Advertisement

Generating Random Hyperbolic Graphs in Subquadratic Time

  • Moritz von Looz
  • Henning Meyerhenke
  • Roman Prutkin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)

Abstract

Complex networks have become increasingly popular for modeling various real-world phenomena. Realistic generative network models are important in this context as they simplify complex network research regarding data sharing, reproducibility, and scalability studies. Random hyperbolic graphs are a very promising family of geometric graphs with unit-disk neighborhood in the hyperbolic plane. Previous work provided empirical and theoretical evidence that this generative graph model creates networks with many realistic features.

In this work we provide the first generation algorithm for random hyperbolic graphs with subquadratic running time. We prove a time complexity of \(O((n^{3/2}+m) \log n)\) with high probability for the generation process. This running time is confirmed by experimental data with our implementation. The acceleration stems primarily from the reduction of pairwise distance computations through a polar quadtree, which we adapt to hyperbolic space for this purpose and which can be of independent interest. In practice we improve the running time of a previous implementation (which allows more general neighborhoods than the unit disk) by at least two orders of magnitude this way. Networks with billions of edges can now be generated in a few minutes.

Keywords

Complex networks Hyperbolic geometry Efficient range query Polar quadtree Generative graph model 

References

  1. 1.
    Aiello, W., Chung, F., Lu, L.: A random graph model for massive graphs. In: Proceedings of the 32nd ACM Symposium on Theory of Computing, pp. 171–180. ACM (2000)Google Scholar
  2. 2.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aldecoa, R., Orsini, C., Krioukov, D.: Hyperbolic graph generator. Comput. Phys. Commun. 196, 492–496 (2015). doi: 10.1016/j.cpc.2015.05.028. http://www.sciencedirect.com/science/article/pii/S0010465515002088
  4. 4.
    Anderson, J.W.: Hyperbolic Geometry. Springer Undergraduate Mathematics Series, 2nd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  5. 5.
    Bader, D.A., Berry, J., Kahan, S., Murphy, R., Riedy, E.J., Willcock, J.: Graph 500 benchmark 1 (“search”), version 1.1. Technical report, Graph 500 (2010)Google Scholar
  6. 6.
    Batagelj, V., Brandes, U.: Efficient generation of large random networks. Phys. Rev. E 71(3), 036113 (2005)CrossRefGoogle Scholar
  7. 7.
    Bode, M., Fountoulakis, N., Müller, T.: On the giant component of random hyperbolic graphs. In: The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol. 16, pp. 425–429. Scuola Normale Superiore (2013)Google Scholar
  8. 8.
    Bode, M., Fountoulakis, N., Müller, T.: The probability that the hyperbolic random graph is connected (2014). http://web.mat.bham.ac.uk/N.Fountoulakis/BFM.pdf. Preprint
  9. 9.
    Chakrabarti, D., Faloutsos, C.: Graph mining: laws, generators, and algorithms. ACM Comput. Surv. (CSUR) 38(1), 2 (2006)CrossRefGoogle Scholar
  10. 10.
    Chakrabarti, D., Zhan, Y., Faloutsos, C.: R-MAT: a recursive model for graph mining. In Proceedings of the 4th SIAM International Conference on Data Mining (SDM), Orlando, FL. SIAM, April 2004Google Scholar
  11. 11.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks: from Biological Nets to the Internet and WWW. Oxford University Press, Oxford (2003)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gugelmann, L., Panagiotou, K., Peter, U.: Random hyperbolic graphs: degree sequence and clustering. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 573–585. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  13. 13.
    Kiwi, M., Mitsche, D.: A bound for the diameter of random hyperbolic graphs. In: 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 26–39. SIAM, January 2015Google Scholar
  14. 14.
    Kolda, T.G., Pinar, A., Todd, P., Seshadhri, C.: A scalable generative graph model with community structure. SIAM J. Sci. Comput. 36(5), C424–C452 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82(3), 036106 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lancichinetti, A., Fortunato, S., Radicchi, F.: Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78(4), 046110 (2008)CrossRefGoogle Scholar
  17. 17.
    Miller, J.C., Hagberg, A.: Efficient generation of networks with given expected degrees. In: Frieze, A., Horn, P., Prałat, P. (eds.) WAW 2011. LNCS, vol. 6732, pp. 115–126. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  18. 18.
    Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010) CrossRefzbMATHGoogle Scholar
  19. 19.
    Samet, H.: Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann Publishers Inc., San Francisco (2005) zbMATHGoogle Scholar
  20. 20.
    Seshadhri, C., Kolda, T.G., Pinar, A.: Community structure and scale-free collections of Erdős-Rényi graphs. Phys. Rev. E 85(5), 056109 (2012)CrossRefGoogle Scholar
  21. 21.
    Seshadhri, C., Pinar, A., Kolda, T.G.: The similarity between stochastic Kronecker and Chung-Lu graph models. In: Proceedings of the 2012 SIAM International Conference on Data Mining (SDM), pp. 1071–1082 (2012)Google Scholar
  22. 22.
    Staudt, C.L., Sazonovs, A., Meyerhenke, H.: NetworKit: an interactive tool suite for high-performance network analysis (2014). arXiv preprint arXiv:1403.3005
  23. 23.
    von Looz, M., Meyerhenke, H.: Querying probabilistic neighborhoods in spatial data sets efficiently, September 2015. ArXiv preprint arXiv:1509.01990
  24. 24.
    von Looz, M., Meyerhenke, H., Prutkin, R.: Generating random hyperbolic graphs in subquadratic time, September 2015. ArXiv preprint arXiv:1501.03545

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Moritz von Looz
    • 1
  • Henning Meyerhenke
    • 1
  • Roman Prutkin
    • 1
  1. 1.Karlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations