Local majority dynamics on preferential attachment graphs

Abstract

Suppose in a graph G vertices can be either red or blue. Let k be odd. At each time step, each vertex v in G polls k random neighbours and takes the majority colour. If it doesn’t have k neighbours, it simply polls all of them, or all less one if the degree of v is even. We study this protocol on the preferential attachment model of Albert and Barabási [3], which gives rise to a degree distribution that has roughly power-law (Formula Presented.), as well as generalisations which give exponents larger than 3. The setting is as follows: Initially each vertex of G is red independently with probability (Formula Presented.), and is otherwise blue. We show that if α is sufficiently biased away from (Formula Presented.), then with high probability, consensus is reached on the initial global majority within O(logd logd t) steps. Here t is the number of vertices and d ≥ 5 is the minimum of k and m (or m−1 if m is even), m being the number of edges each new vertex adds in the preferential attachment generative process. Additionally, our analysis reduces the required bias of α for graphs of a given degree sequence studied in [1] (which includes, e.g., random regular graphs).