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Bayesian approach to multilayer stochastic blockmodel and network changepoint detection

Published online by Cambridge University Press:  06 June 2017

YUNKYU SOHN  and
JONG HEE PARK
YUNKYU SOHN
Affiliation:
Department of Politics, Princeton University, Princeton, NJ 08540, USA Department of Political Science, University of California San Diego, La Jolla, CA 92093, USA (e-mail: ysohn@princeton.edu)
JONG HEE PARK
Affiliation:
Department of Political Science and International Relations, Seoul National University, Seoul 151-742, Republic of Korea (e-mail: jongheepark@snu.ac.kr)
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Abstract

Network scholars commonly encounter multiple networks, each of which is possibly governed by distinct generation rules while sharing a node group structure. Although the stochastic blockmodeling—detecting such latent group structures with group-specific connection profiles—has been a major topic of recent research, the focus has been given to the assortative group discovery of a single network. Despite its universality, concepts, and techniques for simultaneous characterization of node traits of multilayer networks, constructed by stacking multiple networks into layers, have been limited. Here, we propose a Bayesian multilayer stochastic blockmodeling framework that uncovers layer-common node traits and factors associated with layer-specific network generating functions. Without assuming a priori layer-specific generation rules, our fully Bayesian treatment allows probabilistic inference of latent traits. We extend the approach to detect changes in block structures embedded in temporal layers of network time series. We demonstrate the method using synthetic multilayer networks with assortative, disassortative, core-periphery, and overlapping community structures. Finally, we apply the method to empirical social network datasets, and find that it detects significant latent traits and structural changepoints. In particular, we uncover endogenous historical regimes associated with distinct constellations of states in United States Senate roll call vote similarity patterns.

Keywords

multilayer networkstochastic blockmodelnetwork changepoint detectiontensor decompositionnetwork time serieshidden Markov model
Type
Research Article
Information
Network Science , Volume 5 , Special Issue 2: Modeling, Analysis, and Mining of Multilayer Networks , June 2017 , pp. 164 - 186
Copyright
Copyright © Cambridge University Press 2017 

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References

Ahn, Y.-Y., Bagrow, J. P., & Lehmann, S. (2010). Link communities reveal multiscale complexity in networks. Nature, 466 (7307), 761–764.Google Scholar
Airoldi, E. M., Blei, D. M., Fienberg, S. E., & Xing, E. P. (2009). Mixed membership stochastic blockmodels. Advances in Neural Information Processing Systems, 9, 33–40.Google Scholar
Akoglu, L., Tong, H., & Koutra, D. (2015). Graph-based anomaly detection and description: A survey. Data Mining and Knowledge Discovery, 29 (3), 626–688.Google Scholar
Araujo, M., Papadimitriou, S., Głünnemann, S., Faloutsos, C., Basu, P., Swami, A., . . . Koutra, D. (2014). Com2: Fast automatic discovery of temporal (‘comet’) communities. In Tseng, V. S., Ho, T. B., Zhou, Z.-H., Chen, A. L. P., & Kao, H.-Y. (Eds.), Advances in Knowledge Discovery and Data Mining: 18th Pacific-Asia Conference, PAKDD 2014, Tainan, Taiwan, May 13–16, 2014. Proceedings, Part II (pp. 271–283). New York, NY: Springer International Publishing.Google Scholar
Bickel, P. J., & Chen, A. (2009). A nonparametric view of network models and Newman–Girvan and other modularities. Proceedings of the National Academy of Sciences, 106 (50), 21068–21073.Google Scholar
Boccaletti, S., Bianconi, G., Criado, R., Del Genio, C., Gómez-Gardeñes, J., Romance, M., . . . Zanin, M. (2014). The structure and dynamics of multilayer networks. Physics Reports, 544 (1), 1–122.CrossRefGoogle ScholarPubMed
Borgatti, S. P., & Everett, M. G. (2000). Models of core/periphery structures. Social Networks, 21 (4), 375–395.Google Scholar
Chaudhuri, K., Chung, F., & Tsiatas, A. (2012). Spectral clustering of graphs with general degrees in the extended planted partition model. Journal of Machine Learning Research, 23 (35), 1–23.Google Scholar
Chib, S. (1996). Calculating posterior distributions and modal estimates in markov mixture models. Journal of Econometrics, 75, 79–98.CrossRefGoogle Scholar
Chib, S. (1998). Estimation and comparison of multiple change-point models. Journal of Econometrics, 86 (2), 221–241.Google Scholar
De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000). A multilinear singular value decomposition. SIAM Journal on Matrix Analysis and Applications, 21 (4), 1253–1278.Google Scholar
Expert, P., Evans, T. S., Blondel, V. D., & Lambiotte, R. (2011). Uncovering space-independent communities in spatial networks. Proceedings of the National Academy of Sciences, 108 (19), 7663–7668.Google Scholar
Faust, K. (1988). Comparison of methods for positional analysis: Structural and general equivalences. Social Networks, 10 (4), 313–341.Google Scholar
Fishkind, D. E., Sussman, D. L., Tang, M., Vogelstein, J. T., & Priebe, C. E. (2013). Consistent adjacency-spectral partitioning for the stochastic block model when the model parameters are unknown. SIAM Journal on Matrix Analysis and Applications, 34 (1), 23–39.CrossRefGoogle Scholar
Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486 (3), 75–174.Google Scholar
Fortunato, S., & Barthelemy, M. (2007). Resolution limit in community detection. Proceedings of the National Academy of Sciences, 104 (1), 36–41.Google Scholar
Gelfand, A. E., & Smith, A. F. M. (1990, June). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85 (410), 398–409.Google Scholar
Girvan, M., & Newman, M. E. (2002). Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99 (12), 7821–7826.CrossRefGoogle ScholarPubMed
Gopalan, P. K., & Blei, D. M. (2013). Efficient discovery of overlapping communities in massive networks. Proceedings of the National Academy of Sciences, 110 (36), 14534–14539.Google Scholar
Guimera, R., & Amaral, L. A. N. (2005). Functional cartography of complex metabolic networks. Nature, 433 (7028), 895–900.Google Scholar
Han, Q., Xu, K., & Airoldi, E. (2015). Consistent estimation of dynamic and multi-layer block models. Proceedings of the 32nd International Conference on Machine Learning, 37, 1511–1520.Google Scholar
Handcock, M. S., Raftery, A. E., & Tantrum, J. M. (2007). Model-based clustering for social networks. Journal of the Royal Statistical Society: Series A (Statistics in Society), 170 (2), 301–354.CrossRefGoogle Scholar
Hoff, P. D. (2005). Bilinear mixed-effects models for dyadic data. Journal of the American Statistical Association, 100 (469), 286–295.CrossRefGoogle Scholar
Hoff, P. D. (2008). Modeling homophily and stochastic equivalence in symmetric relational data. In Platt, J., Koller, D., Singer, Y., & Roweis, S. (Eds.), Advances in neural information processing systems (vol. 20, pp. 657–664). Cambridge: Cambridge University Press.Google Scholar
Hoff, P. D. (2009). 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.Google Scholar
Hoff, P. D. (2011). Hierarchical multilinear models for multiway data. Computational Statistics & Data Analysis, 55, 530–543.Google Scholar
Hoff, P. D., Raftery, A. E., & Handcock, M. S. (2002). Latent space approaches to social network analysis. Journal of the American Statistical Association, 97 (460), 1090–1098.Google Scholar
Hofman, J. M., & Wiggins, C. H. (2008). Bayesian approach to network modularity. Physical Review Letters, 100 (25), 258701.Google Scholar
Holland, P. W., Laskey, K. B., & Leinhardt, S. (1983). Stochastic blockmodels: First steps. Social Networks, 5 (2), 109–137.Google Scholar
Holme, P., & Saramäki, J. (2012). Temporal networks. Physics Reports, 519 (3), 97–125.CrossRefGoogle Scholar
Karrer, B., & Newman, M. E. (2011). Stochastic blockmodels and community structure in networks. Physical Review E, 83 (1), 016107.Google Scholar
Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y., & Porter, M. A. (2014). Multilayer networks. Journal of Complex Networks, 2 (3), 203–271.Google Scholar
Kolar, M., Song, L., Ahmed, A., & Xing, E. P. (2010). Estimating time-varying networks. The Annals of Applied Statistics, 4 (1), 94–123.CrossRefGoogle Scholar
Kolda, T. G., & Bader, B. W. (2009). Tensor decompositions and applications. SIAM Review, 51 (3), 455–500.Google Scholar
Krzakala, F., Moore, C., Mossel, E., Neeman, J., Sly, A., & Zdeborová, L. (2013). Spectral redemption in clustering sparse networks. Proceedings of the National Academy of Sciences, 110 (52), 20935–20940.CrossRefGoogle ScholarPubMed
Lancichinetti, A., Radicchi, F., Ramasco, J. J., & Fortunato, S. (2011). Finding statistically significant communities in networks. PloS One, 6 (4), e18961.CrossRefGoogle ScholarPubMed
Lee, D. D., & Seung, H. S. (1999). Learning the parts of objects by non-negative matrix factorization. Nature, 401 (6755), 788–791.Google Scholar
Lei, J., & Rinaldo, A. (2014). Consistency of spectral clustering in stochastic block models. The Annals of Statistics, 43 (1), 215–237.Google Scholar
Liu, J. S. (1994). The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem. Journal of the American Statistical Association, 89 (427), 958–966.Google Scholar
Magnani, M., Micenkova, B., & Rossi, L. (2013). Combinatorial analysis of multiple networks. preprint, arXiv:1303.4986.Google Scholar
McCarty, N., Poole, K. T., & Rosenthal, H. (2006). Polarized America: The dance of ideology and unequal riches, Vol. 5. Cambridge, MA: MIT Press.Google Scholar
Meila, M. (2003). Comparing clusterings by the variation of information. In Schoelkopf, B. & Warmuth, M. K. (Eds.), Learning Theory and Kernel Machines: 16th Annual Conference on Computational Learning Theory and 7th Kernel workshop, CLOT/Kernel 2003, Washington, DC, USA, August 24–27, 2003, Proceedings (vol. 2777, pp. 173–187). Berlin: Springer.Google Scholar
Moody, J., & White, D. R. (2003). Structural cohesion and embeddedness: A hierarchical concept of social groups. American Sociological Review, 68 (1), 103–127.Google Scholar
Mucha, P. J., Richardson, T., Macon, K., Porter, M. A., & Onnela, J.-P. (2010). Community structure in time-dependent, multiscale, and multiplex networks. Science, 328 (5980), 876–878.Google Scholar
Nadakuditi, R. R., & Newman, M. E. (2013). Spectra of random graphs with arbitrary expected degrees. Physical Review E, 87 (1), 012803.Google Scholar
Newman, M. E. (2006). Modularity and community structure in networks. Proceedings of the National Academy of Sciences, 103 (23), 8577–8582.Google Scholar
Newman, M. E. (2013, Oct). Spectral methods for community detection and graph partitioning. Physical Review E, 88, 042822. doi: 10.1103/Phys-RevE.88.042822 Google Scholar
Newman, M. E., & Leicht, E. A. (2007). Mixture models and exploratory analysis in networks. Proceedings of the National Academy of Sciences, 104 (23), 9564–9569.Google Scholar
Ng, A. Y., Jordan, M. I., & Weiss, Y. (2002). On spectral clustering: Analysis and an algorithm. Advances in Neural Information Processing Systems, 2, 849–856.Google Scholar
Nowicki, K., & Snijders, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96 (455), 1077–1087.Google Scholar
Palla, G., Barabási, A.-L., & Vicsek, T. (2007). Quantifying social group evolution. Nature, 446 (7136), 664–667.CrossRefGoogle ScholarPubMed
Paul, S., & Chen, Y. (2016). Consistent community detection in multi-relational data through restricted multi-layer stochastic blockmodel. Electronic Journal of Statistics, 10 (2), 3807–3870.Google Scholar
Peel, L., & Clauset, A. (2015). Detecting change points in the large-scale structure of evolving networks. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence (pp. 2914–2920). Austin, TX: AAAI Press.Google Scholar
Peixoto, T. P. (2013). Eigenvalue spectra of modular networks. Physical Review Letters, 111 (9), 098701.CrossRefGoogle ScholarPubMed
Peixoto, T. P. (2015). Inferring the mesoscale structure of layered, edge-valued, and time-varying networks. Physical Review E, 92 (4), 042807.Google Scholar
Riolo, M. A., & Newman, M. E. (2014). First-principles multiway spectral partitioning of graphs. Journal of Complex Networks, 2 (2), 121–140.Google Scholar
Rohe, K., Chatterjee, S., & Yu, B. (2011). Spectral clustering and the high-dimensional stochastic blockmodel. The Annals of Statistics, 39 (4), 1878–1915.Google Scholar
Rosvall, M., & Bergstrom, C. T. (2008). Maps of random walks on complex networks reveal community structure. Proceedings of the National Academy of Sciences, 105 (4), 1118–1123.CrossRefGoogle ScholarPubMed
Smith, A. F. M., & Roberts, G. O. (1993). Bayesian computation via the gibbs sampler and related markov chain monte carlo methods. Journal of the Royal Statistical Society. Series B, Statistical Methodology, 55 (1), 3–23.Google Scholar
Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society. Series B, Statistical Methodology, 64 (4), 583–639.Google Scholar
Stanley, N., Shai, S., Taylor, D., & Mucha, P. (2016). Clustering network layers with the strata multilayer stochastic block model. IEEE Transactions on Network Science and Engineering, 3 (2), 95–105.Google Scholar
Sussman, D. L., Tang, M., Fishkind, D. E., & Priebe, C. E. (2012). A consistent adjacency spectral embedding for stochastic blockmodel graphs. Journal of the American Statistical Association, 107 (499), 1119–1128.Google Scholar
Ward, M. D., Ahlquist, J. S., & Rozenas, A. (2013). Gravity's rainbow: A dynamic latent space model for the world trade network. Network Science, 1 (1), 95–118.Google Scholar
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