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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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Correspondence to Moritz von Looz .

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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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