Fixation and Consensus Times on a Network: A Unified Approach

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.