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 $\alpha = \Omega\left( (\log \log n)^{-1} \right)$. We prove that if initially each vertex is red with probability greater than $1/2+\delta$, and blue otherwise, where $\delta \geq (\log d)^{-C}$ 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.