On the optimization of vectors of Katz-Bonacich's centralities related to Nash equilibria for a class of non-cooperative network games

Evaluating the importance or centrality of nodes in a graph is a fundamental issue in network analysis and design. One possible measure of centrality is Katz-Bonacich's centrality, which arises both at the steady state of a dynamical system involving local communication within an undirected graph, and at the pure Nash equilibrium of a non-cooperative game on that graph, whose players are its nodes. Given this context, the present article explores the connection between vectors of Katz-Bonacich's centralities arising, respectively, in the deterministic and stochastic cases, and its consequences for network design and for finding the so-called key player, i.e. a player/node that mostly contributes to raising the equilibrium aggregate level in the network. The theoretical analysis is complemented by numerical results and by a discussion of its possible extensions to other measures of node centrality.