Resolution of a conjecture on majority dynamics: rapid stabilisation in dense random graphs

Abstract

We study majority dynamics on the binomial random graph $G(n,p)$ with $p=d/n$ and $d>\lambda n^{1/2}$, for some large $\lambda >0$. In this process, each vertex has a state in $\{-1,+1\}$ and at each round every vertex adopts the state of the majority of its neighbours, retaining its state in the case of a tie. We show that with high probability the process reaches unanimity in at most four rounds. This confirms a conjecture of Benjamini, Chan, O' Donnel, Tamuz and Tan.

21 pages