Nonlinear voter models: the transition from invasion to coexistence

A general stochastic framework for frequency dependent processes is developed from which the macroscopic dynamics for key variables, such as global frequencies and correlations, are derived.

Abstract

\n In nonlinear voter models the transitions between two states\n depend in a nonlinear manner on the frequencies of these states in the\n neighborhood. We investigate the role of these nonlinearities on the\n global outcome of the dynamics for a homogeneous network where each\n node is connected to m = 4 neighbors. The paper unfolds in two\n directions. We first develop a general stochastic framework for\n frequency dependent processes from which we derive the macroscopic\n dynamics for key variables, such as global frequencies and\n correlations. Explicit expressions for both the mean-field limit and\n the pair approximation are obtained. We then apply these equations to\n determine a phase diagram in the parameter space that distinguishes\n between different dynamic regimes. The pair approximation allows us to\n identify three regimes for nonlinear voter models: (i) complete\n invasion; (ii) random coexistence; and – most interestingly – (iii)\n correlated coexistence. These findings are contrasted with predictions\n from the mean-field phase diagram and are confirmed by extensive\n computer simulations of the microscopic dynamics. \n\n \n