Fixation and consensus times on a network: a unified approach

TL;DR

This paper presents a unified framework for analyzing fixation and consensus times across various stochastic network models, revealing that in some cases these times are independent of network structure.

Contribution

It introduces a common approach to estimate fixation and consensus times in diverse models, including biodiversity, genetics, language evolution, and opinion dynamics.

Findings

Derived an approximate expression for fixation/consensus times.

Found that in certain models, times are network-structure independent.

Applicable to models with diverse interaction strengths.

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.