Divergence and Consensus in Majority Rule

TL;DR

This paper studies how majority rule dynamics lead to consensus or polarization in a two-class population with tunable interactions, revealing critical thresholds and scaling behaviors.

Contribution

It introduces a model with two classes and probabilistic majority rule, analyzing the conditions for consensus and polarization, including the critical interaction threshold.

Findings

Consensus time scales logarithmically with population size for psilon 

Below the critical psilon, the population can become trapped in polarized states

Escape from polarization scales exponentially with population size

Abstract

We investigate majority rule dynamics in a population with two classes of people, each with two opinion states $\pm 1$, and with tunable interactions between people in different classes. In an update, a randomly selected group adopts the majority opinion if all group members belong to the same class; if not, majority rule is applied with probability $\epsilon$. Consensus is achieved in a time that scales logarithmically with population size if $\epsilon\geq \epsilon_c=\frac{1}{9}$. For $\epsilon <\epsilon_c$, the population can get trapped in a polarized state, with one class preferring the $+1$ state and the other preferring $-1$. The time to escape this polarized state and reach consensus scales exponentially with population size.