Nonlinear Voter Models: The Transition from Invasion to Coexistence

TL;DR

This paper explores how nonlinear interactions in voter models influence the transition from invasion to coexistence, revealing three distinct regimes and emphasizing the importance of correlations in the dynamics.

Contribution

It develops a stochastic framework and derives macroscopic equations for nonlinear voter models, highlighting the role of correlations in phase transitions.

Findings

Identifies three regimes: invasion, random coexistence, correlated coexistence.

Derives explicit mean-field and pair approximation equations.

Confirms phase diagram predictions with computer simulations.

Abstract

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