Fast Convergence to Unanimity in Dense Erd\H{o}s-R\'enyi Graphs

TL;DR

This paper proves that in dense Erdős-Rényi graphs, a majority dynamics process reaches unanimity in three rounds with high probability, improving previous results that required up to four rounds, and shows three rounds are necessary.

Contribution

The authors demonstrate that adding randomness to the graph in each round reduces the consensus time from four to three rounds, establishing both sufficiency and necessity.

Findings

Consensus is reached in three rounds with high probability.

Three rounds are both necessary and sufficient for unanimity.

Adding randomness accelerates convergence compared to fixed graphs.

Abstract

Majority dynamics on the binomial Erd\H{o}s-R\'enyi graph $\mathsf{G}(n,p)$ with $p=\lambda/\sqrt{n}$ is studied. In this process, each vertex has a state in $\{0,1\}$ and at each round, every vertex adopts the state of the majority of its neighbors, retaining its state in the case of a tie. It was conjectured by Benjamini et al. and proved by Fountoulakis et al. that this process reaches unanimity with high probability in at most four rounds. By adding some extra randomness and allowing the underlying graph to be drawn anew in each communication round, we improve on their result and prove that this process reaches consensus in only three communication rounds with probability approaching $1$ as $n$ grows to infinity. We also provide a converse result, showing that three rounds are not only sufficient, but also necessary.