Dynamics of Majority Rule

TL;DR

This paper introduces a majority rule opinion dynamics model, analyzing how consensus is reached in different network structures, revealing that consensus time scales logarithmically in mean-field and varies with dimension in lattices.

Contribution

The study presents a new 2-state opinion model with detailed analysis of consensus times across different network topologies and dimensions.

Findings

Consensus time scales as ln N in mean-field.

On lattices, consensus time varies with dimension and shows strong fluctuations.

Final opinion usually matches initial majority, except in one dimension.

Abstract

We introduce a 2-state opinion dynamics model where agents evolve by majority rule. In each update, a group of agents is specified whose members then all adopt the local majority state. In the mean-field limit, where a group consists of randomly-selected agents, consensus is reached in a time that scales ln N, where N is the number of agents. On finite-dimensional lattices, where a group is a contiguous cluster, the consensus time fluctuates strongly between realizations and grows as a dimension-dependent power of N. The upper critical dimension appears to be larger than 4. The final opinion always equals that of the initial majority except in one dimension.