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The question of why modular hierarchical (MH) organizations are so ubiquitous in nature, however, has received less attention. One hypothesis is that MH topologies may provide an optimal structure for certain dynamical processes. We revisit a MH network model that interpolates, using a single parameter, between two known network topologies: from strong hierarchical modularity to an Erd\u0151s\u2013R\u00e9nyi random connectivity structure. We show that this model displays a similar small-world effect as the Kleinberg model, where the connection probability between nodes decays algebraically with distance. We find that there is an optimal structure, in both models, for which the pair-averaged first passage time (FPT) and mean cover time of a discrete-time random walk are minimal, and provide a heuristic explanation for this effect. Finally, we show that analytic predictions for the pair-averaged FPT based on an effective medium approximation fail to reproduce these minima, which implies that their presence is due to a network structure effect.<\/jats:p>","DOI":"10.1093\/comnet\/cnz010","type":"journal-article","created":{"date-parts":[[2019,2,18]],"date-time":"2019-02-18T13:02:32Z","timestamp":1550494952000},"page":"865-895","source":"Crossref","is-referenced-by-count":8,"title":["Modular hierarchical and power-law small-world networks bear structural optima for minimal first passage times and cover time"],"prefix":"10.1093","volume":"7","author":[{"given":"Benjamin F","family":"Maier","sequence":"first","affiliation":[{"name":"Robert Koch Institute, Nordufer 20, D-13353 Berlin, Germany and Department of Physics, Humboldt Universit\u00e4t zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cristi\u00e1n","family":"Huepe","sequence":"additional","affiliation":[{"name":"CHuepe Labs, 2713 West Haddon Ave #1, Chicago, IL 60622, USA and Northwestern Institute on Complex Systems & ESAM, Northwestern University, Evanston, IL 60208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dirk","family":"Brockmann","sequence":"additional","affiliation":[{"name":"Robert Koch Institute, Nordufer 20, D-13353 Berlin, Germany and Institute for Theoretical Biology, Humboldt Universit\u00e4t zu Berlin, Philippstr. 13, D-10115 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2019,4,8]]},"reference":[{"issue":"3\u20135","key":"2019121010372894100_B1","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.physrep.2009.11.002","article-title":"Community detection in graphs","volume":"486","author":"Fortunato,","year":"2010","journal-title":"Phys. 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