Consider a distributed graph where each vertex holds one of two distinct opinions. In this paper, we are interested in synchronous voting processes where each vertex updates its opinion according to a predefined common local updating rule. For example, each vertex adopts the majority opinion among 1) itself and two randomly picked neighbors in best-of-two or 2) three randomly picked neighbors in best-of-three. Previous works intensively studied specific rules including best-of-two and best-of-three individually. In this paper, we generalize and extend previous works of best-of-two and best-of-three on expander graphs by proposing a new model, quasi-majority functional voting. This new model contains best-of-two and best-of-three as special cases. We show that, on expander graphs with sufficiently large initial bias, any quasi-majority functional voting reaches consensus within $O(\log n)$ steps with high probability. Moreover, we show that, for any initial opinion configuration, any quasi-majority functional voting on expander graphs with higher expansion (e.g., Erd\H{o}s-R\'enyi graph $G(n,p)$ with $p=\Omega(1/\sqrt{n})$) reaches consensus within $O(\log n)$ with high probability. Furthermore, we show that the consensus time is $O(\log n/\log k)$ of best-of-$(2k+1)$ for $k=o(n/\log n)$.

中文翻译：

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扩展图上的准多数功能投票

考虑一个分布式图，其中每个顶点持有两种不同意见之一。在本文中，我们对同步投票过程感兴趣，其中每个顶点根据预定义的公共本地更新规则更新其意见。例如，每个顶点在 1) 自身和两个最佳的两个随机选择的邻居或 2) 三个最佳的三个随机选择的邻居之间采用多数意见。以前的作品深入研究了具体规则，包括分别最好的两个和最好的三个。在本文中，我们通过提出一个新模型，准多数函数投票，概括和扩展了先前关于扩展器图的最佳二选一和三选一的工作。这个新模型包含最好的两个和最好的三个作为特殊情况。我们表明，在具有足够大初始偏差的扩展图上，任何准多数功能投票都以高概率在 $O(\log n)$ 步骤内达成共识。此外，我们表明，对于任何初始意见配置，对具有更高扩展的扩展图（例如，Erd\H{o}sR\'enyi 图 $G(n,p)$ 和 $p= \Omega(1/\sqrt{n})$) 在 $O(\log n)$ 内以高概率达成共识。此外，我们表明，当 $k=o(n/\log n)$ 时，共识时间是 $O(\log n/\log k)$ of best-of-$(2k+1)$。