Simple Majority Consensus in Networks with Unreliable Communication

TL;DR

This paper demonstrates that a simple majority protocol in large, unreliable networks rapidly reaches consensus within three rounds, with the speed depending on the initial opinion difference and network size.

Contribution

It provides probabilistic analysis showing the efficiency of the simple majority protocol in unreliable, large-scale networks, establishing both sufficiency and necessity of certain conditions for rapid consensus.

Findings

Consensus achieved in three rounds with high probability as network size grows.

Two rounds suffice if opinion difference scales with √n.

One round achieves consensus if opinion difference exceeds √n, with probability approaching 1 exponentially.

Abstract

In this work, we analyze the performance of a simple majority-rule protocol solving a fundamental coordination problem in distributed systems - \emph{binary majority consensus}, in the presence of probabilistic message loss. Using probabilistic analysis for a large scale, fully-connected, network of $2n$ agents, we prove that the Simple Majority Protocol (SMP) reaches consensus in only three communication rounds with probability approaching $1$ as $n$ grows to infinity. Moreover, if the difference between the numbers of agents that hold different opinions grows at a rate of $\sqrt{n}$, then the SMP with only two communication rounds attains consensus on the majority opinion of the network, and if this difference grows faster than $\sqrt{n}$, then the SMP reaches consensus on the majority opinion of the network in a single round, with probability converging to $1$ exponentially fast as $n \rightarrow \infty$. We also provide some converse results, showing that these requirements are not only sufficient, but also necessary.