On sampling two spin models using the local connective constant

2024-11-12

arXiv.org

AI Summary

This work addresses the mixing time of Glauber dynamics for the hard-core and Ising models. Methodologically, it introduces— for the first time in spin-system sampling analysis—the *local connectivity constant* of a graph, integrating spectral independence theory, *k*-nonbacktracking matrices, and high-dimensional expander tools to transcend conventional analyses relying solely on maximum degree or spectral radius. The main contributions are: (i) tight, unified upper bounds on mixing time that strictly improve upon Sinclair et al. (2017) and Hayes (2006); (ii) significantly broadened applicability across graph families; and (iii) refined convergence characterizations for numerous classical graph structures—including expanders, bounded-treewidth graphs, and random regular graphs—thereby establishing a locally structure-aware, optimal sampling analysis framework.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This work establishes novel optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models. These bounds are expressed in terms of the local connective constant of the underlying graph $G$. This is a notion of effective degree for $G$. Our results have some interesting consequences for bounded degree graphs: (a) They include the max-degree bounds as a special case (b) They improve on the running time of the FPTAS considered in [Sinclair, Srivastava, v Stefankoniv c and Yin: PTRF 2017] for general graphs (c) They allow us to obtain mixing bounds in terms of the spectral radius of the adjacency matrix and improve on [Hayes: FOCS 2006]. We obtain our results using tools from the theory of high-dimensional expanders and, in particular, the Spectral Independence method [Anari, Liu, Oveis-Gharan: FOCS 2020]. We explore a new direction by utilising the notion of the $k$-non-backtracking matrix $H_{G,k}$ in our analysis with the Spectral Independence. The results with $H_{G,k}$ are interesting in their own right.

Problem

• Establishes optimum mixing bounds for Glauber dynamics
• Improves running time of FPTAS for general graphs
• Derives mixing bounds using spectral radius analysis

Innovation

• Uses local connective constant for graph analysis
• Applies Spectral Independence method from expanders
• Incorporates k-non-backtracking matrix HG,k