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

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 qi-susceptible voter (i = 1, 2) consults a group of qi neighbors at each time step, and adopts their opinion if all group members agree. We show that this model violates the detailed balance whenever q1 = q2 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. editor’s choice Copyright c © EPLA, 2016 Introduction. – Since Schelling’s pioneering work [1] there has been increasing interest in using simple theoretical models to describe social phenomena such as the dynamics of opinions [2]. In this context, individual-based models commonly used in statistical physics are particularly insightful, as they reveal the micro-macro connection in social dynamics [1,2]. The voter model (VM) [3] serves as a reference to describe the evolution of opinions in socially interacting populations. (See, e.g., [2,4] and references therein.) In spite of its paradigmatic role, the VM relies on a number of unrealistic assumptions, such as the total lack of self-confidence of all voters and their perfect conformity, which invariably leads to a consensus. In fact, it has been shown that members of a society respond differently to stimuli and this greatly influences the underlying social dynamics [5–7]. An approach to model a population with different levels of confidence is to assume that some agents are “zealots” who favor one opinion [8] or maintain a fixed opinion [9]. Since the introduction of these simple types of behavior in the VM, a large variety of zealot models have been studied, see, e.g., refs. [10]. In this letter, we focus on a variant of the VM known as the two-state nonlinear q-voter model (qVM) [11] which has attracted much interest [12]. In the qVM, a voter can be influenced by a group of q neighbors. The version with q = 2 is closely related to the well-known models of refs. [13]. Motivated by important psychology and sociology tenets [5,6], the basic ideas underlying the qVM and zealotry have been combined into the q-voter model with inflexible zealots (qVMZ) [14]. Indeed, social scientists have established that conformity by imitation, an important mechanism for collective actions, is observed only when the group size is large and can be altered by individuals who are able to resist the group pressure [6,7]. It is understandable that social conformity is unlikely for small groups and can be significantly suppressed by the presence of zealots. In the qVMZ dynamics, both group size limited conformity and zealotry are accounted for. Furthermore, in a well-mixed setting, this dynamics obeys detailed balance, so that the exact stationary distribution is easily found [14]. The system resembles one in thermal equilibrium, characterized by two phases: As zealotry is lowered