Global majority consensus by local majority polling on graphs of a given degree sequence

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 does not 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 graphs of a given degree sequence, in the following setting: initially each vertex of G is red independently with probability alpha < 1/2, and is otherwise blue. We show that if a is sufficiently biased, then with high probability consensus is reached on the initial global majority within O(log(k) log(k) n) steps if 5 <= k <= d, and O(log(d) log(d) n) steps if k > d. Here, d >= 5 is the effective minimum degree, the smallest integer which occurs Theta(n) times in the degree sequence. We further show that on such graphs, any local protocol in which a vertex does not change colour if all its neighbours have that same colour, takes time at least Omega (log(d) log(d) n), with high probability. Additionally, we demonstrate how the technique for the above sparse graphs can be applied in a straightforward manner to get bounds for the Erdos-Renyi random graphs in the connected regime. (C) 2014 Elsevier B.V. All rights reserved.