The cavity method for counting spanning subgraphs subject to local constraints

TL;DR

This paper proves the validity of the cavity method for counting constrained spanning subgraphs in large, sparse, tree-like graphs, providing explicit formulas for specific cases like Erdős-Rényi graphs.

Contribution

It establishes the cavity method's validity for counting constrained subgraphs in asymptotically tree-like graphs using negative association and random weak limits.

Findings

Convergence of free entropy density in large sparse graphs.

Explicit solution for cavity equations on Galton-Watson trees.

Formula for the b-matching number in Erdős-Rényi graphs.

Abstract

Using the theory of negative association for measures and the notion of random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically tree-like graphs. Specifically, the corresponding free entropy density is shown to converge along any sequence of graphs whose random weak limit is a tree, and the limit is directly expressed in terms of the unique solution to a limiting cavity equation. On a Galton-Watson tree, the latter simplifies into a recursive distributional equation which can be solved explicitely. As an illustration, we provide an explicit-limit formula for the $b-$matching number of an Erd\H{o}s-R\'enyi random graph with fixed average degree and diverging size, for any $b\in\mathbb N$.