Spectral Entropy via Random Spanning Forests

TL;DR

This paper introduces a novel method linking random spanning forests to the heat-kernel partition function, enabling efficient estimation of spectral properties and entropy without expensive computations, validated on synthetic networks.

Contribution

It establishes an exact relation between spanning forests and heat-kernel functions, providing scalable estimation techniques for spectral and thermodynamic graph descriptors.

Findings

Accurate estimation of partition functions and entropy via Wilson sampling.

Validation of inverse-Laplace reconstructions with spectral-density regularization.

Method is scalable and applicable to local graph descriptors.

Abstract

We establish an exact analytic relation between random spanning forests and the heat-kernel partition function. This identity enables estimation of partition functions, energies, and the Von Neumann entropy by Wilson sampling of forests, avoiding costly Laplacian eigendecompositions. We validate inverse-Laplace reconstructions stabilized by a Stieltjes spectral-density regularization on synthetic networks. The approach is scalable and yields local node and edge thermodynamic descriptors.