稠密Erdens-Rényi图一致性的快速收敛
Fast Convergence to Unanimity in Dense Erdős-Rényi Graphs
摘要：
研究了具有$p=λ/\sqrt{n}$的二项Erdens-Rényi图$\mathsf{G}(n,p)$的多数动力学.在这个过程中，每个顶点在$\{0,1\}$中有一个状态，并且在每一轮中，每个顶点采用其大多数邻居的状态，在平局的情况下保留其状态。Benjamini等人提出并由Fountoulakis等人证明，该过程在最多四轮中以高概率达到连续性。通过添加一些额外的随机性，并允许底层图形在每个通信轮中重新绘制，我们改进了他们的结果，并证明了这个过程仅在三个通信轮中达到共识，随着$n$增长到无穷大，概率接近$1$。我们还提供了一个相反的结果，表明三轮不仅是充分的，而且是必要的。
英文摘要：
Majority dynamics on the binomial Erdős-Rényi graph $\mathsf{G}(n,p)$ with $p=λ/\sqrt{n}$ 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.
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