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

Read more

Summary

Introduction

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

Previous work
Our results
Result
Strategy
Related work
Best-of-three voting process
Concentration result for the stochastic block model
Induced dynamical system
Derive Theorems 2 and 4
Polynomial voting processes
Results of general induced dynamical systems
Sink point
Fast consensus
Escape from a fixed point
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call