Mixing Network Extremal Optimization for Community Structure Detection

Approximation of Nash equilibria and the network community structure detection problem

“…The first game-based models use the modularity function [15] to derive payoffs for players [21, 22]. Game theoretic models have been proposed mostly for networks with overlapping communities [21, 23, 24] and for dynamic networks [2, 25]; models based on cooperative games theory have been proposed in [26–28]; other game based approaches can be found in [1, 5, 29]. …”

“…Given a network G = ( V , E ), where V is the set of vertices and E is the set of edges, consider the following game [5], where: N is the set of players represented by network nodes, N = V = {1, 2, …, n }; where n is the network size; S = S 1 × S 2 × ⋯ × S n the set of strategy profiles, and S i the set of strategies of player i ; in this case S i represents the set of possible communities node i may choose from; an element s ∈ S is a possible community structure; we write s = ( C 1 , C 2 , …, C n ), C i represents the community of node i , i ∈ {1, 2, …, n }; we denote strategies C i by using integers, C i ∈ {1, …, C max } where C max is a maximum value expected for the number of communities in the network, and with , w = 1, …, C max the community containing all nodes having C i = w , i ∈ {1, …, n }; for example, C i = 2 indicates that player i belongs to community 2 with all other nodes j ∈ N with C j = 2, and we will denote by the community containing all nodes i ∈ N with C i = 2. U is the payoff function, U = ( u 1 , u 2 , …, u n ) where is the payoff function of player i ∈ N , computed as the node’s contribution to its community , where C i = w [1, 17]. We compute the payoff u i ( C 1 , C 2 , …, C i , …, C n ) of node i relative to community as the difference between the community fitness of with node i included in it and its fitness when i is excluded: The fitness of community in Eq (7) [17] is: where is the internal degree of node j in community (the number of links connecting node j to other nodes in ), is the external degree of node j with respect to community (the number of links connecting node j to nodes outside of ), and α is a parameter controlling the size of the community.…”

“…U is the payoff function, U = ( u 1 , u 2 , …, u n ) where is the payoff function of player i ∈ N , computed as the node’s contribution to its community , where C i = w [1, 17]. We compute the payoff u i ( C 1 , C 2 , …, C i , …, C n ) of node i relative to community as the difference between the community fitness of with node i included in it and its fitness when i is excluded: The fitness of community in Eq (7) [17] is: where is the internal degree of node j in community (the number of links connecting node j to other nodes in ), is the external degree of node j with respect to community (the number of links connecting node j to nodes outside of ), and α is a parameter controlling the size of the community.…”

“…In recent years game theory concepts have been used to address the network community structure detection problem as a game [1, 2]. This problem has multiple applications in economics, politics, sociology, biology, physics, and chemistry.…”

“…But this approach leads to multi-player games that are difficult to solve when the payoff functions do not have nice mathematical properties. A heuristic model that computes Nash equilibria for such games has been proposed and adapted for the community detection problem in [1, 5] for static unweighted networks, and in [2] for dynamic networks. The model is based on extremal optimization and a generative relation for strategy profiles that enables their comparison [6], guiding the search towards game equilibria.…”