Majority Model on Random Regular Graphs
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Consider a graph G=(V,E) and an initial random coloring where each vertex v ∈ 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 G_n,d. It is shown that for all ϵ>0, P_b < 1/2-ϵ results in final complete occupancy by red in O(_d n) rounds with high probability, provided that d≥ c/ϵ^2 for a suitable constant c. 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.
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