The Annals of Statistics

Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels

Peter Bickel, David Choi, Xiangyu Chang, and Hai Zhang

Full-text: Open access


Variational methods for parameter estimation are an active research area, potentially offering computationally tractable heuristics with theoretical performance bounds. We build on recent work that applies such methods to network data, and establish asymptotic normality rates for parameter estimates of stochastic blockmodel data, by either maximum likelihood or variational estimation. The result also applies to various sub-models of the stochastic blockmodel found in the literature.

Article information

Ann. Statist., Volume 41, Number 4 (2013), 1922-1943.

First available in Project Euclid: 23 October 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators

Network statistics stochastic blockmodeling variational methods maximum likelihood


Bickel, Peter; Choi, David; Chang, Xiangyu; Zhang, Hai. Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels. Ann. Statist. 41 (2013), no. 4, 1922--1943. doi:10.1214/13-AOS1124.

Export citation


  • Bickel, P. J. and Chen, A. (2009). A nonparametric view of network models and Newman–Girvan and other modularities. Proc. Natl. Acad. Sci. USA 106 21068–21073.
  • Bickel, P. J., Chen, A. and Levina, E. (2011). The method of moments and degree distributions for network models. Ann. Statist. 39 2280–2301.
  • Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
  • Celisse, A., Daudin, J. J. and Pierre, L. (2011). Consistency of maximum-likelihood and variational estimators in the stochastic block model. Available at arXiv:1105.3288.
  • Channarond, A., Daudin, J. J. and Robin, S. (2011). Classification and estimation in the stochastic block model based on the empirical degrees. Available at arXiv:1110.6517.
  • Chatterjee, S. (2012). Matrix estimation by universal singular value thresholding. Preprint. Available at arXiv:1212.1247.
  • Chaudhuri, K., Chung, F. and Tsiatas, A. (2012). Spectral clustering of graphs with general degrees in the extended planted partition model. J. Mach. Learn. Res. 23 1–23.
  • Choi, D. S., Wolfe, P. J. and Airoldi, E. M. (2012). Stochastic blockmodels with a growing number of classes. Biometrika 99 273–284.
  • Coja-Oghlan, A. and Lanka, A. (2008). Partitioning random graphs with general degree distributions. In Fifth IFIP International Conference on Theoretical Computer Science—TCS 2008. IFIP Int. Fed. Inf. Process. 273 127–141. Springer, New York.
  • Daudin, J. J., Picard, F. and Robin, S. (2008). A mixture model for random graphs. Stat. Comput. 18 173–183.
  • Handcock, M. S., Raftery, A. E. and Tantrum, J. M. (2007). Model-based clustering for social networks. J. Roy. Statist. Soc. Ser. A 170 301–354.
  • Hoff, P. D., Raftery, A. E. and Handcock, M. S. (2002). Latent space approaches to social network analysis. J. Amer. Statist. Assoc. 97 1090–1098.
  • Karrer, B. and Newman, M. E. J. (2011). Stochastic blockmodels and community structure in networks. Phys. Rev. E (3) 83 016107, 10.
  • Latouche, P., Birmelé, E. and Ambroise, C. (2011). Overlapping stochastic block models with application to the French political blogosphere. Ann. Appl. Stat. 5 309–336.
  • Lazer, D., Pentland, A. S., Adamic, L., Aral, S., Barabasi, A. L., Brewer, D., Christakis, N., Contractor, N., Fowler, J., Gutmann, M. et al. (2009). Life in the network: The coming age of computational social science. Science 323 721–723.
  • Le Cam, L. and Yang, G. L. (1988). On the preservation of local asymptotic normality under information loss. Ann. Statist. 16 483–520.
  • Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer, New York.
  • Proulx, S. R., Promislow, D. E. L. and Phillips, P. C. (2005). Network thinking in ecology and evolution. Trends in Ecology & Evolution 20 345–353.
  • Rohe, K., Chatterjee, S. and Yu, B. (2011). Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist. 39 1878–1915.
  • Rohe, K., Qin, T. and Fan, H. (2012). The highest dimensional stochastic blockmodel with a regularized estimator. Preprint. Available at arXiv:1206.2380.
  • Snijders, T. A. B. and Nowicki, K. (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classification 14 75–100.
  • Van der Vaart, A. W. (2000). Asymptotic Statistics 3. Cambridge Univ. Press, Cambridge.
  • Zhao, Y., Levina, E. and Zhu, J. (2012). Consistency of community detection in networks under degree-corrected stochastic block models. Ann. Statist. 40 2266–2292.