On Computing the Hyperbolicity of Real-World Graphs

  • Michele Borassi
  • David Coudert
  • Pierluigi Crescenzi
  • Andrea Marino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)

Abstract

The (Gromov) hyperbolicity is a topological property of a graph, which has been recently applied in several different contexts, such as the design of routing schemes, network security, computational biology, the analysis of graph algorithms, and the classification of complex networks. Computing the hyperbolicity of a graph can be very time consuming: indeed, the best available algorithm has running-time \(\mathcal{O}(n^{3.69})\), which is clearly prohibitive for big graphs. In this paper, we provide a new and more efficient algorithm: although its worst-case complexity is \(\mathcal{O}(n^4)\), in practice it is much faster, allowing, for the first time, the computation of the hyperbolicity of graphs with up to 200,000 nodes. We experimentally show that our new algorithm drastically outperforms the best previously available algorithms, by analyzing a big dataset of real-world networks. Finally, we apply the new algorithm to compute the hyperbolicity of random graphs generated with the Erdös-Renyi model, the Chung-Lu model, and the Configuration Model.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albert, R., DasGupta, B., Mobasheri, N.: Topological implications of negative curvature for biological and social networks. Physical Review E 89(3), 32811 (2014)CrossRefGoogle Scholar
  2. 2.
    Alrasheed, H., Dragan, F.F.: Core-Periphery Models for Graphs based on their delta-Hyperbolicity: An Example Using Biological Networks. Studies in Computational Intelligence 597, 65–77 (2015)Google Scholar
  3. 3.
    Bavelas, A.: Communication patterns in task-oriented groups. Journal of the Acoustical Society of America 22, 725–730 (1950)CrossRefGoogle Scholar
  4. 4.
    Boguna, M., Papadopoulos, F., Krioukov, D.: Sustaining the Internet with Hyperbolic Mapping. Nature Communications 62 (October 2010)Google Scholar
  5. 5.
    Borassi, M., Chessa, A., Caldarelli, G.: Hyperbolicity Measures “Democracy” in Real-World Networks. Preprint on arXiv, pp. 1–10 (March 2015)Google Scholar
  6. 6.
    Chen, W., Fang, W., Hu, G., Mahoney, M.: On the hyperbolicity of small-world and treelike random graphs. Internet Mathematics, pp. 1–40 (2013)Google Scholar
  7. 7.
    Chepoi, V., Dragan, F.F., Estellon, B., Habib, M., Vaxès, Y.: Notes on diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs. Electronic Notes in Discrete Mathematics 31, 231–234 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chepoi, V., Dragan, F.F., Estellon, B., Habib, M., Vaxès, Y., Xiang, Y.: Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs. Algorithmica 62(3-4), 713–732 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chepoi, V., Estellon, B.: Packing and covering δ-hyperbolic spaces by balls. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 59–73. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Cohen, N., Coudert, D., Ducoffe, G., Lancin, A.: Applying clique-decomposition for computing Gromov hyperbolicity. Research Report RR-8535, HAL (2014)Google Scholar
  11. 11.
    Cohen, N., Coudert, D., Lancin, A.: On computing the Gromov hyperbolicity. ACM J. Exp. Algor. (2015)Google Scholar
  12. 12.
    Dress, A., Huber, K., Koolen, J., Moulton, V., Spillner, A.: Basic Phylogenetic Combinatorics. Cambridge University Press, Cambridge (2011)Google Scholar
  13. 13.
    Fang, W.: On Hyperbolic Geometry Structure of Complex Networks. Master’s thesis, MPRI at ENS and Microsoft Research Asia (2011)Google Scholar
  14. 14.
    Fournier, H., Ismail, A., Vigneron, A.: Computing the Gromov hyperbolicity of a discrete metric space. Information Processing Letters 115(6-8), 576–579 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, Springer, Heidelberg (1987)Google Scholar
  16. 16.
    Havlin, S., Cohen, R.: Complex networks: structure, robustness and function. Cambridge University Press, Cambridge (2010)MATHGoogle Scholar
  17. 17.
    Hofstad, R.V.D.: Random Graphs and Complex Networks (2014)Google Scholar
  18. 18.
    Jonckheere, E.A., Lohsoonthorn, P.: Geometry of network security. In: American Control Conference, vol. 2, pp. 976–981. IEEE, Boston (2004)Google Scholar
  19. 19.
    Kennedy, W.S., Narayan, O., Saniee, I.: On the Hyperbolicity of Large-Scale Networks. CoRR, abs/1307.0031:1–22 (2013)Google Scholar
  20. 20.
    Krauthgamer, R., Lee, J.R.: Algorithms on negatively curved spaces. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 119–132 (2006)Google Scholar
  21. 21.
    Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A.: Hyperbolic Geometry of Complex Networks. Physical Review E 82(3), 36106 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mitche, D., Pralat, P.: On the hyperbolicity of random graphs. The Electronic Journal of Combinatorics 21(2), 1–24 (2014)MathSciNetGoogle Scholar
  23. 23.
    Narayan, O., Saniee, I.: The Large Scale Curvature of Networks. Physical Review E 84, 66108 (2011)CrossRefGoogle Scholar
  24. 24.
    Narayan, O., Saniee, I., Tucci, G.H.: Lack of Hyperbolicity in Asymptotic Erdös–Renyi Sparse Random Graphs. Internet Mathematics, 1–10 (2015)Google Scholar
  25. 25.
    Newman, M.E.J.: The Structure and Function of Complex Networks. SIAM Review 45(2), 167–256 (2003)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Papadopoulos, F., Krioukov, D., Boguna, M., Vahdat, A.: Greedy forwarding in scale-free networks embedded in hyperbolic metric spaces. ACM SIGMETRICS Performance Evaluation Review 37(2), 15–17 (2009)CrossRefGoogle Scholar
  27. 27.
    Shang, Y.: Non-Hyperbolicity of Random Graphs with Given Expected Degrees. Stochastic Models 29(4), 451–462 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Soto Gómez, M.A.: Quelques propriétés topologiques des graphes et applications à internet et aux réseaux. PhD thesis, Univ. Paris Diderot, Paris 7 (2011)Google Scholar
  29. 29.
    Wu, Y., Zhang, C.: Hyperbolicity and chordality of a graph. The Electronic Journal of Combinatorics 18(1), 1–22 (2011)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Michele Borassi
    • 1
  • David Coudert
    • 2
  • Pierluigi Crescenzi
    • 3
  • Andrea Marino
    • 4
  1. 1.IMT Institute for Advanced Studies LuccaLuccaItaly
  2. 2.InriaSophia AntipolisFrance
  3. 3.Dipartimento di Ingegneria dell’InformazioneUniversità di FirenzeFirenzeItaly
  4. 4.Dipartimento di InformaticaUniversità di PisaPisaItaly

Personalised recommendations