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arXiv · 1805.10112

Fairest edge usage and minimum expected overlap for random spanning trees

Nathan Albin·Jason Clemens·Derek Hoare·Pietro Poggi-Corradini·Brandon Sit·Sarah Tymochko
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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.

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Nathan Albin, Jason Clemens, Derek Hoare, Pietro Poggi-Corradini, Brandon Sit, Sarah Tymochko. 2018-05-25. Fairest edge usage and minimum expected overlap for random spanning trees. https://doi.org/10.1016/j.disc.2020.112282

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