Asynchronous 3-Majority Dynamics with Many Opinions

TL;DR

This paper analyzes the asynchronous 3-Majority consensus process on complete graphs with multiple opinions, establishing tight bounds on the time needed for all vertices to agree, which was previously unknown for large opinion counts.

Contribution

It provides the first tight bounds on consensus time for asynchronous 3-Majority dynamics with many opinions, extending analysis beyond the case of two opinions.

Findings

Consensus time is ((nk),n^{1.5}) for all k.

Bounds are tight up to polylogarithmic factors.

Analysis extends understanding of asynchronous majority dynamics with many opinions.

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 $\tilde{\Theta}( \min(nk,n^{1.5}) )$ for all $k$. These are the first bounds for all $k$ that are tight up to a polylogarithmic factor.