Cover and Hitting Times of Hyperbolic Random Graphs

Kiwi, M., Schepers, M. and Sylvester, J. (2022) Cover and Hitting Times of Hyperbolic Random Graphs. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022), 19-21 September 2022, 30:1-30:19. (doi: 10.4230/LIPIcs.APPROX/RANDOM.2022.30)

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We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range (2, 3). In particular, we focus on the expected times for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that up to multiplicative constants: the cover time is n(log n) 2 , the maximum hitting time is n log n, and the average hitting time is n. The first two results hold in expectation and a.a.s. and the last in expectation (with respect to the HRG). We prove these results by determining the effective resistance either between an average vertex and the well-connected “center” of HRGs or between an appropriately chosen collection of extremal vertices. We bound the effective resistance by the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane on which we overlay a forest-like structure.

Item Type:Conference Proceedings
Glasgow Author(s) Enlighten ID:Sylvester, Dr John
Authors: Kiwi, M., Schepers, M., and Sylvester, J.
College/School:College of Science and Engineering > School of Computing Science
Published Online:15 September 2022
Copyright Holders:Copyright © Marcos Kiwi, Markus Schepers, and John Sylvester
Publisher Policy:Reproduced under a Creative Commons license
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
305944Multilayer Algorithmics to Leverage Graph StructureKitty MeeksEngineering and Physical Sciences Research Council (EPSRC)EP/T004878/1M&S - Statistics