The Betweenness Centrality Game for Strategic Network Formations

In computer networks and social networks, the betweenness centrality of a node measures the amount of information passing through the node when all pairs are conducting shortest-path exchanges. In this paper, we introduce a strategic network-formation game based on the betweenness centrality. In this game, nodes are selfish players, each of which has some budget to build connections. The goal of each player is to fully utilize her budget to strategically build connections so that her betweenness centrality is as large as possible. We refer to this game as the bounded budget betweenness centrality game or the B3C game. We present both theoretical and experimental results about this game. Theoretically, we show that a general B3C game may not have any nontransient Nash equilibrium and it is in fact NP-hard to determine whether nontransient Nash equilibria exist. Experimentally, we studied the family of uniform B3C games, in which there is an equal amount of information exchange between every pair of nodes, every connection has the same construction cost and every node has enough budget to build k connections. We have discovered several interesting Nash equilibria when k = 2. Our experiments have also inspired us to establish some theoretical results about uniform B3C games. For example, we show that, when k is a variable, it is NP-hard to compute a best response of a node in uniform B³C games; we also prove that a family of symmetric graphs called Abelian Cayley graphs cannot be Nash equilibria for k=2 when the graph is large enough, but there is a unique nontransient Nash equilibrium when k=1. We conjecture, based on our experiments, that every uniform B3C game has a Nash equilibrium, and more strongly, has a Nash equilibrium that is Eulerian.