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Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring

Authors Samuel Baguley , Yannic Maus , Janosch Ruff , George Skretas

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Author Details

Samuel Baguley
  • Hasso Plattner Institute, University of Potsdam, Germany
Yannic Maus
  • TU Graz, Austria
Janosch Ruff
  • Hasso Plattner Institute, University of Potsdam, Germany
George Skretas
  • Hasso Plattner Institute, University of Potsdam, Germany

Funding

This research was partially funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) – project number 390859508, and by Austrian Science Fund (FWF) https://doi.org/10.55776/P36280, https://doi.org/10.55776/I6915, and https://doi.org/10.55776/DOC183.

Acknowledgements

The authors would like to thank Thomas Bläsius for enriching discussions and Maximilian Katzmann for running experiments indicating a gap between core and maximum inner-neighbourhood during a research visit of JR in Karlsruhe.

Cite As Get BibTex

Samuel Baguley, Yannic Maus, Janosch Ruff, and George Skretas. Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.13

Abstract

Hyperbolic random graphs inherit many properties that are present in real-world networks. The hyperbolic geometry imposes a scale-free network with a strong clustering coefficient. Other properties like a giant component, the small world phenomena and others follow. This motivates the design of simple algorithms for hyperbolic random graphs. 
In this paper we consider threshold hyperbolic random graphs (HRGs). Greedy heuristics are commonly used in practice as they deliver a good approximations to the optimal solution even though their theoretical analysis would suggest otherwise. A typical example for HRGs are degeneracy-based greedy algorithms [Bläsius, Fischbeck; Transactions of Algorithms '24]. In an attempt to bridge this theory-practice gap we characterise the parameter of degeneracy yielding a simple approximation algorithm for colouring HRGs. The approximation ratio of our algorithm ranges from (2/√3) to 4/3 depending on the power-law exponent of the model. We complement our findings for the degeneracy with new insights on the clique number of hyperbolic random graphs. We show that degeneracy and clique number are substantially different and derive an improved upper bound on the clique number. Additionally, we show that the core of HRGs does not constitute the largest clique. 
Lastly we demonstrate that the degeneracy of the closely related standard model of geometric inhomogeneous random graphs behaves inherently different compared to the one of hyperbolic random graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random network models
Keywords
  • hyperbolic random graphs
  • scale-free networks
  • power-law graphs
  • cliques
  • degeneracy
  • vertex colouring
  • chromatic number

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References

  1. Samuel Baguley, Yannic Maus, Janosch Ruff, and George Skretas. Hyperbolic random graphs: Clique number and degeneracy with implications for colouring. CoRR, 2024. URL: https://doi.org/10.48550/arXiv.2410.11549.
  2. Leonid Barenboim and Michael Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers, 2013. URL: https://doi.org/10.1007/978-3-031-02009-4.
  3. Thomas Bläsius and Philipp Fischbeck. On the external validity of average-case analyses of graph algorithms. ACM Trans. Algorithms, 2024. URL: https://doi.org/10.1145/3633778.
  4. Thomas Bläsius, Philipp Fischbeck, Tobias Friedrich, and Maximilian Katzmann. Solving vertex cover in polynomial time on hyperbolic random graphs. Theory Comput. Syst., 2023. URL: https://doi.org/10.1007/S00224-021-10062-9.
  5. Thomas Bläsius, Tobias Friedrich, and Maximilian Katzmann. Efficiently approximating vertex cover on scale-free networks with underlying hyperbolic geometry. Algorithmica, 2023. URL: https://doi.org/10.1007/S00453-023-01143-X.
  6. Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, Ulrich Meyer, Manuel Penschuck, and Christopher Weyand. Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs. In ESA, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.21.
  7. Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, and Daniel Stephan. Strongly Hyperbolic Unit Disk Graphs. In STACS, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.13.
  8. Thomas Bläsius, Tobias Friedrich, and Anton Krohmer. Cliques in hyperbolic random graphs. Algorithmica, 2017. URL: https://doi.org/10.1007/s00453-017-0323-3.
  9. Thomas Bläsius, Maximillian Katzmann, and Clara Stegehuis. Maximal cliques in scale-free random graphs. Network Science, 2024. URL: https://doi.org/10.1017/nws.2024.13.
  10. Thomas Bläsius, Tobias Friedrich, and Anton Krohmer. Hyperbolic random graphs: Separators and treewidth. In ESA, 2016. URL: https://doi.org/10.4230/LIPIcs.ESA.2016.15.
  11. Karl Bringmann, Ralph Keusch, and Johannes Lengler. Geometric inhomogeneous random graphs. Theoretical Computer Science, 2019. URL: https://doi.org/10.1016/j.tcs.2018.08.014.
  12. Aleksander Bjørn Grodt Christiansen, Krzysztof Nowicki, and Eva Rotenberg. Improved dynamic colouring of sparse graphs. In STOC, 2023. URL: https://doi.org/10.1145/3564246.3585111.
  13. Stephen A. Cook. The complexity of theorem-proving procedures. In STOC, 1971. URL: https://doi.org/10.1145/800157.805047.
  14. B. V. Dekster. The Jung theorem for spherical and hyperbolic spaces. Acta Mathematica Hungarica, 1995. URL: https://doi.org/10.1007/bf01874495.
  15. B. V. Dekster. The Jung theorem in metric spaces of curvature bounded above. Proceedings of the American Mathematical Society, 1997. URL: https://doi.org/10.1090/s0002-9939-97-03842-2.
  16. Tobias Friedrich, Andreas Göbel, Maximilian Katzmann, and Leon Schiller. Cliques in high-dimensional geometric inhomogeneous random graphs. SIAM Journal on Discrete Mathematics, 2024. URL: https://doi.org/10.1137/23m157394x.
  17. Tobias Friedrich and Anton Krohmer. On the Diameter of Hyperbolic Random Graphs. SIAM Journal on Discrete Mathematics, 2018. URL: https://doi.org/10.1137/17M1123961.
  18. Mohsen Ghaffari and Christoph Grunau. Dynamic o(arboricity) coloring in polylogarithmic worst-case time. In STOC, 2024. URL: https://doi.org/10.1145/3618260.3649782.
  19. Luca Gugelmann, Konstantinos Panagiotou, and Ueli Peter. Random Hyperbolic Graphs: Degree Sequence and Clustering. In ICALP, 2012. URL: https://doi.org/10.1007/978-3-642-31585-5_51.
  20. Heinrich Jung. Ueber die kleinste Kugel, die eine räumliche Figur einschliesst. Journal für die reine und angewandte Mathematik, 1901. URL: http://eudml.org/doc/149122.
  21. Richard M. Karp. Reducibility among Combinatorial Problems, pages 85-103. Springer US, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  22. Maximilian Katzmann. About the analysis of algorithms on networks with underlying hyperbolic geometry. doctoralthesis, Universität Potsdam, 2023. URL: https://doi.org/10.25932/publishup-58296.
  23. Ralph Keusch. Geometric Inhomogeneous Random Graphs and Graph Coloring Games. PhD thesis, ETH Zurich, 2018. URL: https://doi.org/10.3929/ethz-b-000269658.
  24. Marcos Kiwi and Dieter Mitsche. A bound for the diameter of random hyperbolic graphs. In ANALCO, 2015. Google Scholar
  25. Marcos Kiwi and Dieter Mitsche. Spectral gap of random hyperbolic graphs and related parameters. The Annals of Applied Probability, 2018. URL: https://doi.org/10.1214/17-aap1323.
  26. Marcos Kiwi and Dieter Mitsche. On the second largest component of random hyperbolic graphs. SIAM Journal on Discrete Mathematics, 2019. URL: https://doi.org/10.1137/18m121201x.
  27. Marcos Kiwi, Markus Schepers, and John Sylvester. Cover and hitting times of hyperbolic random graphs. Random Structures & Algorithms, 2024. URL: https://doi.org/10.1002/rsa.21249.
  28. Júlia Komjáthy and Bas Lodewijks. Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs. Stochastic Processes and their Applications, 2020. URL: https://doi.org/10.1016/j.spa.2019.04.014.
  29. Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguñá. Hyperbolic geometry of complex networks. Phys. Rev. E, 2010. URL: https://doi.org/10.1103/PhysRevE.82.036106.
  30. Anton Krohmer. Structures & algorithms in hyperbolic random graphs. doctoralthesis, Universität Potsdam, 2016. URL: https://publishup.uni-potsdam.de/frontdoor/index/index/docId/39597.
  31. David W. Matula and Leland L. Beck. Smallest-last ordering and clustering and graph coloring algorithms. Journal of the ACM, 1983. URL: https://doi.org/10.1145/2402.322385.
  32. Michael Mitzenmacher and Eli Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, 2005. URL: https://doi.org/10.1017/CBO9780511813603.
  33. Tobias Müller and Merlijn Staps. The Diameter of KPKVB Random Graphs. Advances in Applied Probability, 2019. URL: https://doi.org/10.1017/apr.2019.23.
  34. Fragkiskos Papadopoulos, Dmitri Krioukov, Marian Boguna, and Amin Vahdat. Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces. In INFOCOM, 2010. URL: https://doi.org/10.1109/infcom.2010.5462131.
  35. Jose L. Walteros and Austin Buchanan. Why is maximum clique often easy in practice? Operations Research, 2020. URL: https://doi.org/10.1287/opre.2019.1970.
  36. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 2007. URL: https://doi.org/10.4086/toc.2007.v003a006.
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