Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asynchronous 3-Majority Dynamics with Many Opinions

Colin Cooper, Frederik Mallmann-Trenn, Tomasz Radzik, Nobutaka Shimizu, Takeharu Shiraga

TL;DR

It is proved that the consensus time is $\tilde{\Theta}( \min(nk,n^{1.5}) )$ for all $k$.

Abstract

We consider 3-Majority, a probabilistic consensus dynamics on a complete graph with $n$ vertices, each vertex starting with one of $k$ initial opinions. At each discrete time step, a vertex $u$ is chosen uniformly at random. The selected vertex $u$ chooses three neighbors $v_1,v_2,v_3$ uniformly at random with replacement and takes the majority opinion held by the three, where ties are broken in favor of the opinion of $v_3$. The main quantity of interest is the consensus time, the number of steps required for all vertices to hold the same opinion. This asynchronous version turns out to be considerably harder to analyze than the synchronous version and so far results have only been obtained for $k=2$. Even in the synchronous version the results for large $k$ are far from tight. In this paper we prove that the consensus time is $\tildeΘ( \min(nk,n^{1.5}) )$ for all $k$. These are the first bounds for all $k$ that are tight up to a polylogarithmic factor.

Asynchronous 3-Majority Dynamics with Many Opinions

TL;DR

It is proved that the consensus time is for all .

Abstract

We consider 3-Majority, a probabilistic consensus dynamics on a complete graph with vertices, each vertex starting with one of initial opinions. At each discrete time step, a vertex is chosen uniformly at random. The selected vertex chooses three neighbors uniformly at random with replacement and takes the majority opinion held by the three, where ties are broken in favor of the opinion of . The main quantity of interest is the consensus time, the number of steps required for all vertices to hold the same opinion. This asynchronous version turns out to be considerably harder to analyze than the synchronous version and so far results have only been obtained for . Even in the synchronous version the results for large are far from tight. In this paper we prove that the consensus time is for all . These are the first bounds for all that are tight up to a polylogarithmic factor.

Paper Structure

This paper contains 28 sections, 39 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Result
  3. Related Work
  4. Proof Overview and Technical Contribution
  5. Key Results
  6. Many Opinion Case
  7. Small Opinion Case
  8. Our Tool: Drift Analysis via the Freedman Inequality
  9. Overview of Gap Emerging Phase
  10. Organization
  11. Preliminaries
  12. Unique Strong Opinion Lemma
  13. Weak cannot become strong
  14. Pair of non-weak opinions
  15. From Unique Strong Opinion to Consensus
  16. ...and 13 more sections

Key Result

Theorem 1.1

For any $k\ge 2$, the asynchronous 3-Majority dynamics with $k$ opinions on an $n$-vertex complete graph reaches consensus within $\tilde{O}\lparen*\rparen{\min\lparen*\rparen{kn, n^{1.5} }}$ steps with high probability. Moreover, there exists an initial configuration on which the dynamics requires

Figures (1)

  • Figure 1: An overview of the proof of \ref{['thm:all k']}.

Theorems & Definitions (86)

  • Theorem 1.1: Main Theorem
  • Theorem 2.1
  • Theorem 2.2
  • proof : Proof of \ref{['thm:all k']} using \ref{['thm:many opinions', 'thm:small k']}.
  • Lemma 2.3
  • Definition 3.1
  • Lemma 3.1
  • proof : Proof of \ref{['item:expectation of alpha']}.
  • Lemma 4.1: Unique Strong Opinion Lemma
  • Lemma 4.2: Weak cannot become strong
  • ...and 76 more