An exact algorithm for time-dependent variational inference for the dynamic stochastic block model

Abstract An exact algorithm for estimating the dynamic stochastic block model is proposed. This model assumes a hidden Markov chain for the evolution of the social behavior of a group of individuals at repeated time occasions and may be used to assign these individuals to the latent blocks in a dynamic fashion. For the estimation of this model, the proposed exact algorithm maximizes the target function introduced by Matias and Miele [7]. This function is derived from a variational approximation of the model log-likelihood, based on the assumption that the latent variables identifying the blocks are a posteriori independent across individuals, but not across time occasions. A simulation study is performed to compare the exact algorithm with the approximate maximization algorithm proposed by Matias and Miele [7]. Results show that there is a certain advantage of the first in terms of dynamic assignment of individuals to the latent blocks in comparison to the true blocking structure, as measured by the adjusted Rand index.

[1]  Kathryn B. Laskey,et al.  Stochastic blockmodels: First steps , 1983 .

[2]  W. Zucchini,et al.  Hidden Markov Models for Time Series: An Introduction Using R , 2009 .

[3]  Alfred O. Hero,et al.  Dynamic Stochastic Blockmodels for Time-Evolving Social Networks , 2014, IEEE Journal of Selected Topics in Signal Processing.

[4]  Kevin S. Xu.  Stochastic Block Transition Models for Dynamic Networks , 2014, AISTATS.

[5]  H. Chipman,et al.  A Continuous-time Stochastic Block Model for Basketball Networks , 2015, 1507.01816.

[6]  Padhraic Smyth,et al.  Stochastic blockmodeling of relational event dynamics , 2013, AISTATS.

[7]  Vincent Miele,et al.  Statistical clustering of temporal networks through a dynamic stochastic block model , 2015, 1506.07464.

[8]  Yihong Gong,et al.  Detecting communities and their evolutions in dynamic social networks—a Bayesian approach , 2011, Machine Learning.

[9]  Franck Picard,et al.  A mixture model for random graphs , 2008, Stat. Comput..

[10]  C. Matias,et al.  A semiparametric extension of the stochastic block model for longitudinal networks , 2015, Biometrika.

[11]  T. Snijders,et al.  Estimation and Prediction for Stochastic Blockmodels for Graphs with Latent Block Structure , 1997 .

[12]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[13]  Xiao Zhang,et al.  Random graph models for dynamic networks , 2016, The European Physical Journal B.

[14]  Francesco Bartolucci,et al.  Latent Markov Models for Longitudinal Data , 2012 .

[15]  Nial Friel,et al.  Choosing the number of groups in a latent stochastic blockmodel for dynamic networks , 2017, Network Science.

[16]  T. Snijders,et al.  Estimation and Prediction for Stochastic Blockstructures , 2001 .

[17]  Francesco Bartolucci,et al.  Dealing with reciprocity in dynamic stochastic block models , 2018, Comput. Stat. Data Anal..

[18]  Riccardo Rastelli.  Exact integrated completed likelihood maximisation in a stochastic block transition model for dynamic networks , 2017, 1710.03551.