Best-of-Three Voting on Dense Graphs

TL;DR

This paper analyzes the Best-of-three voting process on dense graphs, showing rapid convergence to consensus when initial bias exceeds 50%, with high probability within logarithmic time bounds.

Contribution

It provides a rigorous proof of fast consensus emergence in dense graphs under the Best-of-three voting dynamics, extending understanding of opinion spread in complex networks.

Findings

Consensus reached within O(log log n) steps

Initial bias greater than 1/2 + δ leads to rapid convergence

High probability of correct final opinion under specified conditions

Abstract

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.