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

TL;DR

This paper introduces a game theoretic model explaining how social networks form navigable small-world structures, balancing distance and reciprocity, and demonstrates the stability and social benefits of such networks through theoretical and empirical analysis.

Contribution

It presents a novel Distance-Reciprocity Balanced (DRB) game model that explains the emergence and stability of navigable small-world networks, supported by theoretical proofs and empirical simulations.

Findings

Navigable small-world network is a stable Nash equilibrium.

The model shows the network naturally converges to navigability.

Navigable networks have higher social welfare than random networks.

Abstract

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 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. We further show that the navigable small world 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 move the network to the navigable network as soon as every node takes one best-response step. Moreover, we show that navigable small world has much better social welfare than the random network, and provide the price-of-anarchy and price-of-stability results of the game. Our empirical evaluation 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 the real-world network leads to navigability characteristic that is very close to that of the real network. Our theoretical and empirical analyses provide important new insight on the connection between distance, reciprocity and navigability in social networks.