Characterization of the nonequilibrium steady state of a heterogeneous nonlinear q-voter model with zealotry

¹ Department of Applied Mathematics, School of Mathematics, University of Leeds - Leeds LS2 9JT, UK
² Department of Physics & Astronomy, Iowa State University - Ames, IA 50011, USA
³ Department of Physics, Virginia Polytechnic Institute & State University - Blacksburg, VA 24061, USA

Received: 14 January 2016
Accepted: 23 February 2016

Abstract

We introduce a 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 q₁ or q₂. While here zealots never change their opinion, a q_i-susceptible voter 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 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.