Analytical Solution of the Voter Model on Disordered Networks

TL;DR

This paper provides an analytical framework for understanding the voter model dynamics on disordered networks, revealing how network heterogeneity influences the time to reach consensus and the nature of intermediate states.

Contribution

It introduces a mathematical description of the voter model on heterogeneous networks, deriving explicit formulas for quasistationary states and consensus times based on network degree moments.

Findings

For μ ≤ 2, the system reaches complete order exponentially fast.

For μ > 2, the system exhibits a quasistationary active state before ordering.

Analytical results agree well with numerical simulations on various random networks.

Abstract

We present a mathematical description of the voter model dynamics on heterogeneous networks. When the average degree of the graph is $\mu \leq 2$ the system reaches complete order exponentially fast. For $\mu >2$, a finite system falls, before it fully orders, in a quasistationary state in which the average density of active links (links between opposite-state nodes) in surviving runs is constant and equal to $\frac{(\mu-2)}{3(\mu-1)}$, while an infinite large system stays ad infinitum in a partially ordered stationary active state. The mean life time of the quasistationary state is proportional to the mean time to reach the fully ordered state $T$, which scales as $T \sim \frac{(\mu-1) \mu^2 N}{(\mu-2) \mu_2}$, where $N$ is the number of nodes of the network, and $\mu_2$ is the second moment of the degree distribution. We find good agreement between these analytical results and numerical simulations on random networks with various degree distributions.