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Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs

Authors Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, Ulrich Meyer, Manuel Penschuck, Christopher Weyand



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Author Details

Thomas Bläsius
  • Hasso Plattner Institute, Potsdam, Germany
Tobias Friedrich
  • Hasso Plattner Institute, Potsdam, Germany
Maximilian Katzmann
  • Hasso Plattner Institute, Potsdam, Germany
Ulrich Meyer
  • Goethe University, Frankfurt, Germany
Manuel Penschuck
  • Goethe University, Frankfurt, Germany
Christopher Weyand
  • Hasso Plattner Institute, Potsdam, Germany

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Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, Ulrich Meyer, Manuel Penschuck, and Christopher Weyand. Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 21:1-21:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.21

Abstract

Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent beta, and high clustering that can be controlled via the temperature T. We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to T = 0. We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation. Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input. Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Generating random combinatorial structures
Keywords
  • hyperbolic random graphs
  • geometric inhomogeneous random graph
  • efficient network generation

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