Local and global ordering dynamics in multi-state voter models

TL;DR

This paper analyzes how opinions evolve over time in multi-state voter models on networks, revealing exponential decay of active links and entropy, and how zealots can create stable opinion subsets.

Contribution

It provides analytical calculations of active links and entropy in multi-state voter models, and demonstrates how zealots can engineer meta-stable opinion states.

Findings

Active links decay exponentially with network size and structure.

Entropy decreases exponentially at long times with at most two opinions.

Introducing zealots can create stable opinion subsets.

Abstract

We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multi-state voter models with all-to-all interaction and on uncorrelated networks. Individual realisations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a meta-stable state. The density of active links and the entropy in these states varies from realisation to realisation. 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 realisations, the density of active links decays exponentially, with a time scale 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 meta-stable states comprised of only a subset of opinions can be artificially engineered by introducing precisely one zealot in each of the prevailing opinions.