm \\sigma }}_{i} } {\\rm \\; \\; \\; \\; \\; \\; \\; \\; }(1+\\lambda _{i} \\delta )b_{i} (\\hat{{\\bm \\sigma }}_{i} )-\\sum _{j:g_{ij} =1} (1+\\lambda _{j} )\\sigma _{ji}\n\\end{equation}\nIt is easy to see that the optimal solution of these subproblems is also the optimal solution of the relaxed problem \\eqref{ZEqnNum321261}. At the higher level, the master dual problem is in charge of updating the dual variables,\n\\begin{equation} \\label{13)}\n\\begin{array}{l} {\\mathop{{\\rm minimize}}\\limits_{{\\bm \\lambda }} {\\rm \\; \\; \\; \\; \\; \\; \\; \\; }g({\\bm \\lambda })=\\sum _{i}g_{i}  ({\\bm \\lambda })} \\\\ {{\\rm subject\\; to\\; \\; \\; \\; \\; \\; \\; \\; }\\lambda _{i} \\ge 0,\\forall i} \\end{array}\n\\end{equation}\nwhere $g_{i} ({\\bm \\lambda })$ is the maximum value of the Lagrangian \\eqref{ZEqnNum852632} given ${\\bm \\lambda }$ and $g({\\bm \\lambda })$ is the maximum value of the Lagrangian \\eqref{ZEqnNum321261} of the primal problem. The following subgradient method is used to update ${\\bm \\lambda }$,\n\\begin{equation} \\label{ZEqnNum330405}\n\\lambda _{i} (q+1)=\\left[\\lambda _{i} (q)+w({\\bm \\sigma }_{i} -\\delta b_{i} (\\hat{{\\bm \\sigma }}_{i} ))\\right]^{+} ,\\forall i\n\\end{equation}\nwhere $q$ is the iteration index, $w>0$ is a sufficiently small positive step-size. Because \\eqref{ZEqnNum960030} is a convex optimization, such an iterative algorithm will converge \\cite{Boyd} to the dual optimal ${\\bm \\lambda }^{*} $ as $q\\to \\infty $ and the primal variable ${\\bm \\sigma }^{*} ({\\bm \\lambda }(q))$ will also converge to the primal optimal ${\\bm \\sigma }^{*} $.\n\nThis iterative process can be made fully distributed which requires only limited message exchange between neighboring SNIs. We present the Distributed Computation of the Recommended Strategy (DCRS) Algorithm below which is run locally by each SNI of the agents.\n\n\n\\bigskip\n\\noindent\n\\begin{tabular}{p{6in}} \\hline\n\\textbf{Algorithm}: Distributed Computation of the Recommended Strategy (DCRS)  \\\\ \\hline\n(Run by SNI of agent $i$)\\textit{\\newline Input}: Connectivity and utility function of agent $i$.\\newline \\textit{Output}:  ${\\bm \\sigma }_{i} ({\\boldsymbol{K}})=\\{ \\sigma _{ij} (K)\\} _{j:g_{ij} =1} $ (denoted by ${\\bm \\sigma }_{i} =\\{ \\sigma _{ij} \\} _{j:g_{ij} =1} $ for simplification) \\\\ \\hline\n\\textbf{Initialization}:, $q=0$; $\\lambda _{i} (q)=0$\\newline \\textbf{Repeat}:\\newline Send $\\lambda _{i} (q)$ to neighbor $j$, $\\forall j:g_{ij} =1$. $~~~$(Obtain $\\lambda _{j} (q)$ from $j$, $\\forall j$)\\newline Solve \\eqref{ZEqnNum852632} using $\\lambda _{i} (q)$, $\\{\\lambda _{j} (q)\\}_{j:g_{ij} =1}$ to obtain $\\hat{{\\bm \\sigma }}_{i} ({\\bm \\lambda }(q))$.\\newline Send $\\sigma _{ji} ({\\bm \\lambda }(q))$ to neighbor $j$, $\\forall j:g_{ij} =1$.  $~~~$(Obtain $\\sigma _{ij} ({\\bm \\lambda }(q))$ from $j$, $\\forall j$)\\newline Update $\\lambda _{i} (q+1)$  according to \\eqref{ZEqnNum330405}.\\newline \\textbf{Stop} until $\\|\\lambda _{ji} (q+1)-\\lambda _{ji} (q)\\|_2 <\\varepsilon _{\\lambda } $ \\\\ \\hline\n\\end{tabular}\n\\bigskip\n\nThe above DCRS algorithm has the following interpretation. In each period, each SNI computes the information sharing actions of its neighbors that maximize the social surplus with respect to its own agent (i.e. the benefit obtained by its own agent minus the cost incurred by its neighbors). However, this computation has to take into account whether neighboring agents' incentive constraints are satisfied which are reflected by the Lagrangian multipliers. The larger $\\lambda _{i} $ is, the more likely is that agent $i$'s incentive is being violated. Hence, the neighbors of agent $i$ should acquire less information from it. We note that the DCRS algorithm needs to be run to compute the optimal strategy only once in the static topology case or once in a while in the dynamic topology case.\n\n\n\\subsection{Computing the Remaining Components of the Rating Protocol}\nEven though the DCRS algorithm provides a distributed way to compute the recommended strategy when agents have the highest ratings, the other elements of the rating protocol remain to be determined. There are many possible rating protocols that can constitute PPE given the obtained recommended strategies. In fact, we have already provided one way to compute these remaining elements when we determined the sufficient condition in Theorem 2 by using a constructional method. However, this is not the most efficient design in the imperfect monitoring scenario where ratings will occasionally drop due to monitoring errors. Therefore, the remaining components of the rating protocol should still be smartly chosen in the presence of monitoring errors. In this subsection, we consider a rating protocol with a binary rating set $\\Theta =\\{ 1,2\\} $ and $\\sigma _{ij} (\\theta =1)=0,\\forall i,j:g_{ij} =1$. We design the rating update probabilities $\\alpha _{i,2} ,\\beta _{i,1} ,\\forall i$ to maximize the social welfare when monitoring error exists.\n\n\\begin{proposition}\nGiven a binary rating protocol $\\Theta =\\{ 1,2\\} $, $\\sigma _{ij} (2),\\forall i,j:g_{ij} =1$ determined by the DCRS Algorithm and $\\sigma _{ij}(1)=0,\\forall i,j:g_{ij} =1$, when the monitoring error $\\epsilon>0$, the optimal rating update probability that maximize the social welfare is, $\\forall i$, $\\beta _{i,1}^{*} =1,\\alpha _{i,2}^{*} =\\frac{\\|{\\bm \\sigma }_{i} ({\\bf 2})\\|}{\\delta b_{i} (\\hat{{\\bm \\sigma }}_{i} (2))} $\n\\end{proposition}\n\\begin{proof}\nWe can derive the feasible values of $\\alpha _{i,2} ,\\beta _{i,1} ,\\forall i$ for binary rating protocol, i.e.\n\\begin{equation} \\label{ZEqnNum221816}\n\\beta _{i,1} \\ge \\frac{1-\\delta }{\\delta } \\frac{\\|{\\bm \\sigma }_{i} ({\\bf 2})\\|}{b_{i} (\\hat{{\\bm \\sigma }}_{i} (2))-\\|{\\bm \\sigma }_{i} ({\\bf 2})\\|}\n\\end{equation}\n\\begin{equation} \\label{ZEqnNum2218162}\n\\alpha _{i,2} \\ge \\frac{1-\\delta (1-\\beta _{i,1} )}{\\delta } \\frac{\\|{\\bm \\sigma }_{i} ({\\bf 2})\\|}{b_{i} (\\hat{{\\bm \\sigma }}_{i} (2))}\n\\end{equation}\n\nWhen monitoring is imperfect $\\epsilon>0$, agent $i$ will drop to $\\theta _{i} =1$ with positive probability even if it follows the recommended strategy all the time. According to the rating update rule, we can compute the stationary probability that agent $i$ stays at rating $\\theta _{i} =2$, i.e.\n\\begin{equation} \\label{ZEqnNum259722}\n\\frac{(1-\\epsilon)\\beta _{i,1} }{\\epsilon\\alpha _{i,2} +(1-\\epsilon)\\beta _{i,1} }\n\\end{equation}\n\nBecause agents having low ratings harms the social welfare, we need to select $\\alpha _{i,2} ,\\beta _{i,1} $ that maximizes \\eqref{ZEqnNum259722}. This is equivalent to minimize $\\alpha _{i,2} \/\\beta _{i,1} $.  For any $\\beta _{i,1} $, the optimal value of $\\alpha _{i,2} $ is the binding value of \\eqref{ZEqnNum2218162} and hence, we need to minimize $[1-\\delta (1-\\beta _{i,1} )]\/\\beta _{i,1} $. Because $[1-\\delta (1-\\beta _{i,1} )]\/\\beta _{i,1} $ is decreasing in $\\beta _{i,1} $, the optimal value of $\\beta _{i,1} $ is $\\beta _{i,1}^{*} =1$. Using \\eqref{ZEqnNum221816}  again, the optimal value of $\\alpha _{i,2}^{*} =\\frac{\\|{\\bm \\sigma }_{i} ({\\bf 2})\\|}{\\delta b_{i} (\\hat{{\\bm \\sigma }}_{i} (2))} $.          \\end{proof}\n\nIt is worth noting that these probabilities can be computed locally by the SNIs of the agents which do not require any information from other agents.\n\n\n\\subsection{Example Revisited}\nAt this point, we have showed how the rating protocol can be determined in a distributed manner, given the network structure. It is time to revisit the cooperative estimation example for the ring and star topologies in order to illustrate the impact of topology on agents' incentives and recommended strategies. Figure \\ref{ring-star2} illustrates the optimal recommended strategies computed using the method developed in this section for these two topologies.\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale = 0.7]{ring_star2.pdf}}\n\\caption{Optimal strategies for strategic agents interacting over a ring and a star. (The other elements of the rating protocol can be computed as in Section V(C) )}\\label{ring-star2}\n\\vspace{-15pt}\n\\end{figure}\n\n\nIn the ring topology, agents are homogeneous and links are symmetric. As we can see, the optimal recommended strategy ${\\bm \\sigma }^{*} $ is exactly the same as the optimal action ${\\boldsymbol{a}}^{opt} $ for obedient agent case because ${\\boldsymbol{a}}^{opt} $ already provides sufficient incentive for strategic agents to follow. Therefore, we can easily determine that $PoA=1$. However, the strategic behavior of agents indeed degrades the social welfare in other cases, especially when the network becomes more heterogeneous and asymmetric, e.g. the star topologies. Even though taking ${\\boldsymbol{a}}^{opt} $ maximizes the social welfare $V^{opt} =4$ in the star topology, these actions are not incentive-compatible for all agents. In particular, the maximum welfare $V^{opt} =4$ is achieved by sacrificing the individual utility of the center agent (i.e. agent 1 needs to contribute much more than it obtains). However, when agents are strategic, the center agent will not follow these actions ${\\boldsymbol{a}}^{opt} $ and hence, $V^{opt} =4$ cannot be achieved. More problematically, since the center agent will choose not to participate in the information sharing process, the periphery agents do not obtain benefits and hence, they will also choose not to participate in the information sharing process. This leads to a network collapse. In the proposed rating protocol, the recommended strategies satisfy all agents' incentive constraints, namely $\\delta b_{i} (\\hat{{\\bm \\sigma }}_{i} (K))\\ge \\|{\\bm \\sigma }_{i} ({\\bf K})\\|,\\forall i$. By comparing ${\\boldsymbol{a}}^{opt} $ and ${\\bm \\sigma }^{*} $, we can see that the rating protocol recommends more information sharing from the periphery agents to the center agent and less information sharing from the central agent to the periphery agents than the obedient agent case. In this way, the center agent will obtain sufficient benefits from participating in the information sharing. However, due to this compensation for the center agent, the PoA is increased to $PoA=1.036$.\n\nNote that the optimal recommended strategy for strategic agents is computed in a distributed way by the DCRS algorithm. Figure \\ref{convergence} shows the intermediate values of the recommended strategy $\\sigma _{12},\\sigma _{21} $ by running the DCRS algorithm for the star. (Only the strategies between agents 1 and 2 are shown because the rest are identical due to the homogeneity of periphery agents).\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale = 0.7]{convergence.pdf}}\n\\caption{The recommended strategy obtained by running DCRS for the star topology.}\\label{convergence}\n\\vspace{-15pt}\n\\end{figure}\n\n\n\n\\section{Performance Analysis}\nIn this section, we analyze the performance of the rating protocol and try to answer two important questions: (1) What is the performance loss induced by the strategic behavior of agents? (2) What is the performance improvement compared to other (simple) incentive mechanisms?\n\n\\subsection{Price of Anarchy}\nObserve the social welfare maximization problems \\eqref{ZEqnNum918213} and \\eqref{ZEqnNum960030} for obedient agents and strategic agents (by using rating protocols), respectively. It is clear that the social welfare achieved by the rating system is always no larger than that obtained when agents are obedient due to the equilibrium constraint; hence, i.e. $PoA\\ge 1$. The exact value of PoA will, in general, depend on the specific network structure (topology and individual utility functions). In this subsection, we identify a sufficient condition for the connectivity degree of the topology such that PoA is one. To simplify the analysis, we assume that agents' benefit functions are homogeneous and depend only on the sum information sharing action of the neighboring agents, i.e. $b_{i} (\\hat{{\\boldsymbol{a}}}_{i} )=b(\\sum _{j:g_{ij} =1} a_{ji} )$.  Let $d_{i} =\\sum _{j}g_{ij}  $ be the number of neighbors of agent $i$. The degree of network $G$ is defined as $d=\\mathop{\\max }\\limits_{i} d_{i} $.\n\n\\begin{proposition}\nSuppose benefit function structure $b_{i} (\\hat{{\\boldsymbol{a}}}_{i} )=b(\\sum _{j:g_{ij} =1} a_{ji} ),\\forall i$, if the connectivity degree $d$ is no larger than $\\bar{d}$ such that $\\delta b(\\bar{d})-\\bar{d}=0$, then $V^{*} =V^{opt} $, i.e. PoA is one.\n\\end{proposition}\n\n\\begin{proof}\nDue to the concavity of the benefit function (Assumption), there exists $m^{*} $ such that if $d>m^{*} $, $b(d)-d<0$ and if $d\\le m^{*} $, $b(d)-d\\ge 0$. If the connectivity degree satisfies $d<m^{*} $, then the optimal solution of \\eqref{ZEqnNum918213} is $a_{ij} =1,\\forall i,j:g_{ij} =1$. That is, optimality is achieved when all agents share the maximal amount of information with all their neighbors. Therefore, $\\forall d<m^{*} $, the agent $i$'s benefit is $b(m_{i} )$ and its cost is $m_{i} $ in the optimal solution.\n\nAgain due to the concavity of the benefit function, there exists $\\bar{d}\\le m^{*} $ (inequality is due to $\\delta \\in (0,1]$) such that if $d>\\bar{d}$, $\\delta b(d)-d<0$ and if $d\\le \\bar{d}$, $\\delta b(d)-d\\ge 0$.Therefore, if $d\\le \\bar{d}$, $\\forall i$, agent $i$'s benefit and cost satisfy $\\delta b(m_{i} )-m_{i} \\ge 0$. This satisfies the equilibrium constraint due to Theorem 2. Therefore, the achievable social welfare is the same.            \\end{proof}\n\nProposition 2 states that when the connectivity degree is low, the proposed rating protocol will achieve the optimal performance even when agents are strategic.\n\n\\subsection{Comparison with Direct Reciprocation}\nThe proposed rating protocol is not the only incentive mechanism that can incentivize agents to share information with other agents. A well-known direct reciprocation based incentive mechanism is the Tit-for-Tat strategy, which is widely adopted in many networking applications \\cite{Axelrod}-\\cite{Milan}. The main feature of the Tit-for-Tat strategy is that it exploits the repeated \\textit{bilateral} interactions between connected\\textit{ }agents, which can be utilized to incentivize agents to \\textit{directly} reciprocate to each other. However, when agents do not have bilateral interests, such mechanisms fail to provide such incentives and direct reciprocity algorithms cannot be applied.\n\nNevertheless, even if we assume that interests are bilateral between agents, our proposed rating protocol is still guaranteed to outperform the Tit-for-Tat strategy when the utility function takes a concave form as assumed in this paper. Intuitively, because the marginal benefit from acquiring information from one neighbor is decreasing in the total number of neighbors, agents become less incentivized to cooperate when their deviation towards some neighboring agent would not affect future information acquisition from others, as is the case with the Tit-for-Tat strategy. In the following, we formally compare our proposed rating protocol with the Tit-for-Tat strategy. We assume that an agent $i$ has two sharing actions that it can choose to collaborate with its neighboring agent $j$, i.e. $\\{ 0,\\bar{a}_{ij} \\} $ where $\\bar{a}_{ij} \\in (0,1]$. The Tit-for-Tat strategy prescribes the action for each agent $i$ as follows, $\\forall j:g_{ij} =1$,\n\\begin{equation} \\label{17)}\n\\begin{array}{l} {a_{ij} (0)=\\bar{a}_{ij} } \\\\ {a_{ij} (t+1)=\\left\\{\\begin{array}{l} {\\bar{a}_{ij} ,{\\rm \\; \\; \\; if\\; \\; }a_{ji} (t)=\\bar{a}_{ji} } \\\\ {0,{\\rm \\; \\; \\; \\; \\; if\\; \\; }a_{ji} (t)=0} \\end{array}\\right. ,\\forall t\\ge 0} \\end{array}\n\\end{equation}\n\n\\begin{proposition}\nGiven the network structure and the discount factor, any action profile $\\bar{a}$ that can be sustained by the Tit-for-Tat strategy can also be sustained by the rating protocol.\n\\end{proposition}\n\n\\begin{proof}\nConsider the interactions between any pair of agents $i,j$. In the Tit-for-Tat strategy, the long-term utility of agent $i$ by following the strategy when agent $j$ played $\\bar{a}_{ji} $ in the previous period is $U_{i} =\\frac{\\tilde{b}_{ji} (\\bar{a}_{ji} )-\\bar{a}_{ij} }{1-\\delta } $ where $\\tilde{b}_{ji} (x)=b_{i} (\\hat{a}_{i} |a_{ki} =\\bar{a}_{ki} ,a_{ji} =x)$. If agent $i$ deviates in the current period, Tit-for-Tat induces a continuation history $(\\{ \\bar{a}_{ij} ,0\\} ,\\{ 0,\\bar{a}_{ji} \\} ,\\{ \\bar{a}_{ij} ,0\\} ...)$ where the first components are agent $i$'s actions and the second components is agent $j$'s actions.  The long-term utility of agent $i$ by one-shot deviation is thus\n\\begin{equation} \\label{18)}\nU_{i} '=\\frac{\\tilde{b}_{ji} (\\bar{a}_{ji} )}{1-\\delta ^{2} } +\\delta \\frac{\\tilde{b}_{ji} (0)-\\bar{a}_{ij} }{1-\\delta ^{2} }\n\\end{equation}\n\nIncentive-compatibility requires that $U_{i} \\ge U_{i} '$ and therefore\n\\begin{equation} \\label{ZEqnNum345510}\n\\delta (\\tilde{b}_{ji} (\\bar{a}_{ji} )-\\tilde{b}_{ji} (0))\\ge \\bar{a}_{ij}\n\\end{equation}\n\nDue to the concavity of the benefit function, it is easy to see that \\eqref{ZEqnNum345510} leads to $\\delta b_{i} (\\hat{{\\boldsymbol{a}}}_{i} )\\ge \\|{\\boldsymbol{a}}_{i}\\| $ which is a sufficient condition for the rating protocol to be an equilibrium.\n\\end{proof}\n\nProposition 3 proves that the social welfare achievable by the rating protocol equals or exceeds that of the Tit-for-Tat strategy, which confirms the intuitive argument before that diminishing marginal benefit from information acquisition would result in less incentives to cooperate in an environment with only direct reciprocation than in one allowing indirect reciprocation. We note that different action profiles $\\bar{{\\boldsymbol{a}}}$ will generate different social welfare. However, computing the best $\\bar{{\\boldsymbol{a}}}$ among the incentive-compatible Tit-for-Tat strategies is often intractable since \\eqref{ZEqnNum345510} is a non-convex constraint. Hence, implementing the best Tit-for-Tat strategy to maximize the social welfare is often intractable. In contrast, the proposed rating protocol does not have this problem since the equilibrium constraint established in Theorem 2 is convex and hence, the optimal recommended strategy can be solved distributed by the proposed DCRS algorithm.\n\n\n\n\n\\section{Growing Networks}\nIn Section V, we designed the optimal rating protocol by assuming that the network topology is time-invariant. In practice, the social network topology can also change over time due to, e.g. new agents joining the network and new links being created. Nevertheless, our framework can easily handle such growing networks by adopting a simple extension which refreshes the rating protocol (i.e. re-computes the recommended strategy, rating update rules and re-initializes the ratings of agents) with a certain probability each period. We call this probability the refreshing rate and denote it by $\\rho \\in [0,1]$. When topologies are changing, the refreshing rate will also be an important design parameter of the rating protocol.\n\nConsider that the rating protocol was refreshed at period $T$ the last time.  Denote the probability that the rating protocol is refreshed at time $T+t$ as $p(t)$. Denote the network in period $t$ by $G(t)$. We assume that in each period a number $n(t)$ of new agents join the network and stay forever. Therefore, the network topology $G(t+1)$ will be formed based on $G(t)$ and the new agents. Let $V^{*} (G;\\rho )$ be the social welfare achieved by the rating protocol if the network topology is $G$ and the refreshing is set to be $\\rho $. Since there are no recommended strategy and update rules concerning the new agents before the next refreshment, existing agents have no incentives to share information with the new agents and vice versa, the new agents have no incentives to share information with their neighbors. Hence, the average social welfare achieved by the rating protocol before the next refreshment is $V^{*} (G(T);\\rho )$.  The optimal refreshing rate design problem is thus,\n\\begin{equation} \\label{ZEqnNum714883}\n\\begin{array}{c}\n\\rho ^{*} =\\arg \\mathop{\\max }\\limits_{\\rho } \\bigg(\\underbrace{{\\rm {\\mathbb E}}\\sum _{t=0}^{\\infty }p (t)\\frac{1}{t+1} \\sum _{\\tau =0}^{t}V^{opt} (G(T+\\tau )) }_{{\\rm expected\\; optimal\\; social\\; welfare}}-\\underbrace{V^{*} (G(T);\\rho )}_{{\\rm social\\; welfare\\; achieved\\; by\\; the\\; rating\\; protocol}}\\bigg)\n\\end{array}\n\\end{equation}\nThe first term in \\eqref{ZEqnNum714883} is the expected optimal social welfare and the second term is the social welfare achieved by the rating protocol.\n\nWe first investigate the expected optimal social welfare. Let the social welfare variance be $\\Delta _{V} (t+1)\\triangleq V^{OPT} (G(t+1))-V^{OPT} (G(t))$. It is easy to see that $\\Delta _{V} (t+1)\\ge 0$. We assume that the expected social welfare contribution of new agents is ${\\rm {\\mathbb E}}(\\Delta _{V} (t))=\\Delta _{v} $ which is time-independent. Given the refreshing rate $\\rho $, the expected time-average optimal social welfare from $T$ to the next refreshing period can be computed as\n\\[\\begin{array}{l} {\\rm {\\mathbb E}}\\sum _{t=0}^{\\infty }p (t)\\frac{1}{t+1} \\sum _{\\tau =0}^{t}V^{opt} (G(T+\\tau )) \\\\={\\rm {\\mathbb E}}\\sum _{t=0}^{\\infty }\\rho  (1-\\rho )^{t} \\frac{1}{t+1} \\sum _{\\tau =0}^{t}V^{opt} (G(T+\\tau ))  \\\\ {=V^{opt} (G(T))+\\sum _{t=0}^{\\infty }\\rho (1-\\rho )^{t} \\frac{1}{t+1} {\\rm {\\mathbb E}}\\sum _{\\tau =0}^{t}\\Delta _{V} (T+\\tau )  } \\\\ =V^{opt} (G(T))+\\sum _{t=0}^{\\infty }\\rho (1-\\rho )^{t} \\frac{1}{t+1} \\frac{t(t+1)}{2} \\Delta _{V}  \\\\=V^{opt} (G(T))+\\frac{(1-\\rho )\\Delta _{V} }{2\\rho }  \\end{array}\\]\n\nHence, the expected optimal social welfare is decreasing in the refreshing rate $\\rho $.\n\nNext, we investigate the relation between $V^{*} (G(T);\\rho )$ and $\\rho $. This is established in the proposition below.\n\n\\begin{proposition}\n$V^{*} (G(T);\\rho )$ is non-decreasing in $\\rho $.\n\\end{proposition}\n\n\\begin{proof}\nDue to the refreshing, an agent $i$'s long-term utility becomes\n\\begin{equation} \\label{21)}\nU_{i} (t)=u_{i} ({\\boldsymbol{a}}(t))+(1-\\rho )\\delta u_{i} ({\\boldsymbol{a}}(t+1))+[(1-\\rho )\\delta ]^{2} u_{i} ({\\boldsymbol{a}}(t+2))+...\n\\end{equation}\n\nHence, following the similar proof of Theorem 2, agents' incentives can be provided if and only if $(1-\\rho )\\delta b_{i} (\\hat{{\\bm \\sigma }}_{i} (K))\\ge {\\bm \\sigma }_{i} ({\\boldsymbol{K}}),\\forall i$. Therefore the constraint in the optimal strategy design problem \\eqref{ZEqnNum960030} becomes stronger for the rating protocol with refreshing. Hence, the achievable social welfare becomes (weakly) lower.\n\\end{proof}\n\nSummarizing, the refreshing rate impacts the social welfare gap in two different ways. On one hand, $\\frac{(1-\\rho )\\Delta _{V} }{2\\rho } $ is non-decreasing in $\\rho $ since a larger $\\rho $ leads to a better adaptation of the rating protocol to the changing topology. On the other hand, $V^{*} (G(T);\\rho )$ is also non-decreasing in $\\rho $ since a smaller $\\rho $ provides more incentives for agents to follow the rating protocol designed in period $T$. Therefore, the refreshing rate has to balance these two effects. In the simulations, we will show how different refreshing rates influence the social welfare in various exemplary scenarios.\n\n\n\\section{Illustrative Results}\nIn this section, we provide simulation results to illustrate the performance of the rating protocol. In all simulations, we consider the cooperative estimation problem introduced in Section III (A). Therefore, agents' utility function takes the form of $u_{i} ({\\boldsymbol{a}}(t))=[r^{2} -MSE_{i} (\\hat{{\\boldsymbol{a}}}_{i} (t))]-{\\boldsymbol{a}}_{i} (t)$ \\cite{Chen}. We will investigate different aspects of the rating protocol by varying the underlying topologies and the environment parameters.\n\n\\subsection{Impact of Network Topology}\nNow we investigate in more detail how the agents' connectivity shapes their incentives and influences the resulting social welfare. In the first experiment, we consider the cooperative estimation over star topologies with different sizes (hence, different connectivity degrees). Figure \\ref{stardegree} shows the PoA achieved by the rating protocol for discount factors $\\delta =1,0.9,0.8,0.7$ for the noise variance $r^{2} =8$. As predicted by Proposition 3, when the connectivity degree is small enough, the PoA equals one and hence, the performance gap is zero. As the network size increases (hence the connectivity degree increases in the star topology), the socially optimal action requires the center agent to share more information with the periphery agents. However, it becomes more difficult for the center agent to have incentives to do so since the information sharing cost becomes much larger than the benefit. In order to provide sufficient incentives for the center agent to participate in the information sharing process, the rating protocol recommends less information sharing from the center agent to each periphery agent. However, incentives are provided at a cost of reduced social welfare. Figure \\ref{stardegree} also reveals that when agents' discount factor is lower (agents value less the future utility), incentives are more difficult to provide and hence, the PoA becomes higher. In the next simulation, we study scale-free networks in the imperfect monitoring scenarios. In scale-free networks, the number of neighboring agents is distributed as a power law (denote the power law parameter by $d^{SF} $). Table 3 shows the PoA achieved by the rating protocol developed in Section V(C) for various values of $d^{SF} $ and different monitoring error probabilities $\\epsilon$. As we can see, the proposed rating protocol achieves close-to-optimal social welfare in all the simulated environments\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale = 0.8]{stardegree.pdf}}\n\\caption{Performance of the rating protocol for various connectivity degrees in star topologies.}\\label{stardegree}\n\\vspace{-5pt}\n\\end{figure}\n\n\\begin{table}[t]\n\\centerline{\\includegraphics[scale = 0.9]{Table3.pdf}}\n\\caption{Performance of the rating protocol for various   in scale-free topologies.}\\label{Table3}\n\\vspace{-15pt}\n\\end{table}\n\n\\subsection{Comparison with Tit-for-Tat}\nAs mentioned in the analysis, incentive mechanisms based on direct reciprocation such as Tit-for-Tat do not work in networks lacking bilateral interests between connected agents and hence, reasons to mutually reciprocate. In this simulation, to make possible a direct comparison with the Tit-for-Tat strategy, we consider a scenario where the connected agents do have bilateral interest and show that the proposed rating protocol significantly outperforms the Tit-for-Tat strategy. In general, computing the optimal action profile $\\bar{a}^{*} $ for the Tit-for-Tat strategy is difficult because it involves the non-convex constraint $\\delta (b_{i} (\\{ \\bar{a}_{ki}^{*} \\} _{k:g_{ik} =1} )-b_{i} (\\{ \\bar{a}_{ki}^{*} \\} _{k\\ne j:g_{ik} =1} ,0))\\ge \\bar{a}_{ij}^{*} $, $\\forall i,\\forall j\\ne i:g_{ij} =1$; such a difficulty is not presented in our proposed rating protocol because the constraints in our formulated problem are convex. For tractability, here we consider a symmetric and homogeneous network to enable the computation of the optimal action for the Tit-for-Tat strategy. We consider a number $N=100$ of agents and that the number of neighbors of each agent is the same $d_{i} =d,\\forall i$ and each agent adopts a symmetric action profile $\\bar{a}_{ij} =\\bar{a},\\forall i,j$. The noise variance is set to be $r^{2} =4$ in this simulation. Figure \\ref{tft} illustrates the PoA achieved by the proposed rating protocol and the Tit-for-Tat strategy. As predicted by Proposition 4, any action profile that can be sustained by the Tit-for-Tat strategy can also be sustained by the proposed rating protocol (for the same $\\delta $). Hence, the rating protocol yields at least as much social welfare as the Tit-for-Tat strategy. As the discount factor becomes smaller, agents' incentives to cooperate become less and hence, the PoA is larger.\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale = 0.8]{tft.pdf}}\n\\caption{Performance comparison with Tit-for-Tat.}\\label{tft}\n\\vspace{-5pt}\n\\end{figure}\n\n\n\\subsection{Rating Protocol with Refreshing}\nFinally, we consider the optimal choice of the rating protocol refreshing rate $\\rho $ when the network is growing as considered in section VIII. In this simulation, the network starts with $N=50$ agents. In each period, a new agent joins the network with probability 0.1 and stays in the network forever. Any two agents are connected with \\textit{a priori} probability 0.2. We vary the refreshing rate from 0.005 to 0.14. Table 4 records the PoA achieved the rating protocol with refreshing for $\\delta =0.4$. It shows that the optimal refreshing rate needs to be carefully chosen. If $\\rho $ is too large, the incentives for agents to cooperate is small hence, the incentive-compatible rating protocol achieves less social welfare. If $\\rho $ is too small, the rating protocol is not able to adapt to the changing topology well. This introduces more social welfare loss in the long-term as well. The optimal refreshing rate in the simulated network is around 0.04.\n\n\\begin{table}[t]\n\\centerline{\\includegraphics[scale = 1]{Table4.pdf}}\n\\caption{PoA of rating protocols with different refreshing rates. }\\label{Table4}\n\\vspace{-15pt}\n\\end{table}\n\n\\section{Conclusions}\nIn this paper, we studied how to design distributed incentives protocols (based on ratings) aimed at maximizing the social welfare of repeated information sharing among strategic agents in social networks. We showed that it is possible to exploit the ongoing nature of agents' interactions to build incentives for agents to cooperate based on rating protocols. The proposed design framework of the rating protocol enables an efficient way to implement social reciprocity in distributed information sharing networks with arbitrary topologies and achieve much higher social welfare than existing incentive mechanisms. Our analysis also reveals the impact of different topologies on the achievable social welfare in the presence of strategic agents and hence, it provides guidelines for topology configuration and planning for networks with strategic agents. The proposed rating protocols can be applied in a wide range of applications where selfish behavior arises due to cost-benefit considerations including problems involving interactions over social networks, communications networks, power networks, transportation networks, and computer networks.\n\n\n\n\\section*{Appendix: Proof of Theorem 2}\nAccording to Lemma, we know that it suffices to ensure that agent $i$ has the incentives to following the recommended strategy when other agents' ratings are ${\\boldsymbol{K}}$ (i.e. all other agents have the highest rating $K$). However, we need to ensure this holds for all ratings of agent $i$. We will write ${\\bm \\sigma }_{i} ({\\boldsymbol{K}})$ as ${\\bm \\sigma }_{i} $ and $\\hat{{\\bm \\sigma }}_{i} (K)$ as $\\hat{{\\bm \\sigma }}_{i} $ to keep the notation simple.\n\nWe prove the ``only if'' part first, i.e. if $\\|\\bm \\sigma_i\\| \\geq \\delta b_i(\\hat{\\bm \\sigma_i})$. Consider rating level $k$, if agent $i$ follows the recommended strategy, its long-term utility is\n\\begin{align}\nU_i(k, \\bm\\sigma_i) = u_i(k, \\bm\\sigma_i) + \\delta(\\beta_{i,k} U^*_i(k+1) + (1 - \\beta_{i,k} U^*_i(k))\n\\end{align}\nBy deviation to ${\\bf 0}$, its long-term utility is\n\\begin{align}\nU_i(k, {\\bf 0}) = u_i(k, {\\bf 0}) + \\delta(\\alpha_{i,k} U^*_i(k-1) + (1 - \\alpha_{i,k} U^*_i(k))\n\\end{align}\nEquilibrium requires that $U_i(k, \\bm\\sigma_i)  \\geq U_i(k, {\\bf 0})$. Hence,\n\\begin{equation}\n\\begin{aligned}\n&u_i(k, {\\bf 0}) - u_i(k, \\bm\\sigma_i)\\\\\n\\leq &\\delta[(\\beta_{i,k} U^*_i(k+1) + (1 - \\beta_{i,k}) U^*_i(k)) \\\\\n&- (\\alpha_{i,k} U^*_i(k-1) + (1-\\alpha_{i,k}) U^*_i(k))]\n\\end{aligned}\n\\end{equation}\nBy Lemma (3), $U^*_i(K) \\geq U^*_i(k),\\forall k$. Therefore, PPE requires\n\\begin{align}\nu_i(k, {\\bf 0}) - u_i(k, \\bm\\sigma_i) \\leq \\delta U^*_i(K) \\label{eq34}\n\\end{align}\nBecause $u_i(k, {\\bf 0}) - u_i(k, \\bm\\sigma_i) = \\|\\bm\\sigma_i\\|$ and\n\\begin{align}\nU^*_i(K) = \\frac{1}{1-\\delta} u_i(K, \\bm\\sigma_i) = \\frac{1}{1-\\delta}\\left(b_i(\\hat{\\bm\\sigma}_i) -  \\|\\bm\\sigma_i\\|\\right)\n\\end{align}\n(\\ref{eq34}) becomes,\n\\begin{align}\n\\|\\bm\\sigma_i\\| \\leq \\delta b_i(\\hat{\\bm\\sigma}_i)\n\\end{align}\nHence, if $\\|\\bm\\sigma_i\\| > \\delta b_i(\\hat{\\bm\\sigma}_i)$, then no rating protocol can constitute a PPE.\n\nNext we prove the ``if'' part by construction. We let $\\alpha_{i,K-1} = 0$ and hence, the effect rating set is just a binary set $\\{K-1, K\\}$. The value functions can be determined below,\n\\begin{equation}\nU^*_i(K) = u_i(K, \\bm\\sigma_i) + \\delta U^*_i(K) \\label{UK}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n&U^*_i(K-1) = u_i(K-1, \\bm\\sigma_i)\\\\\n& + \\delta (\\beta_{i,K-1} U^*_i(K) + (1-\\beta_{i,K-1}) U^*_i(K-1) \\label{UK-1}\n\\end{aligned}\n\\end{equation}\nThe long-term utilities by deviation is\n\\begin{equation}\n\\begin{aligned}\n&U_i(K, {\\bf 0}) = u_i(K, {\\bf 0}) \\\\\n&+ \\delta(\\alpha_{i, K} U^*_i(K-1) + (1 - \\alpha_{i, K}) U^*_i(K))\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\nU_i(K-1, {\\bf 0}) = u_i(K-1, {\\bf 0}) + \\delta U^*_i(K-1)\n\\end{equation}\n\nFor agent $i$ to have incentives to following the recommended strategy at $\\theta_i = K$, we need the following to hold\n\\begin{align}\nu_i(K, {\\bf 0}) - u_i(K, \\bm\\sigma_i) \\leq \\delta \\alpha_{i,K}(U^*_i(K) - U^*_i(K-1)) \\label{conditionK}\n\\end{align}\n\nFor agent $i$ to have incentives to following the recommended strategy at $\\theta_i = K-1$, we need the following to hold\n\\begin{align}\nu_i(K-1, {\\bf 0}) - u_i(K-1, \\bm\\sigma_i) \\leq \\delta \\beta_{i,K-1}(U^*_i(K) - U^*_i(K-1))\\label{conditionK-1}\n\\end{align}\n\nIn the above two inequalities, $U^*_i(K) - U^*_i(K-1)$ can be computed using (\\ref{UK}) and (\\ref{UK-1}) and is\n\\begin{align}\nU^*_i(K) - U^*_i(K-1) = \\frac{u_i(K, \\bm\\sigma_i) - u_i(K-1, \\bm\\sigma_i)}{1 - \\delta(1 - \\beta_{i, K-1})}.\n\\end{align}\n\nBy choosing $\\alpha_{i,K} = \\beta_{i, K-1} = 1$, both (\\ref{conditionK}) and (\\ref{conditionK-1}) are satisfied. This means that if $\\|\\bm\\sigma_i\\| \\leq \\delta b_i(\\hat{\\bm\\sigma}_i)$, then we can construct at least one binary rating protocol that constitutes a PPE.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}