Zealots in multistate noisy voter models

First-passage distributions of an asymmetric noisy voter model

This paper explores the first-passage times in an asymmetric noisy voter model through analytical methods. The noise in the model leads to bistable behavior, and the asymmetry arises from heterogeneous rates for spontaneous switching. We obtain exact analytical expressions for the probability distribution for two different initial conditions, first-passage times for switching transitions and first return times to a stable state for all system sizes, offering a deeper understanding of the model's dynamics. Additionally, we derive exact expressions for the mean switching time, mean return time, and their mean square variants. The findings are verified through numerical simulations. To enhance clarity regarding the model's behavior, we also provide approximate solutions, emphasizing the parameter dependence of first-passage times in the small switching parameter regime. An interesting result in this regime is that while the mean switching time in the leading order is independent of system size, the mean return time depends inversely on system size. This study not only advances our analytical understanding of the asymmetric noisy voter model but also establishes a framework for exploring similar phenomena in social and biological systems where the noisy voter model is applicable.

Noisy voter models in switching environments

We study the stationary states of variants of the noisy voter model, subject to fluctuating parameters or external environments. Specifically, we consider scenarios in which the herding-to-noise ratio switches randomly and on different timescales between two values. We show that this can lead to a phase in which polarized and heterogeneous states exist. Second, we analyze a population of noisy voters subject to groups of external influencers, and show how multipeak stationary distributions emerge. Our work is based on a combination of individual-based simulations, analytical approximations in terms of a piecewise-deterministic Markov processes (PDMP), and on corrections to this process capturing intrinsic stochasticity in the linear-noise approximation. We also propose a numerical scheme to obtain the stationary distribution of PDMPs with three environmental states and linear velocity fields.

Local and global ordering dynamics in multistate voter models

We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multistate voter models with all-to-all interaction and on uncorrelated networks. Individual realizations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a metastable state. The density of active links and the entropy in these states varies from realization to realization. Making some simple assumptions we are able to analytically calculate the average density of active links and the average entropy in each of these states. We also show that, averaged over realizations, the density of active links decays exponentially, with a timescale set by the size and geometry of the graph, but independent of the initial number of opinion states. The decay of the average entropy is exponential only at long times when there are at most two opinions left in the population. Finally, we show how metastable states comprising only a subset of opinions can be artificially engineered by introducing precisely one zealot in each of the prevailing opinions.

Consensus, Polarization and Hysteresis in the Three-State Noisy q-Voter Model with Bounded Confidence

In this work, we address the question of the role of the influence of group size on the emergence of various collective social phenomena, such as consensus, polarization and social hysteresis. To answer this question, we study the three-state noisy q-voter model with bounded confidence, in which agents can be in one of three states: two extremes (leftist and rightist) and centrist. We study the model on a complete graph within the mean-field approach and show that, depending on the size q of the influence group, saddle-node bifurcation cascades of different length appear and different collective phenomena are possible. In particular, for all values of q>1, social hysteresis is observed. Furthermore, for small values of q∈(1,4), disagreement, polarization and domination of centrists (a consensus understood as the general agreement, not unanimity) can be achieved but not the domination of extremists. The latter is possible only for larger groups of influence. Finally, by comparing our model to others, we discuss how a small change in the rules at the microscopic level can dramatically change the macroscopic behavior of the model.

Consensus, polarization, and coexistence in a continuous opinion dynamics model with quenched disorder

A model of opinion dynamics is introduced in which each individual's opinion is measured on a bounded continuous spectrum. Each opinion is influenced heterogeneously by every other opinion in the population. It is demonstrated that consensus, polarization and a spread of moderate opinions are all possible within this model. Using dynamic mean-field theory, we are able to identify the statistical features of the interactions between individuals that give rise to each of the aforementioned emergent phenomena. The nature of the transitions between each of the observed macroscopic states is also studied. It is demonstrated that heterogeneity of interactions between individuals can lead to polarization, that mostly antagonistic or contrarian interactions can promote consensus at a moderate opinion, and that mostly reinforcing interactions encourage the majority to take an extreme opinion.