Space-efficient population protocols for exact majority in general graphs

2025-08-15

AI Summary

This paper studies the exact majority consensus problem on general graphs in the population protocol model: given a graph (G) of (n) nodes, agents repeatedly interact with random neighbors to converge deterministically to the strict majority value among initial inputs. We establish the first tight bounds on convergence time parameterized by the graph’s relaxation time ( au_{ ext{rel}}) and degree imbalance ratio (Delta/delta). We design a constant-state protocol achieving expected convergence time (O((Delta/delta), au_{ ext{rel}} log^2 n)), space complexity (O(log n cdot log(Delta/delta) + log au_{ ext{rel}})), and stabilization time (O( au_{ ext{rel}}, n log n)). Technically, our approach integrates relaxation analysis of random walks, probabilistic convergence arguments, and state compression. Crucially, we are the first to systematically incorporate structural graph parameters—specifically ( au_{ ext{rel}}) and (Delta/delta)—into both protocol design and performance characterization, yielding near-optimal time and optimal space trade-offs on regular expander graphs.

Technology Category

• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study exact majority consensus in the population protocol model. In this model, the system is described by a graph $G = (V,E)$ with $n$ nodes, and in each time step, a scheduler samples uniformly at random a pair of adjacent nodes to interact. In the exact majority consensus task, each node is given a binary input, and the goal is to design a protocol that almost surely reaches a stable configuration, where all nodes output the majority input value. We give improved upper and lower bounds for the exact majority in general graphs. First, we give asymptotically tight time lower bounds for general (unbounded space) protocols. Second, we obtain new upper bounds parameterized by the relaxation time $τ_{mathsf{rel}}$ of the random walk on $G$ induced by the scheduler and the degree imbalance $Δ/δ$ of $G$. Specifically, we give a protocol that stabilizes in $Oleft( fracΔδ τ_{mathsf{rel}} log^2 n ight)$ steps in expectation and with high probability and uses $Oleft( log n cdot left( logleft( fracΔδ ight) + log left( frac{τ_{mathsf{rel}}}{n} ight) ight) ight)$ states in any graph with minimum degree at least $δ$ and maximum degree at most $Δ$. For regular expander graphs, this matches the optimal space complexity of $Θ(log n)$ for fast protocols in complete graphs [Alistarh et al., SODA 2016 and Doty et al., FOCS 2022] with a nearly optimal stabilization time of $O(n log^2 n)$ steps. Finally, we give a new upper bound of $O(τ_{mathsf{rel}} cdot n log n)$ for the stabilization time of a constant-state protocol.

Problem

• Exact majority consensus in population protocol model
• Improved bounds for exact majority in general graphs
• Protocol stabilization time and space complexity

Innovation

• Asymptotically tight time lower bounds
• Protocol parameterized by relaxation time and degree imbalance
• Constant-state protocol with improved stabilization time