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

Cliques in Hyperbolic Random Graphs

  • Thomas Bläsius
  • Tobias Friedrich
  • Anton Krohmer


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 \(\beta \).


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


  1. 1.
    Aiello, W., Chung, F.R.K., Lu, L.: A random graph model for massive graphs. In: 32nd Symposium Theory of Computing (STOC), pp. 171–180 (2000)Google Scholar
  2. 2.
    Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bode, M., Fountoulakis, N., Müller, T.: The Probability that the Hyperbolic Random Graph is Connected. (2014)
  4. 4.
    Boguñá, M., Papadopoulos, F., Krioukov, D.: Sustaining the internet with hyperbolic mapping. Nat. Commun. 1, 62 (2010)CrossRefGoogle Scholar
  5. 5.
    Candellero, E., Fountoulakis, N.: Clustering and the hyperbolic geometry of complex networks. In: 11th International Workshop Algorithms and Models for the Web Graph (WAW), pp. 1–12 (2014)Google Scholar
  6. 6.
    Candellero, E., Fountoulakis, N.: Bootstrap percolation and the geometry of complex networks. Stoch. Process. Appl. 126(1), 234–264 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Ann. Discret. Math. 48, 165–177 (1991)CrossRefMATHGoogle Scholar
  8. 8.
    Diestel, R.: Graph Theory, 4th edn. Springer, Berlin (2012)MATHGoogle Scholar
  9. 9.
    Dubhashi, D., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  10. 10.
    Fountoulakis, N.: On a geometrization of the Chung–Lu model for complex networks. J. Complex Netw. 3(3), 361–387 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Friedrich, T., Krohmer, A.: Parameterized clique on scale-free networks. In: 23rd International Symposium Algorithms and Computation (ISAAC), pp. 659–668 (2012)Google Scholar
  12. 12.
    Friedrich, T., Krohmer, A.: Cliques in hyperbolic random graphs. In: 34th IEEE Conference on Computer Communications (INFOCOM), pp. 1544–1552 (2015)Google Scholar
  13. 13.
    Gugelmann, L., Panagiotou, K., Peter, U.: Random hyperbolic graphs: degree sequence and clustering. In: 39th International Colloquium on Automata, Languages and Programming (ICALP), pp. 573–585 (2012)Google Scholar
  14. 14.
    Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Janson, S., Łuczak, T., Norros, I.: Large cliques in a power-law random graph. J. Appl. Probab. 47(4), 1124–1135 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jensen, J.: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30(1), 175–193 (1906)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82(3), 036106 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Leskovec, J., Krevl, A.: SNAP Datasets: Stanford Large Network Dataset Collection. (2014)
  19. 19.
    Li, L., Alderson, D., Doyle, J.C., Willinger, W.: Towards a theory of scale-free graphs: definition, properties, and implications. Internet Math. 2(4), 431–523 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liben-Nowell, D., Kleinberg, J.M.: The link-prediction problem for social networks. J. Assoc. Inf. Sci. Technol. 58(7), 1019–1031 (2007)CrossRefGoogle Scholar
  21. 21.
    Newman, M.E.J.: Clustering and preferential attachment in growing networks. Phys. Rev. E 64, 025102 (2001)CrossRefGoogle Scholar
  22. 22.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Norros, I., Reittu, H.: On a conditionally Poissonian graph process. Adv. Appl. Probab. 38(1), 59–75 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Papadopoulos, F., Krioukov, D., Boguñá, M., Vahdat, A.: Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces. In: 29th IEEE Conference on Computer Communications (INFOCOM), pp. 2973–2981 (2010)Google Scholar
  25. 25.
    Penrose, M.D.: Random Geometric Graphs. Oxford University Press, Oxford (2003)CrossRefMATHGoogle Scholar
  26. 26.
    Peter, U.: Random Graph Models for Complex Systems. PhD thesis, ETH Zürich (2014)Google Scholar
  27. 27.
    Raab, M., Steger, A.: “Balls into bins”—a simple and tight analysis. In: 2nd International Workshop on Randomization and Computation (RANDOM), pp. 159–170 (1998)Google Scholar
  28. 28.
    van der Hofstad, R.: Random Graphs and Complex Networks. Lecture notes (2016)
  29. 29.
    Vázquez, A.: Growing network with local rules: preferential attachment, clustering hierarchy, and degree correlations. Phys. Rev. E 67(5), 056104 (2003)CrossRefGoogle Scholar
  30. 30.
    Wang, Y., Yeh, Y.-N.: Polynomials with real zeros and Pólya frequency sequences. J. Comb. Theory Ser. A 109(1), 63–74 (2005)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Hasso Plattner InstitutePotsdamGermany

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