Games On Networks With Community Structure: Existence, Uniqueness And Stability Of Equilibria

We study games with nonlinear best response functions played on a network consisting of disjoint communities. Prior works on network games have identified conditions to guarantee the uniqueness and stability of Nash equilibria in a network without any community structure. In this paper we are interested in accomplishing the same for networks with a community structure; it turns out that these conditions are much easier to verify with certain community structures. Specifically, we consider multipartite graphs and show that the uniqueness and stability of Nash equilibria are related to matrices which are potentially much lower in dimension, on the order of the number of communities, compared to same-size networks without a multipartite structure, in which case such matrices have a dimension the size of the network. We further introduce a new notion of degree centrality to measure the importance and influence of a community in such a network. We show that this notion enables us to find new conditions for uniqueness and stability of Nash equilibria.