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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 statein $\{0,1\}$ and at each round, every vertex adopts the state of the majorityof its neighbors, retaining its state in the case of a tie. It was conjecturedby Benjamini et al. and proved by Fountoulakis et al. that this process reachesunanimity with high probability in at most four rounds. By adding some extrarandomness and allowing the underlying graph to be drawn anew in eachcommunication round, we improve on their result and prove that this processreaches consensus in only three communication rounds with probabilityapproaching $1$ as $n$ grows to infinity. We also provide a converse result,showing that three rounds are not only sufficient, but also necessary.

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