Random Forests and Networks Analysis

TL;DR

This paper reviews algorithms and applications of spanning forests in network analysis, focusing on sampling methods, metastable dynamics, coarse graining, and wavelet algorithms for graph signals.

Contribution

It consolidates recent theoretical, algorithmic, and numerical advances in spanning forests and their diverse applications in network analysis.

Findings

Efficient sampling procedures for spanning forests are established.

Applications include graph signal processing and metastable dynamics modeling.

Theoretical foundations and practical algorithms are integrated.

Abstract

D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: 1) a random-walk-based notion of well-distributed points in a graph 2) how to describe metastable dynamics in finite settings by means of Markov intertwining dualities 3) coarse graining schemes for networks and associated processes 4) wavelets-like pyramidal algorithms for graph signals.