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Statistical mechanics of the fashion game on random networks

A model of fashion on networks is studied. This model consists of two groups of agents that are located on a network and have opposite viewpoints towards being fashionable: behaving consistently with either the majority or the minority of adjacent agents. Checking whether the fashion game has a pure Nash equilibrium (pure NE) is a non-deterministic polynomial complete problem. Using replica-symmetric mean field theory, the largest proportion of satisfied agents and the region where at least one pure NE should exist are determined for several types of random networks. Furthermore, a quantitive analysis of the asynchronous best response dynamics yields the phase diagram of existence and detectability of pure NE in the fashion game on some random networks.