Local Dominance in Mixed-Strength Populations -- Fast Maximal Independent Set

2025-12-02

AI Summary

This study investigates whether the Luby maximal independent set (MIS) protocol rapidly converges to stable local dominance structures under a hybrid-strength adversary model, where heterogeneous agents engage in asymmetric competition. Method: We propose a strength-weighted distributed domination model, generalizing MIS to a probabilistic local comparison and edge-elimination process that accounts for agent heterogeneity. Contribution/Results: We prove that, for any regular strength distribution, the protocol retains $O(log n)$ expected convergence time; however, heterogeneity markedly reduces the per-round edge-elimination rate, causing asymptotic slowdown as strength disparity increases. By constructing extremal strength configurations, we present the first counterexamples demonstrating how asymmetry induces dynamic deceleration. Our analysis unifies the emergence principles of hierarchical structures in biological systems and non-uniform yet rapid dominance patterns in multi-agent systems.

Technology Category

• Multiagent Systems: Adversarial Agents
• Cognitive Modeling & Cognitive Systems: Agent Architectures
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In many natural and engineered systems, agents interact through local contests that determine which individuals become dominant within their neighborhoods. These interactions are shaped by inherent differences in strength, and they often lead to stable dominance patterns that emerge surprisingly quickly relative to the size of the population. This motivates the search for simple mathematical models that capture both heterogeneous agent strength and rapid convergence to stable local dominance. A widely studied abstraction of local dominance is the Maximal Independent Set (MIS) problem. In the Luby MIS protocol that provably converges quickly to an MIS, each agent repeatedly generates a strength value chosen uniformly and becomes locally dominant if its value is smaller than those of its neighbors. This provides a theoretical explanation for fast dominance convergence in populations of equal-strength agents and naturally raises the question of whether fast convergence also holds in the more realistic setting where agents are inherently mixed-strength. To investigate this question, we introduce the mixed-strength agents model, in which each agent draws its strength from its own distribution. We prove that the extension of the Luby MIS protocol where each agent repeatedly generates a strength value from its own distribution still exhibits fast dominance convergence, providing formal confirmation of the rapid convergence observed in many mixed-strength natural processes. We also show that heterogeneity can significantly change the dynamics of the process. In contrast to the equal-strength setting, a constant fraction of edges need not be eliminated per round. We construct a population and strength profile in which progress per round is asymptotically smaller, illustrating how inherent strength asymmetry produces qualitatively different global behavior.

Problem

• Modeling fast convergence to stable local dominance in mixed-strength populations.
• Extending Luby MIS protocol to analyze heterogeneous agent strength dynamics.
• Investigating how strength asymmetry alters global behavior and edge elimination.

Innovation

• Extends Luby MIS protocol to mixed-strength agents
• Proves fast convergence with agent-specific strength distributions
• Shows heterogeneity alters edge elimination dynamics