Fairest edge usage and minimum expected overlap for random spanning trees

TL;DR

This paper investigates how to select probability distributions for random spanning trees to ensure fair edge usage, introducing homogeneous graphs and a deflation process to identify hierarchical structures, with applications of spanning tree modulus.

Contribution

It introduces the concept of homogeneous graphs and a deflation process to analyze hierarchical structures in graphs for fair edge usage in random spanning trees.

Findings

Defined homogeneous graphs with equal edge usage probabilities.

Developed a deflation process to identify homogeneous cores within graphs.

Provided an algorithm based on minimum spanning tree methods for spanning tree modulus.

Abstract

Random spanning trees of a graph $G$ are governed by a corresponding probability mass distribution (or "law"), $\mu$, defined on the set of all spanning trees of $G$. This paper addresses the problem of choosing $\mu$ in order to utilize the edges as "fairly" as possible. This turns out to be equivalent to minimizing, with respect to $\mu$, the expected overlap of two independent random spanning trees sampled with law $\mu$. In the process, we introduce the notion of homogeneous graphs. These are graphs for which it is possible to choose a random spanning tree so that all edges have equal usage probability. The main result is a deflation process that identifies a hierarchical structure of arbitrary graphs in terms of homogeneous subgraphs, which we call homogeneous cores. A key tool in the analysis is the spanning tree modulus, for which there exists an algorithm based on minimum spanning tree algorithms, such as Kruskal's or Prim's.