Stabilizing Consensus with Many Opinions

TL;DR

This paper proves that the 3-majority consensus protocol reliably converges to a valid opinion in polynomial time, even with multiple opinions and adversarial interference, extending previous results beyond binary opinions.

Contribution

It demonstrates the robustness and efficiency of the 3-majority dynamics for multiple opinions under adversarial conditions, a significant extension of prior binary-only results.

Findings

Convergence occurs in polynomial time for up to n^α opinions.

The protocol tolerates adversaries affecting o(√n) nodes per round.

Extends robustness results from binary to multiple opinions.

Abstract

We consider the following distributed consensus problem: Each node in a complete communication network of size $n$ initially holds an \emph{opinion}, which is chosen arbitrarily from a finite set $\Sigma$. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be \emph{valid}, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of an adaptive, malicious adversary who can modify the opinions of a bounded number of nodes in every round. We consider the \emph{3-majority dynamics}: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let $k$ be the number of valid opinions. We show that, if $k \leqslant n^{\alpha}$, where $\alpha$ is a suitable positive constant, the 3-majority dynamics converges in time polynomial in $k$ and $\log n$ with high probability even in the presence of an adversary who can affect up to $o(\sqrt{n})$ nodes at each round. Previously, the convergence of the 3-majority protocol was known for $|\Sigma| = 2$ only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for $|\Sigma| > 2$.