Solution of voter model dynamics on annealed small-world networks

TL;DR

This paper analytically investigates the voter model dynamics on annealed small-world networks, revealing a temporal scale that separates disordered and ordered states, with implications for understanding consensus formation.

Contribution

It introduces an analytical approach to the voter model on annealed small-world networks, capturing the dynamics and temporal scales involved.

Findings

The dynamics on annealed networks mirror those on original small-world networks.

A characteristic time scale τ proportional to network size separates disordered and ordered states.

In the thermodynamic limit, the system remains disordered over time.

Abstract

An analytical study of the behavior of the voter model on the small-world topology is performed. In order to solve the equations for the dynamics, we consider an annealed version of the Watts-Strogatz (WS) network, where long-range connections are randomly chosen at each time step. The resulting dynamics is as rich as on the original WS network. A temporal scale $\tau$ separates a quasi-stationary disordered state with coexisting domains from a fully ordered frozen configuration. $\tau$ is proportional to the number of nodes in the network, so that the system remains asymptotically disordered in the thermodynamic limit.