An Almost Tight Lower Bound for Plurality Consensus with Undecided State Dynamics in the Population Protocol Model

TL;DR

This work analyzes plurality consensus in population protocols with Undecided State Dynamics, focusing on how long it takes for the system to stabilize to the initial majority opinion. The authors develop a drift- and random-walk-based framework to bound the evolution of undecided nodes and pairwise opinion gaps, yielding an almost tight lower bound: stabilization requires $\Omega\left(kn\log \frac{\sqrt n}{k\log n}\right)$ interactions for $k = o\left(\frac{\sqrt n}{\log n}\right)$, equivalent to $\Omega\left(k\log \frac{\sqrt n}{k\log n}\right)$ parallel time; the bound is tight for $k\le n^{1/2-\varepsilon}$ due to a matching $O(k\log n)$ upper bound. The analysis hinges on precise control of the undecided population and careful handling of random-walk-type fluctuations in opinion counts, building on drift-analysis techniques and recent advances in population protocols. The results clarify the time complexity landscape for plurality with undecided dynamics and delineate model-specific limitations compared to the Gossip setting, while suggesting directions for improving memory or synchronization to potentially surpass the bound.

Abstract

We revisit the majority problem in the population protocol communication model, as first studied by Angluin et al. (Distributed Computing 2008). We consider a more general version of this problem known as plurality consensus, which has already been studied intensively in the literature. In this problem, each node in a system of $n$ nodes, has initially one of $k$ different opinions, and they need to agree on the (relative) majority opinion. In particular, we consider the important and intensively studied model of Undecided State Dynamics. Our main contribution is an almost tight lower bound on the stabilization time: we prove that there exists an initial configuration, even with bias $Δ= ω(\sqrt{n\log n})$, where stabilization requires $Ω(kn\log \frac {\sqrt n} {k \log n})$ interactions, or equivalently, $Ω(k\log \frac {\sqrt n} {k \log n})$ parallel time for any $k = o\left(\frac {\sqrt n}{\log n}\right)$. This bound is tight for any $ k \le n^{\frac 1 2 - ε}$, where $ε>0$ can be any small constant, as Amir et al.~(PODC'23) gave a $O(k\log n)$ parallel time upper bound for $k = O\left(\frac {\sqrt n} {\log ^2 n}\right)$.

Figures (1)

• Figure 1: Intuition for stabilization of plurality consensus in undecided state dynamics. On the left figure, we scale up the minority opinions by a $k$ factor for visibility. In this experiment, $n = 1,000,000$ agents and $k = \frac{\sqrt{n}}{\log n \log \log n}$. In the initial configuration $k-1$ opinions had the same support, while opinion $1$ had an additive advantage of $\sqrt{n \log n}$. Out of all the minority opinions, we plot one with a darker color for visibility.

Theorems & Definitions (12)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Theorem 3.5
• proof