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Escape times for subgraph detection and graph partitioning

Zachary M. BoydNicolas FraimanJeremy L. MarzuolaPeter J. MuchaBraxton Osting
Dec 2022
We provide a rearrangement based algorithm for fast detection of subgraphs of$k$ vertices with long escape times for directed or undirected networks.Complementing other notions of densest subgraphs and graph cuts, our method isbased on the mean hitting time required for a random walker to leave adesignated set and hit the complement. We provide a new relaxation of thisnotion of hitting time on a given subgraph and use that relaxation to constructa fast subgraph detection algorithm and a generalization to $K$-partitioningschemes. Using a modification of the subgraph detector on each component, wepropose a graph partitioner that identifies regions where random walks live forcomparably large times. Importantly, our method implicitly respects thedirected nature of the data for directed graphs while also being applicable toundirected graphs. We apply the partitioning method for community detection toa large class of model and real-world data sets.