Solution of the Voter Model by Spectral Analysis

Abstract

An exact spectral analysis of the Markov Propagator for the Voter model is presented for the complete graph, and extended to the complete bipartite graph and uncorrelated random networks. Using a well-defined Martingale approximation in diffusion-dominated regions of phase space, which is almost everywhere for the Voter model, this method is applied to compute analytically several key quantities such as exact expressions for the $m$ time step propagator of the Voter model, all moments of consensus times, and the local times for each macrostate. This spectral method is motivated by a related method for solving the Ehrenfest Urn problem and by formulating the Voter model on the complete graph as an Urn model. Comparisons of the analytical results from the spectral method and numerical results from Monte-Carlo simulations are presented to validate the spectral method.