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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 article, we present a game theoretic model for the formation of navigable small-world networks. We model the network formation as a game called the Distance-Reciprocity Balanced (DRB) game in which people seek for both high reciprocity and long-distance relationships. We show that the game has only two Nash equilibria: One is the navigable small-world network, and the other is the random network in which each node connects with each other node with equal probability, and any other network state can reach the navigable small world via a sequence of best-response moves of nodes. We further show that the navigable small-world equilibrium is very stable—(a) no collusion of any size would benefit from deviating from it; and (b) after an arbitrary deviations of a large random set of nodes, the network would return to the navigable small world as soon as every node takes one best-response step. In contrast, for the random network, a small group collusion or random perturbations is guaranteed to bring the network out of the random-network equilibrium and move to the navigable network as soon as every node takes one best-response step. Moreover, we show that navigable small-world equilibrium has much better social welfare than the random network, and we provide the price-of-anarchy and price-of-stability results of the game. 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, and the DRB game simulated on real-world networks leads to navigability characteristic that is very close to that of the real networks, even though the real-world networks have non-uniform population distributions different from Kleinberg’s small-world model. Our theoretical and empirical analyses provide important new insight on the connection between distance, reciprocity, and navigability in social networks.

A Game Theoretic Model for the Formation of Navigable Small-World Networks—The Tradeoff between Distance and Reciprocity