Phase transitions of Best‐of‐two and Best‐of‐three on stochastic block models

Abstract

This paper is concerned with voting processes on graphs where each vertex holds one of two different opinions. In particular, we study the Best-of-two and the Best-of-three. Here at each synchronous and discrete time step, each vertex updates its opinion to match the majority among the opinions of two random neighbors and itself (the Best-of-two) or the opinions of three random neighbors (the Best-of-three). Previous studies have explored these processes on complete graphs and expander graphs, but we understand significantly less about their properties on graphs with more complicated structures. In this paper, we study the Best-of-two and the Best-of-three on the stochastic block model G(2n, p, q), which is a random graph consisting of two distinct Erdős-Rényi graphs G(n, p) joined by random edges with density q ≤ p. We obtain two main results. First, if p = ω(log n/n) and r = q/p is a constant, we show that there is a phase transition in r with threshold r * (specifically, r * = √ 5 − 2 for the Best-of-two, and r * = 1/7 for the Best-of-three). If r > r * , the process reaches consensus within O(log log n + log n/ log(np)) steps for any initial opinion configuration with a bias of Ω(n). By contrast, if r < r * , then there exists an initial opinion configuration with a bias of Ω(n) from which the process requires at least 2 Ω(n) steps to reach consensus. Second, if p is a constant and r > r * , we show that, for any initial opinion configuration, the process reaches consensus within O(log n) steps. To the best of our knowledge, this is the first result concerning multiple-choice voting for arbitrary initial opinion configurations on non-complete graphs.

Quasi-majority Functional Voting on Expander Graphs

“…w.h.p., where C > 0 is a constant depending only on f . In Corollary 1.8(i), we stress that the worst-case consensus time on G(n, p) was known for p = Ω(1) [39].…”

“…Doerr, Goldberg, Minder, Sauerwald, and Scheideler [21] showed that, on the complete graph K n , the consensus time of best-of-two is O(log n) with high probability 1 for an arbitrary initial opinion configuration. Since best-of-two is simple and is faster than pull voting on the complete graphs, this model gathers special attention in distributed computing and related area [26,15,16,17,19,20,39]. There is a line of works that study best-of-two on expander graphs [15,16,17], which we discuss later.…”

“…Kang and Rivera [29] considered the consensus time of best-of-three on graphs with large minimum degree starting from a random initial configuration. Shimizu and Shiraga [39] showed that, for any initial configurations, best-of-two and best-of-three reach consensus in O(log n) steps w.h.p. if the graph is an Erdős-Rényi graph G(n, p) 2 of p = Ω (1).…”

“…Recently, Cruciani, Natale, and Scornavacca [20] studied best-of-two with a random initial configuration on a clustered regular graph. Shimizu and Shiraga [39] obtained phase-transition results of best-of-two and best-of-three on stochastic block models.…”

Phase Transition of the k-Majority Dynamics in Biased Communication Models

“…From a high level perspective, our results show that adding a bias to k-majority affects the dynamics in a non-trivial way. In particular, the arise of a metastable phase makes the framework suitable to design distributed algorithms to recover planted partitions in networks [CNS19,BCPR19,SS19]. In this direction, we discuss potential applications of the mathematical framework presented in this paper in Section 7.…”