Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs

TL;DR

This paper studies how opinions evolve in majority dynamics on various graph types, showing convergence or oscillation on unimodular graphs, and analyzing consensus behavior on Erdős-Rényi and random regular graphs.

Contribution

It extends understanding of majority dynamics to infinite unimodular graphs and random graphs, revealing conditions for convergence, oscillation, and disagreement.

Findings

Agents almost surely converge or oscillate on unimodular transitive graphs.

On Erdős-Rényi graphs, agents tend to reach the initial majority opinion.

On random 4-regular graphs, agents often converge to different opinions.

Abstract

In majority dynamics, agents located at the vertices of an undirected simple graph update their binary opinions synchronously by adopting those of the majority of their neighbors. On infinite unimodular transitive graphs (e.g., Cayley graphs), when initial opinions are chosen from a distribution that is invariant with respect to the graph automorphism group, we show that the opinion of each agent almost surely either converges, or else eventually oscillates with period two; this is known to hold for finite graphs, but not for all infinite graphs. On Erd\H{o}s-R\'enyi random graphs with degrees $\Omega(\sqrt{n})$, we show that when initial opinions are chosen i.i.d. then agents all converge to the initial majority opinion, with constant probability. Conversely, on random 4-regular finite graphs, we show that with high probability different agents converge to different opinions.