Abstract
AbstractThis 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 round, 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). In this study, we consider 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 a density q ≤ p. We prove phase transition results for these processes: there is a threshold r∗ such that, if q/p > r∗ then the process reaches consensus within rounds and the process requires rounds if q/p < r∗. For the Best‐of‐two and Best‐of‐three, the thresholds are and r∗ = 1/7, respectively.
Highlights
This paper is concerned with voting processes on distributed networks
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
A voting process is defined by a local updating rule: Each vertex updates its opinion according to the rule
Summary
This paper is concerned with voting processes on distributed networks. Consider an undirected connected graph G = (V, E) where each vertex v ∈ V initially holds an opinion from a finite set. A voting process is defined by a local updating rule: Each vertex updates its opinion according to the rule. Voting processes appear as simple mathematical models in a wide range of fields, e.g. social behavior, physical phenomena and biological systems [32, 30, 4]. In distributed computing, voting processes are known as a simple approach for consensus problems [20, 23]. Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany. 32:2 Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have