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.

翻译：Benjami et al. 和 Fountoulakis 等人 的推论和 Fountulakis 等人 证明, 这一过程在最多四轮中达到一致的可能性很高。 通过增加一些额外的随机性,允许在每轮通信中重新绘制基本图表,我们改进了结果,并证明这一进程只在三轮通信中达成共识,其可能性随着美元增长到无限程度而接近1美元。我们还提供了一个反向结果,显示三轮不仅足够,而且必要。