Quasi-majority Functional Voting on Expander Graphs

Abstract

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).

Quasi‐majority functional voting on expander graphs

“…Note that f = H f if f is symmetric. Similar results mentioned in this subsection holds for non-symmetric f (see Lemma 3.5 and 3.6 of the full version [38]). For a C 2 function h : R → R, let 1] |h (x)| be constants 5 The following technical result enables us to estimate E[π(A )] and Var[π(A )] of functional voting.…”

“…See the full version [38] for the proof of Theorem 1.4. Theorem 1.5 (Fast consensus for H f (0) = 0).…”

“…For simplicity, we focus on best-of-(2k + 1) and prove the following result (see the full version [38] for the proof).…”

“…See the full version [38] for other specific results and examples of quasi-majority functional voting.…”

“…In this section, we prove Lemma 2.4 by showing Lemmas 3.2 and 3.3, which are generalizations of Lemma 2.4 in terms of reversible Markov chain. This enables us to evaluate E[π(A )] and Var[π(A )] for functional voting with respect to a C 2 function f (see the full version [38] for functional voting with respect to non-symmetric f ).…”