Fixation and consensus times on a network: a unified approach

Abstract

We investigate a set of stochastic models of biodiversity, population genetics, language evolution and opinion dynamics on a network within a common framework. Each node has a state, 0 < x_i < 1, with interactions specified by strengths m_{ij}. For any set of m_{ij} we derive an approximate expression for the mean time to reach fixation or consensus (all x_i=0 or 1). Remarkably in a case relevant to language change this time is independent of the network structure.Comment: 4+epsilon pages, two-column, RevTeX4, 3 eps figures; version accepted by Phys. Rev. Let