Abstract
If a friend called you 50 times last month, how many times did you call him back? Does the answer change if we ask about SMS, or e-mails? We want to quantify reciprocity between individuals in weighted networks, and we want to discover whether it depends on their topological features (like degree, or number of common neighbors). Here we answer these questions, by studying the call- and SMS records of millions of mobile phone users from a large city, with more than 0.5 billion phone calls and 60 million SMSs, exchanged over a period of six months. Our main contributions are: (1) We propose a novel distribution, the Triple Power Law (3PL), that fits the reciprocity behavior of all 3 datasets we study, with a better fit than older competitors, (2) 3PL is parsimonious; it has only three parameters and thus avoids over-fitting, (3) 3PL can spot anomalies, and we report the most surprising ones, in our real networks, (4) We observe that the degree of reciprocity between users is correlated with their local topological features; reciprocity is higher among mutual users with larger local network overlap and greater degree similarity.
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References
Albert, R., Barabasi, A.-L.: Emergence of scaling in random networks. Science, 509–512 (1999)
Amaral, L.A.N., Scala, A., Barthélémy, M., Stanley, H.E.: Classes of small-world networks. Proceeding of the National Academy of Sciences (2000)
Arnold, B.C.: Bivariate distributions with pareto conditionals. Statistics & Probability Letters 5(4), 263–266 (1987)
Bi, Z., Faloutsos, C., Korn, F.: The ”DGX” distribution for mining massive, skewed data. In: KDD (2001)
Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., Wiener, J.: Graph structure in the web: experiments and models. In: WWW (2000)
Cesar, C.R.-S., Hidalgo, A.: The dynamics of a mobile phone network. Physica A: Statistical Mechanics and its Applications 387(12), 3017–3024 (2008)
Chakrabarti, D., Zhan, Y., Faloutsos, C.: R-MAT: A recursive model for graph mining. In: SDM (2004)
Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703 (2009)
Eagle, N., Pentland, A., Lazer, D.: Inferring social network structure using mobile phone data. Proceedings of the National Academy of Sciences (PNAS) 106, 15274–15278 (2009)
Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: SIGCOMM, pp. 251–262 (August-September 1999)
Garlaschelli, D., Loffredo, M.I.: Patterns of Link Reciprocity in Directed Networks. Phys. Rev. Lett. 93, 268701 (2004)
Granovetter, M.: The strength of weak ties. Amer. Jour. of Sociology 78, 1360–1380 (1973)
Kotz, S., Balakrishnan, N., Johnson, N.L.: Continuous multivariate distributions, 2nd edn. Models and Applications, vol. 1 (2000)
Leskovec, J., McGlohon, M., Faloutsos, C., Glance, N., Hurst, M.: Cascading behavior in large blog graphs: Patterns and a model. In: SDM (2007)
Nadarajah, S.: A bivariate pareto model for drought. Stochastic Environmental Research and Risk Assessment 23, 811–822 (2009)
Nanavati, A.A., Gurumurthy, S., Das, G., Chakraborty, D., Dasgupta, K., Mukherjea, S., Joshi, A.: On the structural properties of massive telecom call graphs: findings and implications. In: CIKM 2006, pp. 435–444. ACM, New York (2006)
Newman, M.E.J.: Power laws, Pareto distributions and Zipf’s law. Contemporary Physics 46(5), 323–351 (2005)
Nguyen, V.-A., Lim, E.-P., Tan, H.-H., Jiang, J., Sun, A.: Do you trust to get trust? a study of trust reciprocity behaviors and reciprocal trust prediction. In: SDM, pp. 72–83 (2010)
Nussbaum, R., Esfahanian, A.-H., Tan, P.-N.: Clustering social networks using distance-preserving subgraphs. In: ASONAM (2010)
Satuluri, V., Parthasarathy, S.: Scalable graph clustering using stochastic flows: applications to community discovery. In: KDD, pp. 737–746 (2009)
Seshadri, M., Machiraju, S., Sridharan, A., Bolot, J., Faloutsos, C., Leskovec, J.: Mobile call graphs: beyond power-law and lognormal distributions. In: KDD, pp. 596–604 (2008)
Tantipathananandh, C., Berger-Wolf, T., Kempe, D.: A framework for community identification in dynamic social networks. In: KDD, pp. 717–726 (2007)
Vaz de Melo, P.O.S., Akoglu, L., Faloutsos, C., Loureiro, A.A.F.: Surprising Patterns for the Call Duration Distribution of Mobile Phone Users. In: Balcázar, J.L., Bonchi, F., Gionis, A., Sebag, M. (eds.) ECML PKDD 2010, Part III. LNCS, vol. 6323, pp. 354–369. Springer, Heidelberg (2010)
Vuong, Q.H.: Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57, 307–333 (1989)
Xekalaki, E.: The bivariate yule distribution and some of its properties. Statistics 17(2), 311–317 (1986)
Xiang, R., Neville, J., Rogati, M.: Modeling relationship strength in online social networks. In: WWW, pp. 981–990 (2010)
Yang, X., Asur, S., Parthasarathy, S., Mehta, S.: A visual-analytic toolkit for dynamic interaction graphs. In: KDD, pp. 1016–1024 (2008)
Yue, S.: The bivariate lognormal distribution to model a multivariate flood episode. Hydrological Processes 14, 2575–2588 (2000)
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Akoglu, L., Vaz de Melo, P.O.S., Faloutsos, C. (2012). Quantifying Reciprocity in Large Weighted Communication Networks. In: Tan, PN., Chawla, S., Ho, C.K., Bailey, J. (eds) Advances in Knowledge Discovery and Data Mining. PAKDD 2012. Lecture Notes in Computer Science(), vol 7302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30220-6_8
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DOI: https://doi.org/10.1007/978-3-642-30220-6_8
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