Greedy Selfish Network Creation

We introduce and analyze greedy equilibria (GE) for the well-known model of selfish network creation by Fabrikant et al.[PODC'03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategy-changes, (2) GE are outcomes found by agents which do not have enough computational resources to play optimally. In the model of Fabrikant et al. agents correspond to Internet Service Providers which buy network links to improve their quality of network usage. It is known that computing a best response in this model is NP-hard. Hence, poly-time agents are likely not to play optimally. But how good are networks created by such agents? We answer this question for very simple agents. Quite surprisingly, naive greedy play suffices to create remarkably stable networks. Specifically, we show that in the SUM version, where agents attempt to minimize their average distance to all other agents, GE capture Nash equilibria (NE) on trees and that any GE is in 3-approximate NE on general networks. For the latter we also provide a lower bound of 3/2 on the approximation ratio. For the MAX version, where agents attempt to minimize their maximum distance, we show that any GE-star is in 2-approximate NE and any GE-tree having larger diameter is in 6/5-approximate NE. Both bounds are tight. We contrast these positive results by providing a linear lower bound on the approximation ratio for the MAX version on general networks in GE. This result implies a locality gap of $\Omega(n)$ for the metric min-max facility location problem, where n is the number of clients.Comment: 28 pages, 8 figures. An extended abstract of this work was accepted at WINE'1

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

Kleinberg proposed a family of small-world networks to ex-plain the navigability of large-scale real-world social net-works. 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 for-mation 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 Equilib-rium 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 collu-sions 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