Analytical solution of the voter model on uncorrelated networks

Abstract

We present a mathematical description of the voter model dynamics on heterogeneous networks. When the average degree of the graph is μ ≤ 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 [(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 ~ [(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.