Nonlinear q-voter model from the quenched perspective

Abstract

We compare two versions of the nonlinear q-voter model: the original one, with annealed randomness, and the modified one, with quenched randomness. In the original model, each voter changes its opinion with a certain probability ǫ if the group of influence is not unanimous. In contrast, the modified version introduces two types of voters that act in a deterministic way in case of disagreement in the influence group: the fraction ǫ of voters always change their current opinion, whereas the rest of them always maintain it. Although both concepts of randomness lead to the same average number of opinion changes in the system on the microscopic level, they cause qualitatively distinct results on the macroscopic level. We focus on the mean-field description of these models. Our approach relies on the stability analysis by the linearization technique developed within dynamical system theory. This approach allows us to derive complete, exact phase diagrams for both models. The results obtained in this paper indicate that quenched randomness promotes continuous phase transitions to a greater extent, whereas annealed randomness favors discontinuous ones. The quenched model also creates combinations of continuous and discontinuous phase transitions unobserved in the annealed model, in which the up-down symmetry may be spontaneously broken inside or outside the hysteresis loop. The analytical results are confirmed by Monte Carlo simulations carried out on a complete graph.The subject of the opinion dynamics is studied within social and computer science, mathematics, statistical physics, and engineering, and models of binary opinions are of particular interest within all these disciplines. Among them the q-voter model is one of the most general one, reducing for specific values of parameters to other models, including the famous voter model. In the original q-voter model all agents are identical and can change randomly their behavior in time, which corresponds to the so-called annealed approach. In this paper, we reformulate the model under the quenched approach, i.e., we assign to agents some individual traits that remain constant in time, and ask the question about the role of the approach in shaping macroscopic behavior of the model. To answer the question, we compare two versions of the model, the quenched and the annealed one. Such a comparison is an important issue in statistical physics because the annealed approach is much simpler for the analytical treatment, and it is often used as an approximation of the real quenched system. Moreover, it may be also interesting from the social point of view because it corresponds to the famous personsituation debate.

Switching from a continuous to discontinuous phase transition under the quenched disorder

“…This result is not only suprising and novel but also open doors to new scientific questions. One of them is already hot topic in both social and physics world, namely if there is universality in existence of discontinuous phase transition and hysteresis among different model of opinion dynamics like voter models [1,2,12,14,21,37,[41][42][43][44][45][46][47][48][49] or the majority-vote models [27-30, 34, 35, 50-53]. Such universality can be seen if we compare majorityvote model with q-voter model with independence, where introducing just 3 states result in discontinuous phase transistion.…”

“…The voter with probability p is set as an anticonformist permanently and with complementary probability 1 − p is set as a conformist one. Thus, we can distinguish two types of agents [21,37] and for each type we introduce the concentration of agents in given state α separately, c (A,α) for anticonformist and c (C,α) for conformist agents respectively. Therefore total concentration of voters in state α is given by…”

“…, s−1 respectively similarly as for annealed scheme. The stability of given stationary point can be track by the determinant and trace of appropriate Jacobian matrix with respect to this point [30,37,38]…”

Discontinuous phase transitions in the q-voter model with generalized anticonformity on random graphs

“…From the social point of view, s(x,t) represents an opinion of an agent placed at node x at time t on a given topic measured in the two-point psychometric scale (yes/no, agree/disagree). We use j to describe the state of a single spin for consistency with earlier papers in which PA was used for the q-voter model 26,39 .…”

Pair approximation for the q-voter models with quenched disorder on networks

“…In low-dimensional systems, the presence of quenched disorder leads to the elimination of discontinuous phase transitions known as a rounding effect [9][10][11]. In the mean-field regime, on the other hand, discontinuous phase transitions may survive the introduction of quenched disorder [12,13], or they may appear after its introduction [14]. The change of disorder may also not impact the phase transitions at all [15].…”