Undecided State Dynamics with Many Opinions

2026-03-03

AI Summary

This work investigates the convergence time of the Undecided-State Dynamics (USD) consensus protocol in multi-opinion settings with $k \geq 2$ opinions, establishing the first general upper bounds for arbitrary initial configurations under both the gossip and population protocol models. By carefully characterizing the evolution of undecided states and combining probabilistic analysis with matching upper- and lower-bound techniques, the authors prove that USD reaches consensus with high probability in $\widetilde{O}(\min\{k, \sqrt{n}\})$ rounds in the gossip model and in $\widetilde{O}(\min\{kn, n^{3/2}\})$ interactions in the population protocol model. These bounds are optimal up to polylogarithmic factors, providing the first tight analysis for all $2 \leq k \leq n$.

Technology Category

• Multiagent Systems: Multiagent Systems under Uncertainty
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study the Undecided-State Dynamics (USD), a fundamental consensus process in which each vertex holds one of $k$ decided opinions or the undecided state. We consider both the gossip model and the population protocol model. Prior work established tight bounds on the consensus time of this process only for the regime $k = O(\sqrt{n}/(\log n)^2)$ (for the population protocol model) and $k = O((n/\log n)^{1/3})$ (for the gossip model), often under restrictive assumptions on the initial configuration. In this paper, we obtain the first consensus-time guarantees for USD that hold for \emph{arbitrary} $2\le k\le n$ and for \emph{arbitrary} initial configurations in both the gossip model and the population protocol model. In the gossip model, USD reaches consensus within $\widetilde O(\min\{k,\sqrt n\})$ synchronous rounds with probability $1-p_{\bot}-n^{-c}$, where $p_{\bot}$ is the gossip-specific probability of collapsing to the all-undecided state in the first round. In the population protocol model, USD reaches consensus within $\widetilde O(\min\{kn,n^{3/2}\})$ asynchronous interactions with high probability. We also present lower bounds that match the upper bounds up to polylogarithmic factors for a specific initial configuration and show that our upper bounds are essentially optimal.

Problem

• Undecided-State Dynamics
• consensus time
• gossip model
• population protocol
• arbitrary initial configuration

Innovation

• Undecided-State Dynamics
• consensus time
• gossip model
• population protocol
• arbitrary initial configuration