Majority Model on Random Regular Graphs

Consider a graph G = (V, E) and an initial random coloring where each vertex v is an element of V is blue with probability Pb 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 G(n, d). It is shown that for epsilon > 0, P-b <= 1/2 - epsilon results in final complete occupancy by red in O(log(d) log n) rounds with high probability, provided that d >= c/epsilon(2) for a sufficiently large constant c. We argue that the bound O(log(d) log n) is asymptomatically tight. Furthermore, we show that with high probability, 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].