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

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 $q_1\ne q_2$ 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.

arXiv comment6 pages, 2 figures, supplementary material and movie available at https://dx.doi.org/10.6084/m9.figshare.2060595

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• A Heterogeneous Out-of-Equilibrium Nonlinear $q$-Voter Model with Zealotry
• Experimental metrics for detection of detailed balance violation
• Homogeneous symmetrical threshold model with nonconformity: independence vs. anticonformity
• Zealots in multi-state noisy voter models
• Polarization and Consensus in a Voter Model under Time-Fluctuating Influences
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• Diffusion on hierarchical systems of weakly-coupled networks
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• Spatial patterns emerging from a stochastic process near criticality
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• Zealotry and Influence Maximization in the Voter Model: When to Target Zealots?
• Stochastic line integrals and stream functions as metrics of irreversibility and heat transfer
• Competing local and global interactions in social dynamics: how important is the friendship network?
• Breaking coexistence: Zealotry vs. nonlinear social impact
• Broken Detailed Balance and Non-Equilibrium Dynamics in Noisy Social Learning Models