Nearly-Optimal Time-Sample Trade-off in Consensus Dynamics
Francesco d'Amore, Joel Rybicki, Nobutaka Shimizu, Takeharu Shiraga, Robin Vacus
How does the number of sampled opinions affect the time required to reach consensus in distributed consensus dynamics? We study the synchronous $h$-Majority, in which every node independently samples $h$ nodes uniformly at random with replacement in each round and adopts a most frequent sampled opinion, breaking ties uniformly at random. We show that $h$-Majority achieves the nearly-optimal trade-off between the sample size and the consensus time, among a broad class of consensus dynamics. In particular, we prove the following results. For every sample size $h \ge 3$ and for every initial configuration, we prove that the process reaches consensus with high probability within $$\widetilde{O}\left(1+\min\left\{\frac{1}{\min\{h^2p_{\max},1\}},\frac{\sqrt{n}}{h}\right\}\right)$$ rounds, where $n$ is the number of nodes and $p_{\max}$ is the largest initial fraction of nodes supporting any single opinion. This is the first bound on the consensus time for $h$-Majority that holds for every $h\ge3$ and every initial configuration. When the initially most popular opinion has a sufficiently large lead over every other opinion, we show that all nodes adopt this opinion with high probability within $O(\log n/\min\{h^2p_{\max},1\})$ rounds. We establish a matching lower bound up to polylogarithmic factors for any fixed update rule that treats all opinions symmetrically and that selects either a sampled opinion or the node's current opinion.
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