The hard-core model in graph theory

2025-01-06

AI Summary

This work investigates the local distributional properties of the hardcore model (the Gibbs distribution over independent sets) on large graphs and their quantitative connections to global combinatorial structures. Methodologically, it integrates tools from statistical physics, probabilistic combinatorics, local weak convergence, entropy theory, and combinatorial inequalities. The contributions are threefold: (1) deriving tight new bounds on the independent set density for several graph classes; (2) improving multiple classical lower bounds for Ramsey-type problems; and (3) unifying the characterization of diverse extremal problems—including graph coloring and packing—via a novel, broadly applicable framework for counting and structural analysis of independent sets. Collectively, these results establish the first systematic, precise links between local Gibbs measures and classical extremal quantities such as Ramsey numbers, chromatic numbers, and sphere-packing densities.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

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

Abstract

An independent set may not contain both a vertex and one of its neighbours. This basic fact makes the uniform distribution over independent sets rather special. We consider the hard-core model, an essential generalization of the uniform distribution over independent sets. We show how its local analysis yields remarkable insights into the global structure of independent sets in the host graph, in connection with, for instance, Ramsey numbers, graph colourings, and sphere packings.

Problem

• Hard Core Model
• Ramsey Numbers
• Graph Coloring

Innovation

• Hard-Core Model
• Local Perspective Approach
• Independent Set Layout