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

2025-05-05

AI Summary

This paper studies the plurality consensus problem—reaching stable agreement on the most frequent opinion among $k$ possible opinions—in the population protocol model under undecided-state dynamics. It establishes the first nearly tight lower bound on the parallel stabilization time for initial configurations exhibiting a bias, namely $Omegaleft(k logfrac{sqrt{n}}{k log n} ight)$, which is tight when $k leq n^{1/2 - varepsilon}$ and matches the optimal upper bound from PODC’23, thereby settling the exact time complexity of plurality consensus in this model. Technically, the proof integrates probabilistic analysis, adversarial construction of biased initial configurations, normalization of parallel time, and stochastic process modeling. This resolves a long-standing theoretical gap and reveals the fundamental interplay between the number of opinions $k$ and the population size $n$ in governing convergence speed.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 $Delta = omega(sqrt{nlog n})$, where stabilization requires $Omega(knlog frac {sqrt n} {k log n})$ interactions, or equivalently, $Omega(klog frac {sqrt n} {k log n})$ parallel time for any $k = oleft(frac {sqrt n}{log n} ight)$. This bound is tight for any $ k le n^{frac 1 2 - epsilon}$, where $epsilon>0$ can be any small constant, as Amir et al.~(PODC'23) gave a $O(klog n)$ parallel time upper bound for $k = Oleft(frac {sqrt n} {log ^2 n} ight)$.

Problem

• Studies plurality consensus in population protocol model
• Analyzes stabilization time lower bound for undecided dynamics
• Proves tight bound for k opinions in n nodes

Innovation

• Almost tight lower bound for stabilization time
• Undecided State Dynamics model analysis
• Focus on plurality consensus problem