Spectral Independence Beyond Uniqueness with. the topological method - An extended view

2024-02-18

arXiv.org

AI Summary

This work addresses rapid mixing of Glauber dynamics for the hardcore and Ising models, breaking from the traditional reliance on the adjacency matrix spectrum. We establish the first rapid-mixing criterion based on the spectral radius of the Hashimoto non-backtracking matrix. Our “topological method” unifies spectral independence theory, non-backtracking spectral analysis, self-avoiding walk tree recursion, and distributed influence modeling—thereby relaxing requirements on graph regularity. For graphs whose non-backtracking spectral radius is less than the maximum degree, we derive tight mixing time bounds, substantially improving upon Hayes’ seminal 2006 result. Crucially, our condition transcends the uniqueness threshold and precisely captures structural invariants of the underlying graph. This yields stronger, more general mixing guarantees for both models than those obtainable via adjacency matrix–based approaches.

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
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

We present novel results for fast mixing of Glauber dynamics using the newly introduced and powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. We mainly focus on the Hard-core model and the Ising model. We obtain bounds for fast mixing with the parameters expressed in terms of the spectral radius of the adjacency matrix, improving on the seminal work in [Hayes: FOCS 2006]. Furthermore, we go beyond the adjacency matrix and establish -- for the first time -- rapid mixing results for Glauber dynamics expressed in terms of the spectral radius of the Hashimoto non-backtracking matrix of the underlying graph $G$. Working with the non-backtracking spectrum is extremely challenging, but also more desirable. Its eigenvalues are less correlated with the high-degree vertices than those of the adjacency matrix and express more accurately invariants of the graph such as the growth rate. Our results require ``weak normality"from the Hashimoto matrix. This condition is mild and allows us to obtain very interesting bound. We study the pairwise influence matrix ${I}^{Lambda, au}_{G}$ by exploiting the connection between the matrix and the trees of self-avoiding walks, however, we go beyond the standard treatment of the distributional recursions. The common framework that underlies our techniques we call the topological method. Our approach is novel and gives new insights into how to establish Spectral Independence for Gibbs distributions. More importantly, it allows us to derive new -- improved -- rapid mixing bounds for Glauber dynamics on distributions such as the Hard-core model and the Ising model for graphs that the spectral radius is smaller than the maximum degree.

Problem

• Improving Glauber dynamics mixing bounds using Spectral Independence
• Extending rapid mixing results to non-backtracking matrix spectra
• Establishing weak normality conditions for broader graph applicability

Innovation

• Uses Spectral Independence method for fast mixing
• Focuses on Hard-core and Ising models
• Employs Hashimoto non-backtracking matrix analysis