The Annals of Applied Probability

Change point detection in network models: Preferential attachment and long range dependence

Shankar Bhamidi, Jimmy Jin, and Andrew Nobel

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Inspired by empirical data on real world complex networks, the last few years have seen an explosion in proposed generative models to understand and explain observed properties of real world networks, including power law degree distribution and “small world” distance scaling. In this context, a natural question is how to understand the effect of change points—how abrupt changes in parameters driving the network model change structural properties of the network. We study this phenomenon in one popular class of dynamically evolving networks: preferential attachment models. We derive asymptotic properties of various functionals of the network including the degree distribution as well as maximal degree asymptotics, in essence showing that the change point does effect the degree distribution but does not change the degree exponent. This provides evidence for long range dependence and sensitive dependence of the evolution of the network on the initial evolution of the process. We propose an estimator for the change point and prove consistency properties of this estimator. The methodology developed highlights the effect of the nonergodic nature of the evolution of the network on classical change point estimators.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 1 (2018), 35-78.

Dates
Received: June 2016
Revised: December 2016
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1520046083

Digital Object Identifier
doi:10.1214/17-AAP1297

Mathematical Reviews number (MathSciNet)
MR3770872

Zentralblatt MATH identifier
06873679

Subjects
Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]

Keywords
Branching process Jagers–Nerman stable age distribution point process preferential attachment change point functional central limit theorem statistical estimation

Citation

Bhamidi, Shankar; Jin, Jimmy; Nobel, Andrew. Change point detection in network models: Preferential attachment and long range dependence. Ann. Appl. Probab. 28 (2018), no. 1, 35--78. doi:10.1214/17-AAP1297. https://projecteuclid.org/euclid.aoap/1520046083


Export citation

References

  • [1] Akoglu, L., McGlohon, M. and Faloutsos, C. (2012). Oddball: Spotting anomalies in weighted graphs. In Advances in Knowledge Discovery and Data Mining 410–421. Springer, Berlin.
  • [2] Albert, R. and Barabási, A.-L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 74 47–97.
  • [3] Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Stat. 39 1801–1817.
  • [4] Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509–512.
  • [5] Basseville, M. and Nikiforov, I. V. (1993). Detection of Abrupt Changes: Theory and Application. Prentice Hall, Englewood Cliffs, NJ.
  • [6] Bhamidi, S., Evans, S. N. and Sen, A. (2012). Spectra of large random trees. J. Theoret. Probab. 25 613–654.
  • [7] Bhamidi, S., Steele, J. M. and Zaman, T. (2015). Twitter event networks and the superstar model. Ann. Appl. Probab. 25 2462–2502.
  • [8] Boas, R. P. Jr. (1977). Partial sums of infinite series, and how they grow. Amer. Math. Monthly 84 237–258.
  • [9] Boccaletti, S., Bianconi, G., Criado, R., del Genio, C. I., Gómez-Gardeñes, J., Romance, M., Sendiña-Nadal, I., Wang, Z. and Zanin, M. (2014). The structure and dynamics of multilayer networks. Phys. Rep. 544 1–122.
  • [10] Bollobás, B. (1985). Random Graphs. Academic Press, London.
  • [11] Bollobás, B. and Riordan, O. M. (2003). Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks 1–34. Wiley-VCH, Weinheim.
  • [12] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18 279–290.
  • [13] Brodsky, B. E. and Darkhovsky, B. S. (1993). Nonparametric Methods in Change-Point Problems. Mathematics and Its Applications 243. Kluwer Academic, Dordrecht.
  • [14] Bubeck, S., Devroye, L. and Lugosi, G. (2017). Finding Adam in random growing trees. Random Structures Algorithms 50 158–172.
    Mathematical Reviews (MathSciNet): MR3607120
    Digital Object Identifier: doi:10.1002/rsa.20649
  • [15] Bubeck, S., Mossel, E. and Rácz, M. Z. (2015). On the influence of the seed graph in the preferential attachment model. IEEE Trans. Netw. Sci. Eng. 2 30–39.
  • [16] Carlstein, E. (1988). Nonparametric change-point estimation. Ann. Statist. 16 188–197.
  • [17] Carlstein, E., Müller, H.-G. and Siegmund, D., eds. (1994). Change-Point Problems (South Hadley, MA, 1992). Institute of Mathematical Statistics Lecture Notes—Monograph Series 23. IMS, Hayward, CA.
  • [18] Chandola, V., Banerjee, A. and Kumar, V. (2009). Anomaly detection: A survey. ACM Comput. Surv. 41 Article No. 15.
  • [19] Chung, F. and Lu, L. (2006). Complex Graphs and Networks. CBMS Regional Conference Series in Mathematics 107. Amer. Math. Soc., Providence, RI.
  • [20] Cooper, C. and Frieze, A. (2003). A general model of web graphs. Random Structures Algorithms 22 311–335.
  • [21] Csörgő, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analysis. Wiley, Chichester.
  • [22] Curien, N., Duquesne, T., Kortchemski, I. and Manolescu, I. (2015). Scaling limits and influence of the seed graph in preferential attachment trees. J. Éc. Polytech. Math. 2 1–34.
  • [23] Dorogovtsev, S. N. and Mendes, J. F. F. (2003). Evolution of Networks. Oxford Univ. Press, Oxford.
  • [24] Duan, D., Li, Y., Jin, Y. and Lu, Z. (2009). Community mining on dynamic weighted directed graphs. In Proceedings of the 1st ACM International Workshop on Complex Networks Meet Information & Knowledge Management 11–18.
  • [25] Durrett, R. (2007). Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics 20. Cambridge Univ. Press, Cambridge.
  • [26] Durrett, R. and Resnick, S. I. (1978). Functional limit theorems for dependent variables. Ann. Probab. 6 829–846.
  • [27] Eagle, N. and Pentland, A. (2006). Reality mining: Sensing complex social systems. Personal and Ubiquitous Computing 10 255–268.
  • [28] Eberle, W. and Holder, L. (2007). Discovering structural anomalies in graph-based data. In Seventh IEEE International Conference on Data Mining Workshops (ICDMW 2007) 393–398.
  • [29] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [30] Heard, N. A., Weston, D. J., Platanioti, K. and Hand, D. J. (2010). Bayesian anomaly detection methods for social networks. Ann. Appl. Stat. 4 645–662.
  • [31] Holme, P. and Saramäki, J. (2012). Temporal networks. Phys. Rep. 519 97–125.
  • [32] Huang, Z. and Zeng, D. D. (2006). A link prediction approach to anomalous email detection. In 2006 IEEE International Conference on Systems, Man and Cybernetics 1131–1136.
  • [33] Jagers, P. (1975). Branching Processes with Biological Applications. Wiley-Interscience, London.
  • [34] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16 221–259.
  • [35] Jagers, P. and Nerman, O. (1984). Limit theorems for sums determined by branching and other exponentially growing processes. Stochastic Process. Appl. 17 47–71.
  • [36] Liptser, R. and Shiryayev, A. N. (1989). Theory of Martingales. Mathematics and Its Applications (Soviet Series) 49. Kluwer Academic, Dordrecht.
  • [37] Ma, W.-Y. and Manjunath, B. S. (1997). Edge flow: A framework of boundary detection and image segmentation. In Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition 744–749.
  • [38] Marangoni-Simonsen, D. and Xie, Y. (2015). Sequential changepoint approach for online community detection. IEEE Signal Process. Lett. 22 1035–1039.
  • [39] McCulloh, I. and Carley, K. M. (2011). Detecting change in longitudinal social networks. DTIC Document.
  • [40] Moreno, S. and Neville, J. (2013). Network hypothesis testing using mixed Kronecker product graph models. In 2013 IEEE 13th International Conference on Data Mining 1163–1168.
  • [41] Móri, T. F. (2007). Degree distribution nearby the origin of a preferential attachment graph. Electron. Commun. Probab. 12 276–282.
  • [42] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45 167–256.
    Mathematical Reviews (MathSciNet): MR2010377
    Digital Object Identifier: doi:10.1137/S003614450342480
  • [43] Newman, M. E. J. (2010). Networks: An Introduction. Oxford Univ. Press, Oxford.
  • [44] Noble, C. C. and Cook, D. J. (2003). Graph-based anomaly detection. In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 631–636.
  • [45] Norris, J. R. (1998). Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics 2. Cambridge Univ. Press, Cambridge. Reprint of 1997 original.
  • [46] Peel, L. and Clauset, A. (2014). Detecting change points in the large-scale structure of evolving networks. Available at arXiv:1403.0989.
    arXiv: 1403.0989
  • [47] Priebe, C. E., Conroy, J. M., Marchette, D. J. and Park, Y. (2005). Scan statistics on enron graphs. Comput. Math. Organ. Theory 11 229–247.
  • [48] Resnick, S. I. and Samorodnitsky, G. (2016). Asymptotic normality of degree counts in a preferential attachment model. Adv. in Appl. Probab. 48 283–299.
  • [49] Rudas, A., Tóth, B. and Valkó, B. (2007). Random trees and general branching processes. Random Structures Algorithms 31 186–202.
  • [50] Sharpnack, J., Rinaldo, A. and Singh, A. (2012). Changepoint detection over graphs with the spectral scan statistic. Available at arXiv:1206.0773.
    arXiv: 1206.0773
  • [51] Shiryaev, A. N. (2008). Optimal Stopping Rules. Stochastic Modelling and Applied Probability 8. Springer, Berlin.
  • [52] Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.
  • [53] Simon, H. A. (1955). On a class of skew distribution functions. Biometrika 42 425–440.
  • [54] Širjaev, A. N. (1963). Optimal methods in quickest detection problems. Teor. Verojatnost. i Primenen. 8 26–51.
  • [55] Sun, J., Faloutsos, C., Papadimitriou, S. and Yu, P. S. (2007). Graphscope: Parameter-free mining of large time-evolving graphs. In Proceedings of the 13th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 687–696.
  • [56] Szymański, J. (1987). On a nonuniform random recursive tree. In Random Graphs ’85 (Poznań, 1985). North-Holland Math. Stud. 144 297–306. North-Holland, Amsterdam.
  • [57] Tsybakov, A. B. (1994). Multidimensional change-point problems and boundary estimation. In Change-Point Problems (South Hadley, MA, 1992). Institute of Mathematical Statistics Lecture Notes—Monograph Series 23 317–329. IMS, Hayward, CA.
  • [58] van der Hofstad, R. (2016). Random Graphs and Complex Networks, Vol. 1. Cambridge Univ. Press, Cambridge. Available at http://www.win.tue.nl/rhofstad/NotesRGCN.pdf.
  • [59] Yudovina, E., Banerjee, M. and Michailidis, G. (2015). Changepoint inference for Erdős–Rényi random graphs. In Stochastic Models, Statistics and Their Applications. Springer Proc. Math. Stat. 122 197–205. Springer, Cham.
  • [60] Yule, G. U. (1925). A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F.R.S. Philos. Trans. R. Soc. Lond. Ser. B 213 21–87.
  • [61] Zhang, W., Pan, G., Wu, Z. and Li, S. (2013). Online community detection for large complex networks. In Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence 1903–1909.

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more