Global majority consensus by local majority polling on graphs of a given degree sequence
Suppose in a graph G G vertices can be either red or blue. Let k k be odd. At each time step, each vertex v v in G G polls k k random neighbours and takes the majority colour. If it does not have k k neighbours, it simply polls all of them, or all less one if the degree of v v is even. We study this protocol on graphs of a given degree sequence, in the following setting: initially each vertex of G G is red independently with probability ?  d k > d . Here, d?5 d ? 5 is the effective minimum degree, the smallest integer which occurs ?(n) ? ( 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 ?(logdlogdn) ? ( 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 Erd?sRnyi random graphs in the connected regime.
consensus local majority social networks distributed computing