3-Majority and 2-Choices with Many Opinions

2025-03-04

AI Summary

This paper studies the multi-opinion synchronous consensus problem on the complete graph: $n$ nodes initially hold opinions from ${1,dots,k}$, and in each round update their opinions synchronously via either the 3-Majority or 2-Choices protocol, aiming for global agreement on a single opinion. Using probabilistic analysis, stochastic process modeling, coupling techniques, and potential function methods, we establish the first tight consensus time bounds for arbitrary $k in [2,n]$: $widetilde{Theta}(min{k,sqrt{n}})$ for 3-Majority and $widetilde{Theta}(k)$ for 2-Choices. These results unify and strictly improve all prior piecewise upper bounds, providing the first near-optimal theoretical characterization valid across the entire range of $k$. Our analysis significantly advances the understanding of convergence dynamics in multi-opinion consensus processes.

Technology Category

• Multiagent Systems: Agreement, Argumentation & Negotiation
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We present the first nearly-optimal bounds on the consensus time for the well-known synchronous consensus dynamics, specifically 3-Majority and 2-Choices, for an arbitrary number of opinions. In synchronous consensus dynamics, we consider an $n$-vertex complete graph with self-loops, where each vertex holds an opinion from ${1,dots,k}$. At each discrete-time round, all vertices update their opinions simultaneously according to a given protocol. The goal is to reach a consensus, where all vertices support the same opinion. In 3-Majority, each vertex chooses three random neighbors with replacement and updates its opinion to match the majority, with ties broken randomly. In 2-Choices, each vertex chooses two random neighbors with replacement. If the selected vertices hold the same opinion, the vertex adopts that opinion. Otherwise, it retains its current opinion for that round. Improving upon a line of work [Becchetti et al., SPAA'14], [Becchetti et al., SODA'16], [Berenbrink et al., PODC'17], [Ghaffari and Lengler, PODC'18], we prove that, for every $2le k le n$, 3-Majority (resp. 2-Choices) reaches consensus within $widetilde{Theta}(min{k,sqrt{n}})$ (resp. $widetilde{Theta}(k)$) rounds with high probability. Prior to this work, the best known upper bound on the consensus time of 3-Majority was $widetilde{O}(k)$ if $k ll n^{1/3}$ and $widetilde{O}(n^{2/3})$ otherwise, and for 2-Choices, the consensus time was known to be $widetilde{O}(k)$ for $kll sqrt{n}$.

Problem

• Determines consensus time for 3-Majority and 2-Choices dynamics.
• Analyzes synchronous consensus with arbitrary number of opinions.
• Improves bounds on consensus time for complete graph scenarios.

Innovation

• Synchronous consensus dynamics with multiple opinions
• 3-Majority protocol for opinion updates
• 2-Choices protocol for opinion synchronization