Abstract
Recently, Papadopoulos, Krioukov, Boguñá and Vahdat [Infocom’10] introduced a random geometric graph model that is based on hyperbolic geometry. The authors argued empirically and by some preliminary mathematical analysis that the resulting graphs have many of the desired properties for models of large real-world graphs, such as high clustering and heavy tailed degree distributions. By computing explicitly a maximum likelihood fit of the Internet graph, they demonstrated impressively that this model is adequate for reproducing the structure of such with high accuracy.
In this work we initiate the rigorous study of random hyperbolic graphs. We compute exact asymptotic expressions for the expected number of vertices of degree k for all k up to the maximum degree and provide small probabilities for large deviations. We also prove a constant lower bound for the clustering coefficient. In particular, our findings confirm rigorously that the degree sequence follows a power-law distribution with controllable exponent and that the clustering is nonvanishing.
See http://arxiv.org/abs/1205.1470 for the full version.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aiello, W., Chung, F., Lu, L.: Random evolution in massive graphs. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 510–519. IEEE (2001)
Anderson, J.W.: Hyperbolic geometry, 2nd edn. Springer (2005)
Barabási, A.L., Albert, R.: Emergence of Scaling in Random Networks. Science 286(5439), 509–512 (1999)
Bollobás, B., Riordan, O., Spencer, J., Tusnády, G.: The degree sequence of a scale-free random graph process. Random Structures and Algorithms 18(3), 279–290 (2001)
Bollobás, B., Riordan, O.: Mathematical results on scale-free random graphs. In: Handbook of Graphs and Networks, pp. 1–34 (2002)
Borgs, C., Chayes, J., Daskalakis, C., Roch, S.: First to market is not everything: an analysis of preferential attachment with fitness. In: Proceedings of the 39th ACM Symposium on Theory of Computing, pp. 135–144. ACM (2007)
Buckley, P., Osthus, D.: Popularity based random graph models leading to a scale-free degree sequence. Discrete Mathematics 282(1-3), 53–68 (2004)
Chierichetti, F., Kumar, R., Lattanzi, S., Panconesi, A., Raghavan, P.: Models for the compressible web. In: Proceedings of the 50th IEEE Symposium on Foundations of Computer Science, pp. 331–340. IEEE (2009)
Cooper, C., Frieze, A.: A general model of web graphs. Random Structures & Algorithms 22(3), 311–335 (2003)
Dubhashi, D., Panconesi, A., Press, C.U.: Concentration of measure for the analysis of randomized algorithms. Cambridge University Press (2009)
Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. ACM SIGCOMM Computer Communication Review 29, 251–262 (1999)
Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic Geometry of Complex Networks. Physical Review E 82(3) (2010)
Lattanzi, S., Sivakumar, D.: Affiliation networks. In: Proceedings of the 41st ACM Symposium on Theory of Computing, pp. 427–434. ACM (2009)
Leskovec, J., Kleinberg, J., Faloutsos, C.: Graphs over time: densification laws, shrinking diameters and possible explanations. In: Proceedings of the 11th ACM SIGKDD International Conference on Knowledge Discovery in Data Mining, pp. 177–187. ACM (2005)
Milgram, S.: The small world problem. Psychology Today 2(1), 60–67 (1967)
Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1(2), 226–251 (2004)
Newman, M., Park, J.: Why social networks are different from other types of networks. Physical Review E 68(3), 036122 (2003)
Papadopoulos, F., Krioukov, D., Boguñá, M., Vahdat, A.: Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces. In: Proceedings of the 29th IEEE International Conference on Computer Communications, INFOCOM 2010, pp. 2973–2981 (2010)
Serrano, M., Boguñá, M.: Clustering in complex networks. i. general formalism. Physical Review E 74(5), 056114 (2006)
Travers, J., Milgram, S.: An experimental study of the small world problem. Sociometry, 425–443 (1969)
Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393(6684), 440–442 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gugelmann, L., Panagiotou, K., Peter, U. (2012). Random Hyperbolic Graphs: Degree Sequence and Clustering. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31585-5_51
Download citation
DOI: https://doi.org/10.1007/978-3-642-31585-5_51
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31584-8
Online ISBN: 978-3-642-31585-5
eBook Packages: Computer ScienceComputer Science (R0)