Discrete Hyperbolic Random Graph Model

Authors Dorota Celińska-Kopczyńska , Eryk Kopczyński

Thumbnail PDF


  • Filesize: 1.91 MB
  • 19 pages

Document Identifiers

Author Details

Dorota Celińska-Kopczyńska
  • Institute of Informatics, University of Warsaw, Poland
Eryk Kopczyński
  • Institute of Informatics, University of Warsaw, Poland


We would like to thank all the referees for their comments which have greatly improved the paper.

Cite AsGet BibTex

Dorota Celińska-Kopczyńska and Eryk Kopczyński. Discrete Hyperbolic Random Graph Model. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The hyperbolic random graph model (HRG) has proven useful in the analysis of scale-free networks, which are ubiquitous in many fields, from social network analysis to biology. However, working with this model is algorithmically and conceptually challenging because of the nature of the distances in the hyperbolic plane. In this paper, we propose a discrete variant of the HRG model (DHRG) where nodes are mapped to the vertices of a triangulation; our algorithms allow us to work with this model in a simple yet efficient way. We present experimental results conducted on networks, both real-world and simulated, to evaluate the practical benefits of DHRG in comparison to the HRG model.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random network models
  • Theory of computation → Routing and network design problems
  • Human-centered computing → Social network analysis
  • hyperbolic geometry
  • scale-free networks
  • routing
  • tessellation


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Albert-László Barábasi and Reka Albert. Emergence of scaling in random networks. Science, 286(5439):509-512, 1999. URL: https://doi.org/10.1126/science.286.5439.509.
  2. Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, and Anton Krohmer. Hyperbolic embeddings for near-optimal greedy routing. In Algorithm Engineering and Experiments (ALENEX), pages 199-208, 2018. Google Scholar
  3. Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, Ulrich Meyer, Manuel Penschuck, and Christopher Weyand. Efficiently generating geometric inhomogeneous and hyperbolic random graphs. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 21:1-21:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.21.
  4. Thomas Bläsius, Tobias Friedrich, Anton Krohmer, and Sören Laue. Efficient embedding of scale-free graphs in the hyperbolic plane. In European Symposium on Algorithms (ESA), pages 16:1-16:18, 2016. Google Scholar
  5. Marián Boguñá, Fragkiskos Papadopoulos, and Dmitri Krioukov. Sustaining the internet with hyperbolic mapping. Nature Communications, 1(6):1-8, September 2010. URL: https://doi.org/10.1038/ncomms1063.
  6. Karl Bringmann, Ralph Keusch, and Johannes Lengler. Geometric inhomogeneous random graphs. Theoretical Computer Science, 2018. URL: https://doi.org/10.1016/j.tcs.2018.08.014.
  7. Károly Böröczky. Gömbkitöltések állandó görbületű terekben I. Matematikai Lapok, 25:265-306, 1974. Google Scholar
  8. James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry. Hyperbolic geometry. In In Flavors of geometry, pages 59-115. University Press, 1997. Available online at URL: http://www.msri.org/communications/books/Book31/files/cannon.pdf.
  9. Dorota Celińska. Information and influence in social network of Open Source community. In 9th Annual Conference of the EuroMed Academy of Business, 2016. Google Scholar
  10. Daniel Funke, Sebastian Lamm, Ulrich Meyer, Manuel Penschuck, Peter Sanders, Christian Schulz, Darren Strash, and Moritz von Looz. Communication-free massively distributed graph generation. Journal of Parallel and Distributed Computing, 131:200-217, 2019. URL: https://doi.org/10.1016/j.jpdc.2019.03.011.
  11. Georgios Gousios. The ghtorrent dataset and tool suite. In Proceedings of the 10th Working Conference on Mining Software Repositories, MSR '13, pages 233-236, Piscataway, NJ, USA, 2013. IEEE Press. URL: http://dl.acm.org/citation.cfm?id=2487085.2487132.
  12. Ilya Grigorik. Github Archive. https://www.githubarchive.org/, 2012.
  13. Luca Gugelmann, Konstantinos Panagiotou, and Ueli Peter. Random hyperbolic graphs: Degree sequence and clustering. In Artur Czumaj, Kurt Mehlhorn, Andrew Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming, pages 573-585, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. Google Scholar
  14. Stephen G. Kobourov and Kevin Wampler. Non-euclidean spring embedders, pages 207-214. IEEE Computer Society, 2004. URL: https://doi.org/10.1109/INFVIS.2004.49.
  15. Eryk Kopczyński, Dorota Celińska, and Marek Čtrnáct. HyperRogue: Playing with hyperbolic geometry. In Proceedings of Bridges #1: Mathematics, Art, Music, Architecture, Education, Culture, pages 9-16, Phoenix, Arizona, 2017. Tessellations Publishing. Google Scholar
  16. Eryk Kopczyński and Dorota Celińska-Kopczyńska. RogueViz: non-Euclidean geometry engine for visualizations, games, math art, and research, October 2021. URL: https://github.com/zenorogue/hyperrogue/.
  17. John Lamping, Ramana Rao, and Peter Pirolli. A focus+context technique based on hyperbolic geometry for visualizing large hierarchies. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI '95, pages 401-408, New York, NY, USA, 1995. ACM Press/Addison-Wesley Publishing Co. URL: https://doi.org/10.1145/223904.223956.
  18. Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data, June 2014. Google Scholar
  19. Tamara Munzner. Exploring large graphs in 3d hyperbolic space. IEEE Computer Graphics and Applications, 18(4):18-23, 1998. URL: https://doi.org/10.1109/38.689657.
  20. Fragkiskos Papadopoulos, Rodrigo Aldecoa, and Dmitri Krioukov. Network geometry inference using common neighbors. Phys. Rev. E, 92:022807, August 2015. URL: https://doi.org/10.1103/PhysRevE.92.022807.
  21. Fragkiskos Papadopoulos, Maksim Kitsak, M. Angeles Serrano, Marian Boguñá, and Dmitri Krioukov. Popularity versus Similarity in Growing Networks. Nature, 489:537-540, September 2012. Google Scholar
  22. Manuel Penschuck. Generating practical random hyperbolic graphs in near-linear time and with sub-linear memory. In SEA, 2017. Google Scholar
  23. Frederic Sala, Chris De Sa, Albert Gu, and Christopher Re. Representation tradeoffs for hyperbolic embeddings. In Proc. ICML, pages 4460-4469, Stockholmsmässan, Stockholm Sweden, 2018. PMLR. URL: http://proceedings.mlr.press/v80/sala18a.html.
  24. Zeynab Samei and Mahdi Jalili. Application of hyperbolic geometry in link prediction of multiplex networks. Scientific Reports, 9(1):12604, August 2019. URL: https://doi.org/10.1038/s41598-019-49001-7.
  25. Moritz von Looz. High-Performance Graph Algorithms. PhD thesis, Karlsruher Institut für Technologie (KIT), 2019. URL: https://doi.org/10.5445/IR/1000095908.
  26. Moritz von Looz, Henning Meyerhenke, and Roman Prutkin. Generating Random Hyperbolic Graphs in Subquadratic Time, pages 467-478. Springer Berlin Heidelberg, Berlin, Heidelberg, 2015. URL: https://doi.org/10.1007/978-3-662-48971-0_40.
  27. Moritz von Looz, Mustafa Safa Ozdayi, S. Laue, and Henning Meyerhenke. Generating massive complex networks with hyperbolic geometry faster in practice. 2016 IEEE High Performance Extreme Computing Conference (HPEC), pages 1-6, 2016. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail