The Block Point Process Model for Continuous-time Event-based Dynamic Networks

We consider the problem of analyzing timestamped relational events between a set of entities, such as messages between users of an on-line social network. Such data are often analyzed using static or discrete-time network models, which discard a significant amount of information by aggregating events over time to form network snapshots. In this paper, we introduce a block point process model (BPPM) for continuous-time event-based dynamic networks. The BPPM is inspired by the well-known stochastic block model (SBM) for static networks. We show that networks generated by the BPPM follow an SBM in the limit of a growing number of nodes. We use this property to develop principled and efficient local search and variational inference procedures initialized by regularized spectral clustering. We fit BPPMs with exponential Hawkes processes to analyze several real network data sets, including a Facebook wall post network with over 3,500 nodes and 130,000 events.

[1]  Fabrice Rossi,et al.  Exact ICL maximization in a non-stationary temporal extension of the stochastic block model for dynamic networks , 2016, Neurocomputing.

[2]  Jorge Nocedal,et al.  A trust region method based on interior point techniques for nonlinear programming , 2000, Math. Program..

[3]  Scott W. Linderman,et al.  Discovering Latent Network Structure in Point Process Data , 2014, ICML.

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

[5]  Edoardo M. Airoldi,et al.  A Survey of Statistical Network Models , 2009, Found. Trends Mach. Learn..

[6]  James R. Foulds,et al.  HawkesTopic: A Joint Model for Network Inference and Topic Modeling from Text-Based Cascades , 2015, ICML.

[7]  Rebecca Willett,et al.  Tracking Dynamic Point Processes on Networks , 2014, IEEE Transactions on Information Theory.

[8]  Scott W. Linderman,et al.  Scalable Bayesian Inference for Excitatory Point Process Networks , 2015, 1507.03228.

[9]  Katherine A. Heller,et al.  Modelling Reciprocating Relationships with Hawkes Processes , 2012, NIPS.

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

[11]  Tai Qin,et al.  Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel , 2013, NIPS.

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

[13]  Bin Yu,et al.  Co-clustering directed graphs to discover asymmetries and directional communities , 2016, Proceedings of the National Academy of Sciences.

[14]  Bin Yu,et al.  Spectral clustering and the high-dimensional stochastic blockmodel , 2010, 1007.1684.

[15]  T. Taimre,et al.  Hawkes Processes , 2015, 1507.02822.

[16]  Peter F. Halpin,et al.  Modelling Dyadic Interaction with Hawkes Processes , 2013, Psychometrika.

[17]  Ji Zhu,et al.  Consistency of community detection in networks under degree-corrected stochastic block models , 2011, 1110.3854.

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

[19]  Jure Leskovec,et al.  SEISMIC: A Self-Exciting Point Process Model for Predicting Tweet Popularity , 2015, KDD.

[20]  Shuang Li,et al.  COEVOLVE: A Joint Point Process Model for Information Diffusion and Network Co-evolution , 2015, NIPS.

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

[22]  Yosihiko Ogata,et al.  On Lewis' simulation method for point processes , 1981, IEEE Trans. Inf. Theory.

[23]  Thomas L. Griffiths,et al.  Learning Systems of Concepts with an Infinite Relational Model , 2006, AAAI.

[24]  P. Latouche,et al.  Model selection and clustering in stochastic block models based on the exact integrated complete data likelihood , 2015 .

[25]  David Lazer,et al.  Inferring friendship network structure by using mobile phone data , 2009, Proceedings of the National Academy of Sciences.

[26]  Mark E. J. Newman,et al.  Stochastic blockmodels and community structure in networks , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  E. Xing,et al.  A state-space mixed membership blockmodel for dynamic network tomography , 2008, 0901.0135.

[28]  Kazuo Yano,et al.  Self-exciting point process modeling of conversation event sequences , 2012, ArXiv.

[29]  Andrea L. Bertozzi,et al.  Modeling E-mail Networks and Inferring Leadership Using Self-Exciting Point Processes , 2016 .

[30]  T. Ozaki.  Maximum likelihood estimation of Hawkes' self-exciting point processes , 1979 .

[31]  Xiangyu Chang,et al.  Asymptotic Normality of Maximum Likelihood and its Variational Approximation for Stochastic Blockmodels , 2012, ArXiv.

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

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

[34]  Le Song,et al.  Evolving Cluster Mixed-Membership Blockmodel for Time-Evolving Networks , 2011, AISTATS.

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

[36]  David M. Blei,et al.  Variational Inference: A Review for Statisticians , 2016, ArXiv.

[37]  A. Rinaldo,et al.  Consistency of spectral clustering in stochastic block models , 2013, 1312.2050.

[38]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Le Song,et al.  NetCodec: Community Detection from Individual Activities , 2015, SDM.

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

[41]  Edoardo M. Airoldi,et al.  Consistent estimation of dynamic and multi-layer block models , 2014, ICML.

[42]  Padhraic Smyth,et al.  Modeling relational events via latent classes , 2010, KDD.

[43]  Carey E. Priebe,et al.  A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs , 2011, 1108.2228.

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

[45]  Alain Celisse,et al.  Consistency of maximum-likelihood and variational estimators in the Stochastic Block Model , 2011, 1105.3288.

[46]  Krishna P. Gummadi,et al.  On the evolution of user interaction in Facebook , 2009, WOSN '09.