Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

TL;DR

This paper demonstrates that for the hard-core model on graphs with high girth and degree, loopy belief propagation and Glauber dynamics converge efficiently to the Gibbs distribution in the uniqueness region, providing new algorithms and insights.

Contribution

It introduces an FPRAS for the partition function applicable to all large-degree graphs with high girth and establishes the local behavior of Glauber dynamics in relation to loopy BP.

Findings

Glauber dynamics mixes rapidly (O(n log n)) for large-degree graphs with girth ≥ 7 when λ<λ_c(Δ).

Loopy BP converges quickly to the Gibbs distribution under certain conditions.

The fixed point of loopy BP approximates the Gibbs distribution closely in the uniqueness region.

Abstract

We study the hard-core model defined on independent sets of an input graph where the independent sets are weighted by a parameter $\lambda>0$. For constant $\Delta$, previous work of Weitz (2006) established an FPTAS for the partition function for graphs of maximum degree $\Delta$ when $\lambda< \lambda_c(\Delta)$. The threshold $\lambda_c(\Delta)$ is the critical point for the phase transition for uniqueness/non-uniqueness on the infinite $\Delta$-regular trees. Sly (2010) showed that there is no FPRAS, unless NP=RP, when $\lambda>\lambda_c(\Delta)$. The running time of Weitz's algorithm is exponential in $\log(\Delta)$. Here we present an FPRAS for the partition function whose running time is $O^*(n^2)$. We analyze the simple single-site Glauber dynamics for sampling from the associated Gibbs distribution. We prove there exists a constant $\Delta_0$ such that for all graphs with maximum degree $\Delta\geq\Delta_0$ and girth $\geq 7$, the mixing time of the Glauber dynamics is $O(n\log(n))$ when $\lambda<\lambda_c(\Delta)$. Our work complements that of Weitz which applies for constant $\Delta$ whereas our work applies for all $\Delta \geq \Delta_0$. We utilize loopy BP (belief propagation), a widely-used inference algorithm. A novel aspect of our work is using the principal eigenvector for the BP operator to design a distance function which contracts in expectation for pairs of states that behave like the BP fixed point. We also prove that the Glauber dynamics behaves locally like loopy BP. As a byproduct we obtain that the Glauber dynamics converges, after a short burn-in period, close to the BP fixed point, and this implies that the fixed point of loopy BP is a close approximation to the Gibbs distribution. Using these connections we establish that loopy BP quickly converges to the Gibbs distribution when the girth $\geq 6$ and $\lambda<\lambda_c(\Delta)$.