Time- and Space-Optimal Silent Self-Stabilizing Exact Majority in Population Protocols

2025-03-22

AI Summary

This paper investigates the self-stabilizing exact majority problem in the population protocol model: given an arbitrary initial configuration of agents holding opinions A or B, the goal is for all agents to converge—without external intervention—and permanently stabilize on the true majority opinion. We first prove that the problem is unsolvable without prior knowledge of the population size (n). Then, we present the first silent self-stabilizing protocol that is simultaneously optimal in both time and space: it operates via pairwise random interactions among finite-state agents, leveraging probabilistic analysis and information-theoretic lower bounds. The protocol achieves expected parallel time (O(n)), (O(n log n)) with high probability, and uses only (O(n)) states per agent. Crucially, its silence property guarantees no further state changes after convergence. Our tight matching upper and lower bounds establish theoretical optimality.

Technology Category

• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

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

Abstract

We address the self-stabilizing exact majority problem in the population protocol model, introduced by Angluin, Aspnes, Diamadi, Fischer, and Peralta (2004). In this model, there are $n$ state machines, called agents, which form a network. At each time step, only two agents interact with each other, and update their states. In the self-stabilizing exact majority problem, each agent has a fixed opinion, $mathtt{A}$ or $mathtt{B}$, and stabilizes to a safe configuration in which all agents output the majority opinion from any initial configuration. In this paper, we show the impossibility of solving the self-stabilizing exact majority problem without knowledge of $n$ in any protocol. We propose a silent self-stabilizing exact majority protocol, which stabilizes within $O(n)$ parallel time in expectation and within $O(n log n)$ parallel time with high probability, using $O(n)$ states, with knowledge of $n$. Here, a silent protocol means that, after stabilization, the state of each agent does not change. We establish lower bounds, proving that any silent protocol requires $Omega(n)$ states, $Omega(n)$ parallel time in expectation, and $Omega(n log n)$ parallel time with high probability to reach a safe configuration. Thus, the proposed protocol is time- and space-optimal.

Problem

• Solves self-stabilizing exact majority in population protocols
• Proves impossibility without knowing network size n
• Achieves time- and space-optimal silent stabilization

Innovation

• Silent self-stabilizing exact majority protocol
• O(n) parallel time with O(n) states
• Optimal time and space complexity bounds