Advertisement

Extremal Dependencies and Rank Correlations in Power Law Networks

  • Yana Volkovich
  • Nelly Litvak
  • Bert Zwart
Part of the Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering book series (LNICST, volume 5)

Abstract

We analyze dependencies in complex networks characterized by power laws (Web sample, Wikipedia sample and a preferential attachment graph) using statistical techniques from the extreme value theory and the theory of multivariate regular variation. To the best of our knowledge, this is the first attempt to apply this well developed methodology to comprehensive graph data. The new insights this yields are striking: the three above-mentioned data sets are shown to have a totally different dependence structure between graph parameters, such as in-degree and PageRank. Based on the proposed approach, we suggest a new measure for rank correlations. Unlike most known methods, this measure is especially sensitive to rank permutations for top-ranked nodes. Using the new correlation measure, we demonstrate that the PageRank ranking is not sensitive to moderate changes in the damping factor.

Keywords

Extremal dependencies Statistical analysis Power laws PageRank Web Wikipedia Preferential attachment 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albert, R., Barabási, A.L.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, Chichester (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Boldi, P., Vigna, S.: The WebGraph framework I: Compression techniques. In: Proc. of the Thirteenth International World Wide Web Conference (WWW 2004), pp. 595–601 (2004)Google Scholar
  4. 4.
    Chakrabarti, D., Faloutsos, C.: Graph mining: Laws, generators, and algorithms. ACM Comput. Surv. 38(1), 2 (2006)CrossRefGoogle Scholar
  5. 5.
    de Haan, L., de Ronde, J.: Sea and wind: multivariate extremes at work. Extremes 1(1), 7–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Donato, D., Laura, L., Leonardi, S., Millozi, S.: Large scale properties of the webgraph. Eur. Phys. J. 38, 239–243 (2004)CrossRefGoogle Scholar
  7. 7.
    Doyle, J.C., Alderson, D.L., Li, L., Low, S., Roughan, M., Shalunov, S., Tanaka, R., Willinger, W.: The robust yet fragile nature of the Internet. PNAS 102(41), 14497–14502 (2005)CrossRefGoogle Scholar
  8. 8.
    Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Fortunato, S., Boguñá, M., Flammini, A., Menczer, F.: Approximating pageRank from in-degree. In: Aiello, W., Broder, A., Janssen, J., Milios, E.E. (eds.) WAW 2006. LNCS, vol. 4936, pp. 59–71. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Kleinberg, J.M.: Authoritative sources in a hyperlinked environment. JACM 46(5), 604–632 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Langville, A.N., Meyer, C.D.: Google’s PageRank and beyond: the science of search engine rankings. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  12. 12.
    Li, L., Alderson, D.L., Doyle, J.C., Willinger, W.: Towards a theory of scale-free graphs: definition, properties, and implications. Internet Math. 2(4), 431–523 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Melucci, M.: On rank correlation in information retrieval evaluation. SIGIR Forum 41(1), 18–33 (2007)CrossRefGoogle Scholar
  14. 14.
    Mikosch, T.: Modelling dependence and tails in financial time series. In: Symposium in Honour of Ole E. Barndorff-Nielsen (Aarhus, 2000). Memoirs, vol. 16, pp. 61–73. Univ. Aarhus, Aarhus (2000)Google Scholar
  15. 15.
    Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Math. 1(2), 226–251 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Newman, M.E.J.: Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46, 323–351 (2005)CrossRefGoogle Scholar
  18. 18.
    Page, L., Brin, S., Motwani, R., Winograd, T.: The PageRank citation ranking: Bringing order to the Web. Technical report, Stanford Digital Library Technologies Project (1998)Google Scholar
  19. 19.
    Park, K., Willinger, W.: Self-similar network traffic and performance evaluation. Wiley, Chichester (2000)CrossRefGoogle Scholar
  20. 20.
    Resnick, S.I.: Heavy-tail Phenomena. Springer, New York (2007)zbMATHGoogle Scholar
  21. 21.
    Volkovich, Y., Litvak, N., Zwart, B.: A framework for evaluating statistical dependencies and rank correlations in power law graphs. Memorandum 1868, University of Twente, Enschede (2008)Google Scholar

Copyright information

© ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering 2009

Authors and Affiliations

  • Yana Volkovich
    • 1
  • Nelly Litvak
    • 1
  • Bert Zwart
    • 2
  1. 1.University of TwenteEnschedeThe Netherlands
  2. 2.CWI, Science Park Amsterdam, Kruislaan 413AmsterdamThe Netherlands

Personalised recommendations