Analytical Solution of the Voter Model on Disordered Networks

Abstract

We present a mathematical description of the voter model dynamics on heterogeneous networks. When the average degree of the graph is $μ\leq 2$ the system reaches complete order exponentially fast. For $μ>2$, 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 $\frac{(μ-2)}{3(μ-1)}$, 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 $T \sim \frac{(μ-1) μ^2 N}{(μ-2) μ_2}$, where $N$ is the number of nodes of the network, and $μ_2$ 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.

20 pages, 8 figures