Quasi-majority Functional Voting on Expander Graphs
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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., Erdős-Rényi graph G(n,p) with p=Ω(1/√(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).
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