# Consensus and diversity in multi-state noisy voter models

## TL;DR

This paper analyzes a multi-opinion noisy voter model with all-to-all interactions, revealing a noise-driven transition between consensus and mixed opinion states, with exact and approximate analytical characterizations of the stationary distribution.

## Contribution

It introduces an effective death-birth process to analytically study the stationary distribution and transition dynamics in multi-state noisy voter models.

## Key findings

- Identifies a noise-driven transition between consensus and mixed states.
- Provides exact solutions for homogeneous rates and approximations for heterogeneous rates.
- Derives mean switching times between consensus states.

## Abstract

We study a variant of the voter model with multiple opinions; individuals can imitate each other and also change their opinion randomly in mutation events. We focus on the case of a population with all-to-all interaction. A noise-driven transition between regimes with multi-modal and unimodal stationary distributions is observed. In the former, the population is mostly in consensus states; in the latter opinions are mixed. We derive an effective death-birth process, describing the dynamics from the perspective of one of the opinions, and use it to analytically compute marginals of the stationary distribution. These calculations are exact for models with homogeneous imitation and mutation rates, and an approximation if rates are heterogeneous. Our approach can be used to characterize the noise-driven transition and to obtain mean switching times between consensus states.