Majority Model on Random Regular Graphs

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 $\epsilon >0$, $P_b \le 1/2-\epsilon$ results in final complete occupancy by red in $\mathcal {O}(\log _d\log n)$ rounds with high probability, provided that $d\ge c/\epsilon ^2$ for a sufficiently large constant c. We argue that the bound $\mathcal {O}(\log _d\log n)$ is asymptomatically tight. 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 [22].