Majority Model on Random Regular Graphs

TL;DR

This paper analyzes a majority-based coloring process on random regular graphs, showing rapid convergence to a single color under certain conditions and proving the graph's immunity to small initial influence sets.

Contribution

It provides the first analysis of the majority model on random regular graphs, establishing conditions for rapid consensus and demonstrating immunity to small initial influence sets.

Findings

Red dominates in logarithmic rounds when initial blue probability is below 1/2 minus epsilon.

Random regular graphs are immune, with large minimal influence sets.

Answers an open question of Peleg regarding dynamic monopolies.

Abstract

Consider a graph $G=(V,E)$ and an initial random coloring where each vertex $v \in V$ is blue with probability $P_b$ and red otherwise, independently from all other vertices. In each round, all vertices simultaneously switch their color to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. The main goal of the present paper is to analyze the behavior of this basic and natural process on the random $d$-regular graph $\mathbb{G}_{n,d}$. It is shown that for all $\epsilon>0$, $P_b \le 1/2-\epsilon$ results in final complete occupancy by red in $\mathcal{O}(\log_d\log n)$ rounds with high probability, provided that $d\geq c/\epsilon^2$ for a suitable constant $c$. Furthermore, we show that with high probability, $\mathbb{G}_{n,d}$ is immune; i.e., the smallest dynamic monopoly is of linear size. A dynamic monopoly is a subset of vertices that can take over in the sense that a commonly chosen initial color eventually spreads throughout the whole graph, irrespective of the colors of other vertices. This answers an open question of Peleg.