Tight Analysis for the 3-Majority Consensus Dynamics

TL;DR

This paper provides a tight analysis of the 3-majority consensus dynamics, proving that it converges within an optimal bound of O(k log n) rounds even against strong adversaries, resolving a key open question.

Contribution

The authors establish a tight, optimal bound on the convergence time of the 3-majority consensus process under adversarial conditions, improving upon previous results.

Findings

Convergence occurs within O(k log n) rounds even with strong adversaries.

The bound is proven to be optimal, matching known lower bounds.

The analysis answers an open question from prior research.

Abstract

We present a tight analysis for the well-studied randomized 3-majority dynamics of stabilizing consensus, hence answering the main open question of Becchetti et al. [SODA'16]. Consider a distributed system of n nodes, each initially holding an opinion in {1, 2, ..., k}. The system should converge to a setting where all (non-corrupted) nodes hold the same opinion. This consensus opinion should be \emph{valid}, meaning that it should be among the initially supported opinions, and the (fast) convergence should happen even in the presence of a malicious adversary who can corrupt a bounded number of nodes per round and in particular modify their opinions. A well-studied distributed algorithm for this problem is the 3-majority dynamics, which works as follows: per round, each node gathers three opinions --- say by taking its own and two of other nodes sampled at random --- and then it sets its opinion equal to the majority of this set; ties are broken arbitrarily, e.g., towards the node's own opinion. Becchetti et al. [SODA'16] showed that the 3-majority dynamics converges to consensus in O((k^2\sqrt{\log n} + k\log n)(k+\log n)) rounds, even in the presence of a limited adversary. We prove that, even with a stronger adversary, the convergence happens within O(k\log n) rounds. This bound is known to be optimal.