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.
This work is partially supported by SPP 1736 Algorithms for Big Data of the German Research Foundation (DFG) and by the Ministry of Science, Research and the Arts Baden-Württemberg (MWK) via project Parallel Analysis of Dynamic Networks.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47 (2002)
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
Anderson, J.W.: Hyperbolic Geometry. Springer Undergraduate Mathematics Series, 2nd edn. Springer, Berlin (2005)
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)
Batagelj, V., Brandes, U.: Efficient generation of large random networks. Phys. Rev. E 71(3), 036113 (2005)
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)
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
Chakrabarti, D., Faloutsos, C.: Graph mining: laws, generators, and algorithms. ACM Comput. Surv. (CSUR) 38(1), 2 (2006)
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 2004
Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks: from Biological Nets to the Internet and WWW. Oxford University Press, Oxford (2003)
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)
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 2015
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)
Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82(3), 036106 (2010)
Lancichinetti, A., Fortunato, S., Radicchi, F.: Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78(4), 046110 (2008)
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)
Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010)
Samet, H.: Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann Publishers Inc., San Francisco (2005)
Seshadhri, C., Kolda, T.G., Pinar, A.: Community structure and scale-free collections of Erdős-Rényi graphs. Phys. Rev. E 85(5), 056109 (2012)
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)
Staudt, C.L., Sazonovs, A., Meyerhenke, H.: NetworKit: an interactive tool suite for high-performance network analysis (2014). arXiv preprint arXiv:1403.3005
von Looz, M., Meyerhenke, H.: Querying probabilistic neighborhoods in spatial data sets efficiently, September 2015. ArXiv preprint arXiv:1509.01990
von Looz, M., Meyerhenke, H., Prutkin, R.: Generating random hyperbolic graphs in subquadratic time, September 2015. ArXiv preprint arXiv:1501.03545
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
von Looz, M., Meyerhenke, H., Prutkin, R. (2015). Generating Random Hyperbolic Graphs in Subquadratic Time. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_40
Download citation
DOI: https://doi.org/10.1007/978-3-662-48971-0_40
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48970-3
Online ISBN: 978-3-662-48971-0
eBook Packages: Computer ScienceComputer Science (R0)