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International Symposium on Algorithms and Computation

ISAAC 2015: Algorithms and Computation pp 467–478Cite as

Generating Random Hyperbolic Graphs in Subquadratic Time

Part of the Lecture Notes in Computer Science book series (LNTCS,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

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.

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Notes

  1. 1.

    We consider the name “hyperbolic unit-disk graphs” as more precise, but we use “random hyperbolic graphs” to be consistent with the literature. More general neighborhoods are possible [15] but not considered here since most theoretical works [7, 8, 12] are for unit-disk neighborhoods.

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Authors and Affiliations

  1. Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Moritz von Looz, Henning Meyerhenke & Roman Prutkin

Authors
  1. Moritz von Looz
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  2. Henning Meyerhenke
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  3. Roman Prutkin
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Correspondence to Moritz von Looz .

Editor information

Editors and Affiliations

  1. Masdar Institute, Abu Dhabi, United Arab Emirates

    Khaled Elbassioni

  2. Kyoto University, Kyoto, Japan

    Kazuhisa Makino

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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

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  • DOI: https://doi.org/10.1007/978-3-662-48971-0_40

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