License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.30
URN: urn:nbn:de:0030-drops-171523
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17152/
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Kiwi, Marcos ; Schepers, Markus ; Sylvester, John

Cover and Hitting Times of Hyperbolic Random Graphs

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LIPIcs-APPROX30.pdf (0.8 MB)

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.

BibTeX - Entry

@InProceedings{kiwi_et_al:LIPIcs.APPROX/RANDOM.2022.30,
  author =	{Kiwi, Marcos and Schepers, Markus and Sylvester, John},
  title =	{{Cover and Hitting Times of Hyperbolic Random Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/17152},
  URN =		{urn:nbn:de:0030-drops-171523},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.30},
  annote =	{Keywords: Random walk, hyperbolic random graph, cover time, hitting time, average hitting time, target time, effective resistance, Kirchhoff index}
}

Keywords: Random walk, hyperbolic random graph, cover time, hitting time, average hitting time, target time, effective resistance, Kirchhoff index
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)
Issue Date: 2022
Date of publication: 15.09.2022

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