A Game Theoretic Model for the Formation of Navigable Small-world Networks

Kleinberg proposed a family of small-world networks to explain the navigability of large-scale real-world social networks. However, the underlying mechanism that drives real networks to be navigable is not yet well understood. In this paper, we present a game theoretic model for the formation of navigable small world networks. We model the network formation as a game in which people seek for both high reciprocity and long-distance relationships. We show that the navigable small-world network is a Nash Equilibrium of the game. Moreover, we prove that the navigable small-world equilibrium tolerates collusions of any size and arbitrary deviations of a large random set of nodes, while non-navigable equilibria do not tolerate small group collusions or random perturbations. Our empirical evaluation further demonstrates that the system always converges to the navigable network even when limited or no information about other players’ strategies is available. Our theoretical and empirical analyses provide important new insight on the connection between distance, reciprocity and navigability in social networks.