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Analytical Solution of the Voter Model on Uncorrelated Networks

Vazquez, F.; Eguiluz, V. M.
New Journal of Physics 10 No.6, 063011 (1-19) (2008)

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.