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Spectral Entropy via Random Spanning Forests

Carlo Nicolini

TL;DR

The paper addresses the challenge of linking thermodynamic observables on networks to combinatorial forest statistics without resorting to Laplacian eigendecomposition. It establishes an exact bridge by showing that the expected root count $s(q)$ from random rooted forests is the Laplace transform partner of the heat-trace partition function $Z(\beta)$, enabling reconstruction of spectral quantities via Wilson sampling. Key contributions include deriving $\chi(q)=\det(qI+L)$, $s(q)=q d/dq \log\chi(q)$, introducing local descriptors $\pi_v(q)$ and $\theta_e(q)$, and presenting a robust inverse Laplace approach using Stieltjes regularization. The method yields scalable, decomposition-free estimators for energy, entropy, and spectral moments, with practical utility for network analysis and model selection.

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.

Spectral Entropy via Random Spanning Forests

TL;DR

The paper addresses the challenge of linking thermodynamic observables on networks to combinatorial forest statistics without resorting to Laplacian eigendecomposition. It establishes an exact bridge by showing that the expected root count from random rooted forests is the Laplace transform partner of the heat-trace partition function , enabling reconstruction of spectral quantities via Wilson sampling. Key contributions include deriving , , introducing local descriptors and , and presenting a robust inverse Laplace approach using Stieltjes regularization. The method yields scalable, decomposition-free estimators for energy, entropy, and spectral moments, with practical utility for network analysis and model selection.

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.

Paper Structure

This paper contains 13 sections, 33 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theory and models
  3. Trees, forests, random walks
  4. Sampling random spanning forests
  5. Spectral entropies framework
  6. From random rooted forests to heat trace
  7. Network observables from forest statistics
  8. Numerical experiments
  9. Conclusions
  10. Appendix
  11. Code and data availability
  12. Wilson algorithm
  13. Numerical inverse Laplace transform via Stieltjes spectral density approximation

Figures (2)

  • Figure 1: Wilson sampling of an Erdos-Renyi graph with 100 nodes and edge probability 0.05. Errorbars are computed as standard deviation from $k=24$ independent runs of Wilson algorithm, solid line is the exact spectral value as in Eq \ref{['eq:s_def']}. At low $q$ values the expected number of roots $s(q)$ approaches 1, while at large $q$ it tends to $n=100$.
  • Figure 2: Reconstruction of the partition function $Z(\beta)$ from Wilson sampling on an Erdős-Rényi graph with $n=50$ nodes and edge probability $p=0.1$. The solid blue orange line is the exact spectral value of $Z(\beta)$, the blue solid line is obtained from the Wilson sampling and Stieltjes spectral approximation. Confidence (light blue filled area) are computed as standard deviation on $g(q)$ with $48$ independent Wilson's Monte Carlo samples.