Bayesian Analysis

Mixed Membership Stochastic Blockmodels for Heterogeneous Networks

Weihong Huang, Yan Liu, and Yuguo Chen

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Heterogeneous networks are useful for modeling complex systems that consist of different types of objects. However, there are limited statistical models to deal with heterogeneous networks. In this paper, we propose a statistical model for community detection in heterogeneous networks. We formulate a heterogeneous version of the mixed membership stochastic blockmodel to accommodate heterogeneity in the data and the content dependent property of the pairwise relationship. We also apply a variational algorithm for posterior inference. The proposed procedure is shown to be consistent for community detection under mixed membership stochastic blockmodels for heterogeneous networks. We demonstrate the advantage of the proposed method in modeling overlapping communities and multiple memberships through simulation studies and applications to a real data set.

Article information

Bayesian Anal., Advance publication (2018), 26 pages.

First available in Project Euclid: 19 June 2019

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clustering community detection heterogeneous network mixed membership model stochastic blockmodel variational algorithm

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Huang, Weihong; Liu, Yan; Chen, Yuguo. Mixed Membership Stochastic Blockmodels for Heterogeneous Networks. Bayesian Anal., advance publication, 19 June 2019. doi:10.1214/19-BA1163.

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  • Airoldi, E., Blei, D., Xing, E., and Fienberg, S. (2005). “A latent mixed membership model for relational data.” In Proceedings of the 3rd International Workshop on Link Discovery, 82–89. ACM.
  • Airoldi, E. M., Blei, D. M., Fienberg, S. E., and Xing, E. P. (2008). “Mixed membership stochastic blockmodels.” Journal of Machine Learning Research, 9(Sep): 1981–2014.
  • Gao, J., Liang, F., Fan, W., Sun, Y., and Han, J. (2009). “Graph-based consensus maximization among multiple supervised and unsupervised models.” In Advances in Neural Information Processing Systems, 585–593.
  • Goldenberg, A., Zheng, A. X., Fienberg, S. E., and Airoldi, E. M. (2010). “A survey of statistical network models.” Foundations and Trends in Machine Learning, 2(2): 129–233.
  • Handcock, M. S., Raftery, A. E., and Tantrum, J. M. (2007). “Model-based clustering for social networks.” Journal of the Royal Statistical Society: Series A, 170(2): 301–354.
  • Hoff, P. D., Raftery, A. E., and Handcock, M. S. (2002). “Latent space approaches to social network analysis.” Journal of the American Statistical Association, 97(460): 1090–1098.
  • Huang, W., Liu, Y., and Chen, Y. (2019). “Supplementary Material for “Mixed Membership Stochastic Blockmodels for Heterogeneous Networks”.” Bayesian Analysis.
  • Ji, M., Sun, Y., Danilevsky, M., Han, J., and Gao, J. (2010). “Graph regularized transductive classification on heterogeneous information networks.” In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, 570–586. Springer.
  • Jonsson, P. F., Cavanna, T., Zicha, D., and Bates, P. A. (2006). “Cluster analysis of networks generated through homology: Automatic identification of important protein communities involved in cancer metastasis.” BMC Bioinformatics, 7(1): 2.
  • Lancichinetti, A., Fortunato, S., and Kertész, J. (2009). “Detecting the overlapping and hierarchical community structure in complex networks.” New Journal of Physics, 11(3): 033015.
  • Nowicki, K. and Snijders, T. A. B. (2001). “Estimation and prediction for stochastic blockstructures.” Journal of the American Statistical Association, 96(455): 1077–1087.
  • Sengupta, S. and Chen, Y. (2015). “Spectral clustering in heterogeneous networks.” Statistica Sinica, 25(3): 1081–1106.
  • Sun, Y. and Han, J. (2012). Mining Heterogeneous Information Networks: Principles and Methodologies. Morgan & Claypool Publishers.
  • Wainwright, M. J. and Jordan, M. I. (2008). “Graphical models, exponential families, and variational inference.” Foundations and Trends in Machine Learning, 1(1–2): 1–305.
  • Xie, J., Kelley, S., and Szymanski, B. K. (2013). “Overlapping community detection in networks: The state-of-the-art and comparative study.” ACM Computing Surveys, 45(4): 43:1–43:35.
  • Zhang, J. and Chen, Y. (2019). “Modularity based community detection in heterogeneous networks.” Statistica Sinica, in press.
  • Zhao, Y., Levina, E., and Zhu, J. (2012). “Consistency of community detection in networks under degree-corrected stochastic block models.” The Annals of Statistics, 40(4): 2266–2292.

Supplemental materials

  • Supplementary Material for “Mixed Membership Stochastic Blockmodels for Heterogeneous Networks”. The supplementary material contains the details of the variational posterior inference, variational EM algorithm, and the proofs of theoretical results in Section 4.