The non-linear q-voter model

TL;DR

This paper introduces a non-linear q-voter model, analyzes its phase transitions on a fully connected network, and explores how different parameters influence consensus formation and ordering dynamics.

Contribution

It presents a novel non-linear variant of the voter model, providing analytical solutions and uncovering new types of ordering dynamics specific to mean-field systems.

Findings

Disordered phase exists at high epsilon, ordered at low epsilon.

Three transition scenarios identified: generalized voter, Ising-like and percolation, and initial-condition-dependent regimes.

Fluctuations eliminate the intermediate regime in spatially extended systems.

Abstract

We introduce a non-linear variant of the voter model, the q-voter model, in which q neighbors (with possible repetition) are consulted for a voter to change opinion. If the q neighbors agree, the voter takes their opinion; if they do not have an unanimous opinion, still a voter can flip its state with probability $\epsilon$. We solve the model on a fully connected network (i.e. in mean-field) and compute the exit probability as well as the average time to reach consensus. We analyze the results in the perspective of a recently proposed Langevin equation aimed at describing generic phase transitions in systems with two ($Z_2$ symmetric) absorbing states. We find that in mean-field the q-voter model exhibits a disordered phase for high $\epsilon$ and an ordered one for low $\epsilon$ with three possible ways to go from one to the other: (i) a unique (generalized voter-like) transition, (ii) a series of two consecutive Ising-like and directed percolation transition, and (iii) a series of two transitions, including an intermediate regime in which the final state depends on initial conditions. This third (so far unexplored) scenario, in which a new type of ordering dynamics emerges, is rationalized and found to be specific of mean-field, i.e. fluctuations are explicitly shown to wash it out in spatially extended systems.