3-Majority and 2-Choices with Many Opinions

TL;DR

This work resolves the long-standing question of nearly tight consensus times for the synchronous 3-Majority and 2-Choices dynamics with an arbitrary number of opinions on the complete graph with self-loops. The authors develop a novel multi-step concentration framework based on the Bernstein condition and Freedman-type inequalities to rigorously control the evolution of key statistics, notably the $\,\ell^2$-norm $\gamma_t$ and the pairwise bias $\delta_t$. They prove near-optimal upper and matching lower bounds: 3-Majority achieves consensus in $\tilde{\Theta}(\min\{\sqrt{n}, k\})$ rounds, while 2-Choices achieves consensus in $\tilde{\Theta}(k)$ rounds, for all $2\le k\le n$, with high probability; and they provide plurality-consensus results and lower bounds to confirm the tightness across regimes. The results significantly sharpen previous bounds and extend the analysis to arbitrary $k$, offering a general framework that could be applied to related consensus processes and to asynchronous settings. Overall, the paper advances the theory of fast consensus by combining drift analysis with multi-step concentration techniques to handle large numbers of opinions.

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 $2\le k \le n$, 3-Majority (resp.\ 2-Choices) reaches consensus within $\widetildeΘ(\min\{k,\sqrt{n}\})$ (resp.\ $\widetildeΘ(k)$) 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 $k\ll \sqrt{n}$.

Figures (3)

• Figure 1: Prior to this work
• Figure 2: This work (\ref{['thm:main theorem']})
• Figure 4: Proof outline for 3-Majority in the case where $\gamma_0\geq C(\log n)/\sqrt{n}$. Here, $C>0$ denotes a sufficiently large constant, $c\in (0,1)$ denotes a sufficiently small constant, and $c_*>0$ denotes an arbitrary constant. Throughout this proof outline, we use \ref{['lem:taunormdown is large']} to ensure $\gamma_t\geq C(1-c^{\downarrow}_\gamma)(\log n)/\sqrt{n}$ in a sufficiently long period. The proof for 2-Choices follows a similar outline.

Theorems & Definitions (95)

• Theorem 1.1: Main
• Theorem 2.1: Starting from large $\gamma_0$
• Theorem 2.2: Growth of $\gamma_t$
• Lemma 2.3: Weak Opinion Vanishing; see also \ref{['lem:weakvanish']}
• Lemma 2.4: Strong Opinion Weakening; see also \ref{['lem:gap amplification', 'lem:initial bias weak']}
• Remark 2.5
• Theorem 2.6: Plurality consensus
• Theorem 2.7: Lower bound
• Definition 3.1: 3-Majority and 2-Choices
• Definition 3.2: Basic quantities