Best-of-Three Voting on Dense Graphs

Given a graph $G$ of $n$ vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Best-of-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree $d=n^{\alpha }$,, where $\alpha =\mathrm{\Omega }((\log \log n{)}^{-1})$. We prove that if initially each vertex is red with probability greater than $1\mathrm{/}2+\delta$, and blue otherwise, where $\delta \ge (\log d{)}^{-C}$ for some $C>0$, then with high probability this dynamic reaches a final state where all vertices are red within $O\left( \log \log n\right) + O\left( \log \left( \delta^{-1} \right) \right)$ steps.