Fairest edge usage and minimum expected overlap for random spanning trees
Clemens, Jason R.
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Albin, N., Clemens, J., Hoare, D., Poggi-Corradini, P., Sit, B., & Tymochko, S. (2021). Fairest edge usage and minimum expected overlap for random spanning trees. Discrete Mathematics, 344(5) doi:10.1016/j.disc.2020.112282
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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