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Cover and Hitting Times of Hyperbolic Random Graphs

Authors Marcos Kiwi , Markus Schepers , John Sylvester

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Author Details

Marcos Kiwi
  • Department of Industrial Engineering and Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile
Markus Schepers
  • Institut für Medizinische Biometrie, Epidemiologie und Informatik, Johannes-Gutenberg-University Mainz, Germany
John Sylvester
  • School of Computing Science, University of Glasgow, UK

Funding

  • Kiwi, Marcos: gratefully acknowledges the support of grant GrHyDy ANR-20-CE40-0002 and BASAL funds for centers of excellence from ANID-Chile grants ACE210010 and FB210005.
  • Schepers, Markus: gratefully acknowledges the support of the DIAMANT PhD travel grant for a 2-week research stay in Santiago de Chile.
  • Sylvester, John: was supported by EPSRC project EP/T004878/1: Multilayer Algorithmics to Leverage Graph Structure and the ERC Starting Grant 679660 (DYNAMIC MARCH).

Cite As Get BibTex

Marcos Kiwi, Markus Schepers, and John Sylvester. Cover and Hitting Times of Hyperbolic Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.30

Abstract

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 nlog 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Random network models
  • Mathematics of computing → Stochastic processes
Keywords
  • Random walk
  • hyperbolic random graph
  • cover time
  • hitting time
  • average hitting time
  • target time
  • effective resistance
  • Kirchhoff index

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