Stochastic pair approximation treatment of the noisy voter model

TL;DR

This paper develops a comprehensive stochastic pair approximation framework to analyze the noisy voter model on heterogeneous networks, accurately capturing finite-size effects and critical phenomena.

Contribution

It introduces a full stochastic description for the pair approximation scheme applied to the noisy voter model, including finite-size and correlation effects, validated against simulations.

Findings

Accurate predictions of stationary and dynamic quantities across critical regions.

Finite-size effects cannot be simply modeled by an effective system size.

The approach effectively captures fluctuations and correlations in heterogeneous networks.

Abstract

We present a full stochastic description of the pair approximation scheme to study binary-state dynamics on heterogeneous networks. Within this general approach, we obtain a set of equations for the dynamical correlations, fluctuations and finite-size effects, as well as for the temporal evolution of all relevant variables. We test this scheme for a prototypical model of opinion dynamics known as the noisy voter model that has a finite-size critical point. Using a closure approach based on a system size expansion around a stochastic dynamical attractor we obtain very accurate results, as compared with numerical simulations, for stationary and time dependent quantities whether below, within or above the critical region. We also show that finite-size effects in complex networks cannot be captured, as often suggested, by merely replacing the actual system size $N$ by an effective network dependent size $N_{{\rm eff}}$