Characterization of the Nonequilibrium Steady State of a Heterogeneous Nonlinear $q$-Voter Model with Zealotry

TL;DR

This paper introduces a heterogeneous nonlinear $q$-voter model with zealots, analyzing its non-equilibrium steady state, probability currents, and correlations, revealing complex behaviors when susceptibility parameters differ.

Contribution

The study characterizes the non-equilibrium steady state of a novel $q$-voter model with zealots, highlighting its violation of detailed balance and rich phase behavior.

Findings

Model exhibits non-zero probability currents in NESS.

Analytical Gaussian approximation matches numerical results.

Distinct regimes of zealotry density influence opinion distribution.

Abstract

We introduce an heterogeneous nonlinear $q$-voter model with zealots and two types of susceptible voters, and study its non-equilibrium properties when the population is finite and well mixed. In this two-opinion model, each individual supports one of two parties and is either a zealot or a susceptible voter of type $q_1$ or $q_2$. While here zealots never change their opinion, a $q_i$-susceptible voter ($i=1,2$) consults a group of $q_i$ neighbors at each time step, and adopts their opinion if all group members agree. We show that this model violates the detailed balance whenever $q_1 \neq q_2$ and has surprisingly rich properties. Here, we focus on the characterization of the model's non-equilibrium stationary state (NESS) in terms of its probability distribution and currents in the distinct regimes of low and high density of zealotry. We unveil the NESS properties in each of these phases by computing the opinion distribution and the circulation of probability currents, as well as the two-point correlation functions at unequal times (formally related to a "probability angular momentum"). Our analytical calculations obtained in the realm of a linear Gaussian approximation are compared with numerical results.