Consistent detection and optimal localization of all detectable change points in piecewise stationary arbitrarily sparse network-sequences

We consider the offline change point detection and localization problem in the context of piecewise stationary networks, where the observable is a finite sequence of networks. We develop algorithms involving some suitably modified CUSUM statistics based on adaptively trimmed adjacency matrices of the observed networks for both detection and localization of single or multiple change points present in the input data. We provide rigorous theoretical analysis and finite sample estimates evaluating the performance of the proposed methods when the input (finite sequence of networks) is generated from an inhomogeneous random graph model, where the change points are characterized by the change in the mean adjacency matrix. We show that the proposed algorithms can detect (resp. localize) all change points, where the change in the expected adjacency matrix is above the minimax detectability (resp. localizability) threshold, consistently without any a priori assumption about (a) a lower bound for the sparsity of the underlying networks, (b) an upper bound for the number of change points, and (c) a lower bound for the separation between successive change points, provided either the minimum separation between successive pairs of change points or the average degree of the underlying networks goes to infinity arbitrarily slowly. We also prove that the above condition is necessary to have consistency.

[1]  A. Rinaldo,et al.  Optimal change point detection and localization in sparse dynamic networks , 2018, The Annals of Statistics.

[2]  Maria A. Kazandjieva,et al.  A high-resolution human contact network for infectious disease transmission , 2010, Proceedings of the National Academy of Sciences.

[3]  Franck Picard,et al.  A statistical approach for array CGH data analysis , 2005, BMC Bioinformatics.

[4]  Niloy Ganguly,et al.  Time series analysis of temporal networks , 2015, ArXiv.

[5]  C. Priebe,et al.  The Kato–Temple inequality and eigenvalue concentration with applications to graph inference , 2016, 1603.06100.

[6]  Jari Saramäki,et al.  Temporal Networks , 2011, Encyclopedia of Social Network Analysis and Mining.

[7]  Diane J. Cook,et al.  A survey of methods for time series change point detection , 2017, Knowledge and Information Systems.

[8]  Nancy R. Zhang,et al.  Detecting simultaneous changepoints in multiple sequences. , 2010, Biometrika.

[9]  Tanya Y. Berger-Wolf,et al.  Structure Prediction in Temporal Networks using Frequent Subgraphs , 2007, 2007 IEEE Symposium on Computational Intelligence and Data Mining.

[10]  C'eline L'evy-Leduc,et al.  Detection and localization of change-points in high-dimensional network traffic data , 2009, 0908.2310.

[11]  Leto Peel,et al.  Detecting Change Points in the Large-Scale Structure of Evolving Networks , 2014, AAAI.

[12]  M. Carolyn Gates,et al.  Controlling infectious disease through the targeted manipulation of contact network structure , 2015, Epidemics.

[13]  Karl J. Friston,et al.  Structural and Functional Brain Networks: From Connections to Cognition , 2013, Science.

[14]  Olivier Capp'e,et al.  Homogeneity and change-point detection tests for multivariate data using rank statistics , 2011, 1107.1971.

[15]  Lizhen Lin,et al.  Change-point detection in dynamic networks via graphon estimation , 2019, 1908.01823.

[16]  Tiago P. Peixoto,et al.  Change points, memory and epidemic spreading in temporal networks , 2017, Scientific Reports.

[17]  Piotr Sapiezynski,et al.  Measuring Large-Scale Social Networks with High Resolution , 2014, PloS one.

[18]  Fabrice Heitz,et al.  Automatic change detection in multimodal serial MRI: application to multiple sclerosis lesion evolution , 2003, NeuroImage.

[19]  Richard James,et al.  Temporal dynamics and network analysis , 2012 .

[20]  Tiago P. Peixoto.  Inferring the mesoscale structure of layered, edge-valued, and time-varying networks. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  M. Lavielle,et al.  Adaptive Detection of Multiple Change-Points in Asset Price Volatility , 2007 .

[22]  G. Lorden.  PROCEDURES FOR REACTING TO A CHANGE IN DISTRIBUTION , 1971 .

[23]  Steve Harenberg,et al.  Anomaly detection in dynamic networks: a survey , 2015 .

[24]  Zaïd Harchaoui,et al.  Kernel Change-point Analysis , 2008, NIPS.

[25]  Ping Yang,et al.  Adaptive Change Detection in Heart Rate Trend Monitoring in Anesthetized Children , 2006, IEEE Transactions on Biomedical Engineering.

[26]  Petter Holme,et al.  Modern temporal network theory: a colloquium , 2015, The European Physical Journal B.

[27]  Shlomo Havlin,et al.  Dynamic motifs in socio-economic networks , 2014 .

[28]  Robert Lund,et al.  A Review and Comparison of Changepoint Detection Techniques for Climate Data , 2007 .

[29]  M. Srivastava,et al.  Likelihood Ratio Tests for a Change in the Multivariate Normal Mean , 1986 .

[30]  Monika Ritsch-Marte,et al.  A new method for change-point detection developed for on-line analysis of the heart beat variability during sleep , 2005 .

[31]  Francis R. Bach,et al.  Learning smoothing models of copy number profiles using breakpoint annotations , 2013, BMC Bioinformatics.

[32]  A. Carpentier,et al.  Two-sample hypothesis testing for inhomogeneous random graphs , 2017, The Annals of Statistics.

[33]  Carey E. Priebe,et al.  Anomaly Detection in Time Series of Graphs using Fusion of Graph Invariants , 2012, IEEE Journal of Selected Topics in Signal Processing.

[34]  Jari Saramäki,et al.  Effects of time window size and placement on the structure of an aggregated communication network , 2012, EPJ Data Science.

[35]  Piotr Fryzlewicz,et al.  Wild binary segmentation for multiple change-point detection , 2014, 1411.0858.

[36]  E. S. Page.  On problems in which a change in a parameter occurs at an unknown point , 1957 .

[37]  Aaron Clauset,et al.  Assembling thefacebook: Using Heterogeneity to Understand Online Social Network Assembly , 2015, WebSci.

[38]  Christos Faloutsos,et al.  RSC: Mining and Modeling Temporal Activity in Social Media , 2015, KDD.

[39]  Badrinath Roysam,et al.  Image change detection algorithms: a systematic survey , 2005, IEEE Transactions on Image Processing.

[40]  P. Perron,et al.  Estimating and testing linear models with multiple structural changes , 1995 .

[41]  Petter Holme,et al.  Simulated Epidemics in an Empirical Spatiotemporal Network of 50,185 Sexual Contacts , 2010, PLoS Comput. Biol..

[42]  Ciro Cattuto,et al.  Fingerprinting temporal networks of close-range human proximity , 2013, 2013 IEEE International Conference on Pervasive Computing and Communications Workshops (PERCOM Workshops).

[43]  George Michailidis,et al.  Change point estimation in high dimensional Markov random‐field models , 2014, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[44]  George Michailidis,et al.  Change Point Estimation in a Dynamic Stochastic Block Model , 2018, J. Mach. Learn. Res..

[45]  A. Shiryaev.  On Optimum Methods in Quickest Detection Problems , 1963 .

[46]  E. S. Page.  CONTINUOUS INSPECTION SCHEMES , 1954 .

[47]  O. Sporns.  Structure and function of complex brain networks , 2013, Dialogues in clinical neuroscience.

[48]  B. Brodsky,et al.  Nonparametric Methods in Change Point Problems , 1993 .

[49]  Nancy R. Zhang,et al.  Detecting simultaneous variant intervals in aligned sequences , 2011, 1108.3177.

[50]  Zaïd Harchaoui,et al.  A regularized kernel-based approach to unsupervised audio segmentation , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[51]  Soumendu Sundar Mukherjee.  On Some Inference Problems for Networks , 2018 .

[52]  M. A. Girshick,et al.  A BAYES APPROACH TO A QUALITY CONTROL MODEL , 1952 .

[53]  M. Mann,et al.  System-Wide Temporal Characterization of the Proteome and Phosphoproteome of Human Embryonic Stem Cell Differentiation , 2011, Science Signaling.

[54]  Vinko Zlatic,et al.  Extraction of Temporal Networks from Term Co-Occurrences in Online Textual Sources , 2014, PloS one.

[55]  Hao Chen,et al.  Graph-based change-point detection , 2012, 1209.1625.

[56]  Luis E C Rocha,et al.  Information dynamics shape the sexual networks of Internet-mediated prostitution , 2010, Proceedings of the National Academy of Sciences.