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Algorithmica

, Volume 80, Issue 8, pp 2324–2344 | Cite as

Cliques in Hyperbolic Random Graphs

  • Thomas Bläsius
  • Tobias Friedrich
  • Anton Krohmer
Article
  • 169 Downloads

Abstract

Most complex real world networks display scale-free features. This characteristic motivated the study of numerous random graph models with a power-law degree distribution. There is, however, no established and simple model which also has a high clustering of vertices as typically observed in real data. Hyperbolic random graphs bridge this gap. This natural model has recently been introduced by Krioukov et al. (in Phys Rev E 82(3):036106, 2010) and has shown theoretically and empirically to fulfill all typical properties of real world networks, including power-law degree distribution and high clustering. We study cliques in hyperbolic random graphs G and present new results on the expected number of k-cliques \(\mathbb {E}\left[ K_k\right] \) and the size of the largest clique \(\omega (G)\). We observe that there is a phase transition at power-law exponent \(\beta = 3\). More precisely, for \(\beta \in (2,3)\) we prove \(\mathbb {E}\left[ K_k\right] =n^{k (3-\beta )/2} \varTheta (k)^{-k}\) and \(\omega (G)=\varTheta (n^{(3-\beta )/2})\), while for \(\beta \geqslant 3\) we prove \(\mathbb {E}\left[ K_k\right] =n \, \varTheta (k)^{-k} \) and \(\omega (G)=\varTheta (\log (n)/ \log \log n)\). Furthermore, we show that for \(\beta \geqslant 3\), cliques in hyperbolic random graphs can be computed in time \(\mathcal {O}(n)\). If the underlying geometry is known, cliques can be found with worst-case runtime \(\mathcal {O}(m \cdot n^{2.5})\) for all values of \(\beta \).

Keywords

Hyperbolic random graphs Random graphs Scale-free networks Social networks Cliques 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Hasso Plattner InstitutePotsdamGermany

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