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.
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References
Albert, R., Barabási, A.L.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, Chichester (2004)
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)
Chakrabarti, D., Faloutsos, C.: Graph mining: Laws, generators, and algorithms. ACM Comput. Surv. 38(1), 2 (2006)
de Haan, L., de Ronde, J.: Sea and wind: multivariate extremes at work. Extremes 1(1), 7–45 (1998)
Donato, D., Laura, L., Leonardi, S., Millozi, S.: Large scale properties of the webgraph. Eur. Phys. J. 38, 239–243 (2004)
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)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events. Springer, Heidelberg (1997)
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)
Kleinberg, J.M.: Authoritative sources in a hyperlinked environment. JACM 46(5), 604–632 (1999)
Langville, A.N., Meyer, C.D.: Google’s PageRank and beyond: the science of search engine rankings. Princeton University Press, Princeton (2006)
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)
Melucci, M.: On rank correlation in information retrieval evaluation. SIGIR Forum 41(1), 18–33 (2007)
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)
Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Math. 1(2), 226–251 (2004)
Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)
Newman, M.E.J.: Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46, 323–351 (2005)
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)
Park, K., Willinger, W.: Self-similar network traffic and performance evaluation. Wiley, Chichester (2000)
Resnick, S.I.: Heavy-tail Phenomena. Springer, New York (2007)
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)
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© 2009 ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering
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Volkovich, Y., Litvak, N., Zwart, B. (2009). Extremal Dependencies and Rank Correlations in Power Law Networks. In: Zhou, J. (eds) Complex Sciences. Complex 2009. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02469-6_43
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DOI: https://doi.org/10.1007/978-3-642-02469-6_43
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