Bayesian Analysis

Detecting Structural Changes in Longitudinal Network Data

Jong Hee Park and Yunkyu Sohn

Full-text: Open access


Dynamic modeling of longitudinal networks has been an increasingly important topic in applied research. While longitudinal network data commonly exhibit dramatic changes in its structures, existing methods have largely focused on modeling smooth topological changes over time. In this paper, we develop a hidden Markov network change-point model (HNC) that combines the multilinear tensor regression model (Hoff, 2011) with a hidden Markov model using Bayesian inference. We model changes in network structure as shifts in discrete states yielding particular sets of network generating parameters. Our simulation results demonstrate that the proposed method correctly detects the number, locations, and types of changes in latent node characteristics. We apply the proposed method to international military alliance networks to find structural changes in the coalition structure among nations.

Article information

Bayesian Anal., Volume 15, Number 1 (2020), 133-157.

First available in Project Euclid: 22 February 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

network latent space hidden Markov model WAIC military alliance

Creative Commons Attribution 4.0 International License.


Park, Jong Hee; Sohn, Yunkyu. Detecting Structural Changes in Longitudinal Network Data. Bayesian Anal. 15 (2020), no. 1, 133--157. doi:10.1214/19-BA1147.

Export citation


  • Bartolucci, F., Marino, M. F., and Pandolfi, S. (2018). “Dealing with Reciprocity in Dynamic Stochastic Block Models.” Computational Statistics & Data Analysis, 123: 86–100.
  • Benson, A. R., Gleich, D. F., and Leskovec, J. (2016). “Higher-order organization of complex networks.” Science, 353(6295): 163–166.
  • Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.
  • Björck, A. (1996). Numerical Methods for Least Squares Problems. SIAM.
  • Burt, R. S. (2009). Structural Holes: The Social Structure of Competition. Harvard University Press.
  • Carlin, B. P. and Polson, N. G. (1991). “Inference for Non-Conjugate Bayesian Models Using the Gibbs Sampler.” Canadian Journal of Statistics, 19: 399–405.
  • Chaudhuri, K., Graham, F. C., and Tsiatas, A. (2012). “Spectral Clustering of Graphs with General Degrees in the Extended Planted Partition Model.” In COLT, volume 23, 35–1.
  • Chib, S. (1995). “Marginal Likelihood From the Gibbs Output.” Journal of the American Statistical Association, 90(432): 1313–1321.
  • Chib, S. (1998). “Estimation and Comparison of Multiple Change-point Models.” Journal of Econometrics, 86(2): 221–241.
  • Cranmer, S. J., Heinrich, T., and Desmarais, B. A. (2014). “Reciprocity and the Structural Determinants of the International Sanctions Network.” Social Networks, 36(January): 5–22.
  • Cribben, I. and Yu, Y. (2016). “Estimating Whole-Brain Dynamics by Using Spectral Clustering.” Journal of the Royal Statistical Society: Series C (Applied Statistics).
  • De Lathauwer, L., De Moor, B., and Vandewalle, J. (2000). “A Multilinear Singular Value Decomposition.” SIAM Journal on Matrix Analysis and Applications, 21(4): 1253–1278.
  • Desmarais, B. A. and Cranmer, S. J. (2012). “Statistical Mechanics of Networks: Estimation and Uncertainty.” Physica A, 391(4): 1865–1876.
  • Drton, M. (2009). “Likelihood Ratio Tests and Singularities.” Ann. Statist., 37(2): 979–1012.
  • Gelman, A., Hwang, J., and Vehtari, A. (2014). “Understanding Predictive Information Criteria for Bayesian Models.” Statistics and Computing, 24(6): 997–1016.
  • Gibler, D. (2008). International Military Alliances, 1648-2008. CQ Press.
  • Goldenberg, A., Zheng, A. X., Fienberg, S. E., and Airoldi, E. M. (2010). “A Survey of Statistical Network Models.” Found. Trends Mach. Learn., 2(2): 129–233.
  • Guhaniyogi, R. and Dunson, D. B. (2015). “Bayesian Compressed Regression.” Journal of the American Statistical Association, 110(512): 1500–1514.
  • Guo, F., Hanneke, S., Fu, W., and Xing, E. P. (2007). “Recovering Temporally Rewiring Networks: A Model-based Approach.” Proceedings of the 24 th International Conference on Machine Learning, 321–328.
  • Han, S. and Dunson, D. B. (2018). “Multiresolution Tensor Decomposition for Multiple Spatial Passing Networks.” CoRR, abs/1803.01203.
  • Hanneke, S., Fu, W., and Xing, E. P. (2010). “Discrete Temporal Models of Social Networks.” Electronic Journal of Statistics, 4: 585–605.
  • Hartigan, J. A. (1985). “A Failure of Likelihood Asymptotics for Normal Mixtures.” In LeCam, L. and Olshen, R. A. (eds.), Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, volume 2, 807–810. Belmont, California: Wadsworth Statistics/Probability Series.
  • Heard, N. A., Weston, D. J., Platanioti, K., and Hand, D. J. (2010). “Bayesian Anomaly Detection Methods for Social Networks.” Annals of Applied Statistics, 4(2): 645–662.
  • Hoff, P. (2007). “Model Averaging and Dimension Selection for the Singular Value Decomposition.” Journal of the American Statistical Association, 102(478): 674–685.
  • Hoff, P. D. (2008). “Modeling Homophily and Stochastic Equivalence in Symmetric Relational Data.” In Platt, J., Koller, D., Singer, Y., and Roweis, S. (eds.), Advances in Neural Information Processing Systems 20, 657–664. Cambridge University Press.
  • Hoff, P. D. (2009a). “Multiplicative Latent Factor Models for Description and Prediction of Social Networks.” Computational & Mathematical Organization Theory, 15(4): 261–272.
  • Hoff, P. D. (2009b). “Simulation of the Matrix Bingham-von Mises-Fisher Distribution, With Applications to Multivariate and Relational Data.” Journal of Computational and Graphical Statistics, 18(2): 438–456.
  • Hoff, P. D. (2011). “Hierarchical Multilinear Models for Multiway Data.” Computational Statistics & Data Analysis, 55: 530–543.
  • Hoff, P. D. (2015). “Multilinear Tensor Regression for Longitudinal Relational Data.” The Annals of Applied Statistics, 9(3): 1169–1193.
  • Holme, P. and Saramäki, J. (2012). “Temporal Networks.” Physics Reports, 519(3): 97–125.
  • Johndrow, J. E., Bhattacharya, A., and Dunson, D. B. (2017). “Tensor Decompositions and Sparse Log-linear Models.” Annals of Statistics, 45(1): 1–38.
  • Karrer, B. and Newman, M. E. (2011). “Stochastic Blockmodels and Community Structure in Networks.” Physical Review E, 83(1): 016107.
  • Liu, J. S., Wong, W. H., and Kong, A. (1994). “Covariance Structure of the Gibbs Sampler with Applications to the Comparisons of Estimators and Augmentation Schemes.” Biometrika, 81(1): 27.
  • Minhas, S., Hoff, P. D., and Ward, M. D. (2016). “A New Approach to Analyzing Coevolving Longitudinal Networks in international relations.” Journal of Peace Research, 53(3): 491–505.
  • Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective. MIT press.
  • Newman, M. E. (2006). “Modularity and Community Structure in Networks.” Proceedings of the National Academy of Sciences, 103(23): 8577–8582.
  • Newman, M. E. (2010). Networks: An Introduction. Oxford University Press.
  • Newman, M. E. and Girvan, M. (2004). “Finding and Evaluating Community Structure in Networks.” Physical Review E, 69(2): 026113.
  • Peixoto, T. P. (2013). “Eigenvalue Spectra of Modular Networks.” Physical Review Letters, 111(9): 098701–5.
  • Rai, P., Wang, Y., and Carin, L. (2015). “Leveraging Features and Networks for Probabilistic Tensor Decomposition.” AAAI’15 Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, 2942–2948.
  • Ridder, S. D., Vandermarliere, B., and Ryckebusch, J. (2016). “Detection and Localization of Change Points in Temporal Networks with the Aid of Stochastic Block Models.” Journal of Statistical Mechanics: Theory and Experiment, 2016(11): 113302.
  • Robins, G. L. and Pattison, P. E. (2001). “Random Graph Models for Temporal Processes in Social Networks.” Journal of Mathematical Sociology, 25(5–41).
  • Rohe, K., Chatterjee, S., and Yu, B. (2011). “Spectral Clustering and the High-Dimensional Stochastic Blockmodel.” The Annals of Statistics, 39(4): 1878–1915.
  • Rothenberg, G. E. (1968). “The Austrian Army in the Age of Metternich.” Journal of Modern History, 40(2): 156–165.
  • Snijders, T. A. B., Steglich, C. E. G., and Schweinberger, M. (2006). Longitudinal Models in the Behavioral and Related Sciences, 41–71. Routledge.
  • Snijders, T. A. B., van de Bunt, G. G., and Steglich, C. E. G. (2010). “Introduction to Stochastic Actor-based Models for Network dynamics.” Social Networks, 32(1): 44–60.
  • Snyder, G. H. (1997). Alliance Politics. Cornell University Press.
  • Sohn, Y. and Park, J. H. (2017). “Bayesian Approach to Multilayer Stochastic Block Model and Network Changepoint Detection.” Network Science, 5(2): 164–186.
  • van Dyk, D. A. and Park, T. (2008). “Partially Collapsed Gibbs Samplers.” Journal of the American Statistical Association, 103(482): 790–796.
  • Vermeiren, J. (2016). The First World War and German National Identity: The Dual Alliance at War. Cambridge University Press.
  • Wang, X., Yuan, K., Hellmayr, C., Liu, W., and Markowetz, F. (2014). “Reconstructing Evolving Signalling Networks by Hidden Markov Nested Effects Models.” Annals of Applied Statistics, 8(1): 448–480.
  • Ward, M. D., Ahlquist, J. S., and Rozenas, A. (2013). “Gravity’s Rainbow: A Dynamic Latent Space Model for the World Trade Network” Network Science, 1(1): 95–118.
  • Wasserman, S. and Faust, K. (1994). Social Network Analysis: Methods and Applications, Cambridge University Press.
  • Watanabe, S. (2010). “Asymptotic Equivalence of Bayes Cross Calidation and Widely applicable information criterion in singular learning theory.” Journal of Machine Learning Research, 11: 3571–3594.
  • Westveld, A. H. and Hoff, P. D. (2011). “A Mixed Effects Model for Longitudinal Relational and Network Data, with Applications to International Trade and Conflict.” Annals of Applied Statistics, 5: 843–872.
  • Zhao, Y., Levina, E., Zhu, J., et al. (2012). “Consistency of Community Detection in Networks Under Degree-corrected Stochastic Block Models.” The Annals of Statistics, 40(4): 2266–2292.