Fairest edge usage and minimum expected overlap for random spanning trees

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. (C) 2021 Elsevier B.V. All rights reserved.