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Improved Bounds for Graph Distances in Scale Free Percolation and Related Models

Authors Kostas Lakis , Johannes Lengler, Kalina Petrova , Leon Schiller

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Author Details

Kostas Lakis
  • ETH Zürich, Department of Computer Science, Zürich, Switzerland
Johannes Lengler
  • ETH Zürich, Department of Computer Science, Zürich, Switzerland
Kalina Petrova
  • ETH Zürich, Department of Computer Science, Zürich, Switzerland
Leon Schiller
  • ETH Zürich, Department of Computer Science, Zürich, Switzerland

Funding

  • Lakis, Kostas: gratefully acknowledges support from the John S. Latsis Public Benefit Foundation and the Onassis Foundation
  • Petrova, Kalina: funded by SNSF grant No. CRSII5 173721
  • Schiller, Leon: funded by SNSF grant No. 197138

Acknowledgements

This research started at the joint workshop of the Combinatorial Structures and Algorithms and Theory of Combinatorial Algorithms groups of ETH Zürich held in Stels, Switzerland, January 2024. We thank the organizers for providing a very pleasant and inspiring working atmosphere.

Cite AsGet BibTex

Kostas Lakis, Johannes Lengler, Kalina Petrova, and Leon Schiller. Improved Bounds for Graph Distances in Scale Free Percolation and Related Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 74:1-74:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.74

Abstract

In this paper, we study graph distances in the geometric random graph models scale-free percolation SFP, geometric inhomogeneous random graphs GIRG, and hyperbolic random graphs HRG. Despite the wide success of the models, the parameter regime in which graph distances are polylogarithmic is poorly understood. We provide new and improved lower bounds. In a certain portion of the parameter regime, those match the known upper bounds. Compared to the best previous lower bounds by Hao and Heydenreich [Hao and Heydenreich, 2023], our result has several advantages: it gives matching bounds for a larger range of parameters, thus settling the question for a larger portion of the parameter space. It strictly improves the lower bounds of [Hao and Heydenreich, 2023] for all parameters settings in which those bounds were not tight. It gives tail bounds on the probability of having short paths, which imply shape theorems for the k-neighbourhood of a vertex whenever our lower bounds are tight, and tight bounds for the size of this k-neighbourhood. And last but not least, our proof is much simpler and not much longer than two pages, and we demonstrate that it generalizes well by showing that the same technique also works for first passage percolation.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Stochastic processes
  • Theory of computation → Random network models
Keywords
  • Mathematics
  • Probability Theory
  • Combinatorics
  • Random Graphs
  • Random Metric Spaces

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