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Unformatted text preview: A Small World Threshold for Economic Network Formation Eyal EvenDar Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 [email protected] Michael Kearns Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 [email protected] Abstract We introduce a gametheoretic model for network formation inspired by earlier stochastic models that mix localized and longdistance connectivity. In this model, players may purchase edges at distance d at a cost of d , and wish to minimize the sum of their edge purchases and their average distance to other players. In this model, we show there is a striking small world threshold phenomenon: in two dimensions, if &lt; 2 then every Nash equilibrium results in a network of constant diameter (independent of network size), and if &gt; 2 then every Nash equilibrium results in a network whose diameter grows as a root of the network size, and thus is unbounded. We contrast our results with those of Kleinberg [8] in a stochastic model, and empirically investigate the navigability of equilibrium networks. Our theoretical results all generalize to higher dimensions. 1 Introduction Research over the last decade from fields as diverse as biology, sociology, economics and computer science has established the frequent empirical appearance of certain structural properties in natu rally occurring networks. These properties include small diameter, local clustering of edges, and heavytailed degree distributions [11]. Not content to simply catalog such apparently universal properties, many researchers have proposed stochastic models of decentralized network formation that can explain their emergence. A typical such model is known as preferential attachment [3], in which arriving vertices are probabilistically more likely to form links to existing vertices with high degree; this generative process is known to form networks with power law degree distributions. In parallel with these advances, economists and computer scientists have examined models in which networks are formed due to rational or gametheoretic forces rather than probabilistic ones. In such models networks are formed via the selfinterested behavior of individuals who benefit from participation in the network [7]. Common examples include models in which a vertex or player can purchase edges, and would like to minimize their average shortestpath distance to all other vertices in the jointly formed network. A players overall utility thus balances the desire to purchase few edges yet still be wellconnected in the network. While stochastic models for network formation define a (possibly complex) distribution over possible networks, the gametheoretic models are typ ically equated with their (possibly complex) set of (Nash) equilibrium networks. It is also common to analyze the socalled Price of Anarchy [9] in such models, which measures how much worse an equilibrium network can be than some measure of social or centralized optimality [6, 2, 5, 1].equilibrium network can be than some measure of social or centralized optimality [6, 2, 5, 1]....
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 Fall '09
 MichaelKearns
 Graph Theory, Nash, grid distance, J. Kleinberg, link stable graph

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