Analytical and numerical study of the non-linear noisy voter model on complex networks

TL;DR

This paper analyzes a non-linear noisy voter model on complex networks, revealing phase transitions and finite-size effects, with exact solutions for all-to-all networks and approximate results for complex networks, confirmed by simulations.

Contribution

It introduces a non-linear dependence of transition rates in the noisy voter model and provides exact and approximate solutions, highlighting the impact of network topology on phase transitions.

Findings

Non-linear interactions induce phase transitions surviving in the thermodynamic limit.

Network heterogeneity shifts transition lines and affects finite-size scaling.

Theoretical predictions are confirmed by numerical simulations.

Abstract

We study the noisy voter model using a specific non-linear dependence of the rates that takes into account collective interaction between individuals. The resulting model is solved exactly under the all-to-all coupling configuration and approximately in some random networks environments. In the all-to-all setup we find that the non-linear interactions induce "bona fide" phase transitions that, contrary to the linear version of the model, survive in the thermodynamic limit. The main effect of the complex network is to shift the transition lines and modify the finite-size dependence, a modification that can be captured with the introduction of an effective system size that decreases with the degree heterogeneity of the network. While a non-trivial finite-size dependence of the moments of the probability distribution is derived from our treatment, mean-field exponents are nevertheless obtained in the thermodynamic limit. These theoretical predictions are well confirmed by numerical simulations of the stochastic process.