Two Applications of Random Spanning Forests

Abstract

We use random spanning forests to find, for any Markov process on a finite set of size n and any positive integer m ≤ n, a probability law on the subsets of size m such that the mean hitting time of a random target that is drawn from this law does not depend on the starting point of the process. We use the same random forests to give probabilistic insights into the proof of an algebraic result due to Micchelli and Willoughby and used by Fill and by Miclo to study absorption times and convergence to equilibrium of reversible Markov chains. We also introduce a related coalescence and fragmentation process that leads to a number of open questions.

Smoothing Graph Signals via Random Spanning Forests

“…Let us also define s ij = 1 if t(i) = t(j), and 0 otherwise. We need the Proposition 2.3 of [11]. Fixing a partition P of V, one has:…”

Random Forests and Networks Analysis

“…The Markov process F in Theorem 5 is based on the construction of a coalescence-fragmentation process with values in F making use of Diaconis-Fulton's stack representation of random walks. For a detailed account on this algorithm and a number of related open questions, we refer the reader to section 2.3 in [4].…”

“…The aim of this paper is to survey some recent results [4,5,2,3] on a certain measure on spanning forests of a given graph and its applications within the context of networks analysis. We call a network on n ∈ N vertices a directed and weighted graph…”

“…Wilson [42] in the 1990's described a simple and efficient algorithm based on looperased 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 [4,5,2,3] 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.…”

Intertwining wavelets or multiresolution analysis on graphs through random forests

“…Lf (x) := y∈X w(x, y)(f (y) − f (x)), (2) where f : X → R is an arbitrary function, and w(x, y) is now viewed as the transition rate from x to y. For x ∈ X , let w(x) := y∈X \{x} w(x, y) .…”