Mukund Seshadri
Christos Faloutsos
ABSTRACT
We analyze a massive social network, gathered from the records of a large mobile phone operator, with more than a million users and tens of millions of calls. We examine the distributions of the number of phone calls per customer; the total talk minutes per customer; and the distinct number of calling partners per customer. We find that these distributions are skewed, and that they significantly deviate from what would be expected by power-law and lognormal distributions.
To analyze our observed distributions (of number of calls, distinct call partners, and total talk time), we propose PowerTrack , a method which fits a lesser known but more suitable distribution, namely the Double Pareto LogNormal (DPLN) distribution, to our data and track its parameters over time. Using PowerTrack , we find that our graph changes over time in a way consistent with a generative process that naturally results in the DPLN distributions we observe. Furthermore, we show that this generative process lends itself to a natural and appealing social wealth interpretation in the context of social networks such as ours. We discuss the application of those results to our model and to forecasting.
Supplemental Material
- L. A. Adamic and B. A. Huberman. The Web-shidden order. Communications of the ACM, 44(9):55--60, 2001. Google Scholar
Digital Library
- L. A. N. Amaral, A. Scala, M. Barthélémy, and H. E. Stanley. Classes of small-world networks. Proceedings of the National Academy of Sciences, 97(21):11149--11152, 2000.Google ScholarCross Ref
- A.-L. Barabási and R. Albert. Emergence of scaling in random networks. Science, 286:509--512, 1999.Google ScholarCross Ref
- Z. Bi, C. Faloutsos, and F. Korn. The DGX distribution for mining massive, skewed data. In Proceedings of ACM KDD, pages 17--26, New York, NY, 2001. ACM Press. Google ScholarDigital Library
- P. Boldi, B. Codenotti, M. Santini, and S. Vigna. Structural properties of the African Web. In International World Wide Web Conference, New York, NY, 2002. ACM Press.Google Scholar
- A. Z. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener. Graph structure in the web: experiments and models. In International World Wide Web Conference, New York, NY, 2000. ACM Press. Google ScholarDigital Library
- A. Clauset. Power law distributions in empirical data. http://www.santafe.edu/~aaronc/powerlaws/ Google ScholarDigital Library
- A. Clauset, C. R. Shalizi, and M. E. J. Newman. Power-law distributions in empirical data. ArXiv e-print 0706.1062v1, 2007.Google Scholar
- C. Cortes, D. Pregibon, and C. Volinsky. Communities of interest. In IDA '01: Proceedings of the 4th International Conference on Advances in Intelligent Data Analysis, pages 105--114, London, UK, 2001. Springer-Verlag. Google ScholarDigital Library
- M. Faloutsos, P. Faloutsos, and C. Faloutsos. On power-law relationships of the Internet topology. In Proceedings of ACM SIGCOMM, pages 251--262, New York, NY, 1999. Google ScholarDigital Library
- R. Govindan and H. Tangmunarunkit. Heuristics for Internet map discovery. In IEEE INFOCOM, pages 1371--1380, Los Alamitos, CA, March 2000. IEEE Computer Society Press.Google ScholarCross Ref
- J.-P. Onnela, J. Saramaäki, J. Hyvöven, G. Szabó, M. Argollo de Menezes, K. Kaski, and A.-L. Barabási. Structure and Tie Strengths in Mobile Communication Networks. New Journal of Physics, 9, 2007.Google Scholar
- S. Keshav. Why cell phones will dominate the future Internet. Computer Communications Review, 35(2), April 2005. Google ScholarDigital Library
- S. R. Kumar, P. Raghavan, S. Rajagopalan, and A. Tomkins. Extracting large-scale knowledge bases from the web. VLDB, pages 639--650, 1999. Google ScholarDigital Library
- M. Mitzenmacher. A Brief History of Generative Models for Power Law and Lognormal Distributions. Internet Mathematics, 1(2):226--251.Google ScholarCross Ref
- M. Mitzenmacher. Dynamic Models for File Sizes and Double Pareto Distributions. Internet Mathematics, 1(3):305--334, 2004.Google ScholarCross Ref
- A. A. Nanavati, S. Gurumurthy, G. Das, D. Chakraborty, K. Dasgupta, S. Mukherjea, and A. Joshi. On the structural properties of massive telecom call graphs : findings and implications. Proc. of 15th ACM Conference on Information and Knowledge Management, pages 435--444, 2006. Google ScholarDigital Library
- M. E. J. Newman. Power laws, pareto distributions and Zipf's law. Contemporary Physics, 46:323--351, 2005.Google ScholarCross Ref
- M. E. J. Newman. Power laws, pareto distributions and Zipf's law. Contemporary Physics, 46:323--351, 2005.Google ScholarCross Ref
- V. Pareto. Oeuvres Completes. Droz, Geneva, 1896.Google Scholar
- D. M. Pennock, G. W. Flake, S. Lawrence, E. J. Glover, and C. L. Giles. Winners don't take all: Characterizing the competition for links on the Web. Proceedings of the National Academy of Sciences, 99(8):5207--5211, 2002.Google ScholarCross Ref
- S. Redner. How popular is your paper? an empirical study of the citation distribution. The European Physics Journal B, 4:131--134, 1998.Google ScholarCross Ref
- W. Reed and M. Jorgensen. The double pareto-lognormal distribution - a new parametric model for size distribution. Communications in Statistics -Theory and Methods, 33(8):1733--1753, 2004.Google ScholarCross Ref
- R.Gibrat. inégalités économiques. Librarie du Recuil Sirey, 1931.Google Scholar
- G. Szabo and A.-L. Barabasi. Network effects in service usage. ArXiv e-prints physics/0611177, November 2006.Google Scholar
Index Terms
- Mobile call graphs: beyond power-law and lognormal distributions
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