Zealots in the mean-field noisy voter model

TL;DR

This paper investigates how zealots influence the noisy voter model at the mean-field level, revealing complex phase transitions and symmetry breaking effects caused by the interplay of noise, herding, and fixed-opinion voters.

Contribution

It provides a systematic analytical and numerical analysis of the noisy voter model with zealots, introducing heterogeneity and uncovering new phase behaviors and symmetry-breaking phenomena.

Findings

Zealots can significantly alter the phase transition from consensus to coexistence.

Heterogeneous parameters can be mapped to a system without zealots, simplifying analysis.

The combination of noise, herding, and zealots leads to non-trivial symmetry breaking and new phases.

Abstract

The influence of zealots on the noisy voter model is studied theoretically and numerically at the mean-field level. The noisy voter model is a modification of the voter model that includes a second mechanism for transitions between states: apart from the original herding processes, voters may change their states because of an intrinsic, noisy in origin source. By increasing the importance of the noise with respect to the herding, the system exhibits a finite-size phase transition from a quasi-consensus state, where most of the voters share the same opinion, to a one with coexistence. Upon introducing some zealots, or voters with fixed opinion, the latter scenario may change significantly. We unveil the new situations by carrying out a systematic numerical and analytical study of a fully connected network for voters, but allowing different voters to be directly influenced by different zealots. We show that this general system is equivalent to a system of voters without zealots, but with heterogeneous values of their parameters characterizing herding and noisy dynamics. We find excellent agreement between our analytical and numerical results. Noise and herding/zealotry acting together in the voter model yields not a trivial mixture of the scenarios with the two mechanisms acting alone: it represents a situation where the global-local (noise-herding) competitions is coupled to a symmetry breaking (zealots). In general, the zealotry enhances the effective noise of the system, which may destroy the original quasi--consensus state, and can introduce a bias towards the opinion of the majority of zealots, hence breaking the symmetry of the system and giving rise to new phases ...