Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion

TL;DR

This paper demonstrates that the complex dynamics of voter models on networks can be exactly reduced to a single-coordinate diffusion process, simplifying analysis of ordering behavior.

Contribution

It introduces a universal diffusion reduction for voter models on arbitrary networks, depending on only two characteristic timescales, regardless of network complexity.

Findings

The reduction applies under certain conditions, simplifying the analysis of ordering.

The method identifies key timescales from pair dynamics.

Degree correlations significantly influence ordering times.

Abstract

We study the voter model and related random-copying processes on arbitrarily complex network structures. Through a representation of the dynamics as a particle reaction process, we show that a quantity measuring the degree of order in a finite system is, under certain conditions, exactly governed by a universal diffusion equation. Whenever this reduction occurs, the details of the network structure and random-copying process affect only a single parameter in the diffusion equation. The validity of the reduction can be established with considerably less information than one might expect: it suffices to know just two characteristic timescales within the dynamics of a single pair of reacting particles. We develop methods to identify these timescales, and apply them to deterministic and random network structures. We focus in particular on how the ordering time is affected by degree correlations, since such effects are hard to access by existing theoretical approaches.