The Annals of Applied Statistics

Maximum likelihood estimation for social network dynamics

Tom A. B. Snijders, Johan Koskinen, and Michael Schweinberger

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A model for network panel data is discussed, based on the assumption that the observed data are discrete observations of a continuous-time Markov process on the space of all directed graphs on a given node set, in which changes in tie variables are independent conditional on the current graph. The model for tie changes is parametric and designed for applications to social network analysis, where the network dynamics can be interpreted as being generated by choices made by the social actors represented by the nodes of the graph. An algorithm for calculating the Maximum Likelihood estimator is presented, based on data augmentation and stochastic approximation. An application to an evolving friendship network is given and a small simulation study is presented which suggests that for small data sets the Maximum Likelihood estimator is more efficient than the earlier proposed Method of Moments estimator.

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Ann. Appl. Stat., Volume 4, Number 2 (2010), 567-588.

First available in Project Euclid: 3 August 2010

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Graphs longitudinal data method of moments stochastic approximation Robbins–Monro algorithm


Snijders, Tom A. B.; Koskinen, Johan; Schweinberger, Michael. Maximum likelihood estimation for social network dynamics. Ann. Appl. Stat. 4 (2010), no. 2, 567--588. doi:10.1214/09-AOAS313.

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  • Airoldi, E., Blei, D. M., Fienberg, S. E., Goldenberg, A., Xing, E. P. and Zheng, A. X. (2007). Statistical Network Analysis: Models, Issues and New Directions (ICML 2006). Lecture Notes in Computer Science 4503. Springer, Berlin.
  • Carrington, P. J., Scott, J. and Wasserman, S., eds. (2005). Models and Methods in Social Network Analysis. Cambridge Univ. Press.
  • Davis, J. A. (1970). Clustering and hierarchy in interpersonal relations: Testing two theoretical models on 742 sociograms. American Sociological Review 35 843–852.
  • de Federico de la Rúa, A. (2003). La dinámica de las redes de amistad. La elección de amigos en el programa Erasmus. REDES 4.3 1–44.
  • Efron, B. (1977). Discussion of Dempster, Laird and Rubin (1977). Maximum likelihood estimation from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 29.
  • Fisher, R. A. (1925). Theory of statistical estimation. Proceedings of the Cambridge Philosophical Society 22 700–725.
  • Frank, O. (1991). Statistical analysis of change in networks. Statist. Neerlandica 45 283–293.
  • Frank, O. and Strauss, D. (1986). Markov graphs. J. Amer. Statist. Assoc. 81 832–842.
  • Gelman, A. and Meng, X.-L. (1998). Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. Statist. Sci. 13 163–185.
  • Gu, M. G. and Kong, F. H. (1998). A stochastic approximation algorithm with Markov chain Monte-Carlo Method for incomplete data estimation problems. Proc. Natl. Acad. Sci. USA 95 7270–7274.
  • Gu, M. G. and Zhu, H.-T. (2001). Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation. J. Roy. Statist. Soc. Ser. B 63 339–355.
  • Hanneke, S. and Xing, E. P. (2007). Discrete temporal models of social networks. In Statistical Network Analysis: Models, Issues and New Directions (ICML 2006) (E. Airoldi, D. M. Blei, S. E. Fienberg, A. Goldenberg, E. P. Xing and A. X. Zheng, eds.). Lecture Notes in Computer Science 4503 115–125. Springer, Berlin.
  • Holland, P. and Leinhardt, S. (1977). A dynamic model for social networks. Journal of Mathematical Sociology 5 5–20.
  • Hunter, D. R. and Handcock, M. S. (2006). Inference in curved exponential family models for networks. J. Graph. Comput. Statist. 15 565–583.
  • Krackhardt, D. and Handcock, M. S. (2007). Heider vs Simmel: Emergent features in dynamic structures. In Statistical Network Analysis: Models, Issues and New Directions (ICML 2006). (E. Airoldi, D. M. Blei, S. E. Fienberg, A. Goldenberg, E. P. Xing and A. X. Zheng, eds.) Lecture Notes in Computer Science 4503 14–27. Springer, Berlin.
  • Koskinen, J. H. and Snijders, T. A. B. (2007). Bayesian inference for dynamic network data. J. Statist. Plan. Inference 137 3930–3938.
  • Kushner, H. J. and Yin, G. G. (2003). Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed. Springer, New York.
  • Leenders, R. T. A. J. (1995). Models for network dynamics: A Markovian framework. Journal of Mathematical Sociology 20 1–21.
  • Louis, T. A. (1982). Finding observed information when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44 226–233.
  • Maddala, G. S. (1983). Limited-dependent and Qualitative Variables in Econometrics. Cambridge Univ. Press.
  • Norris, J. R. (1997). Markov Chains. Cambridge Univ. Press.
  • Orchard, T. and Woodbury, M. A. (1972). A missing information principle: Theory and applications. In Proceedings Sixth Berkeley Sympos. Math. Statist. Probab. 1 697–715. Univ. California Press, Berkeley.
  • Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22 400–407.
  • Robins, G. and Pattison, P. (2001). Random graph models for temporal processes in social networks. Journal of Mathematical Sociology 25 5–41.
  • Schweinberger, M. and Snijders, T. A. B. (2006). Markov models for digraph panel data: Monte Carlo-based derivative estimation. Comput. Statist. Data Anal. 51 4465–4483.
  • Singer, H. (2008). Nonlinear continuous time modeling approaches in panel research. Statist. Neerlandica 62 29–57.
  • Snijders, T. A. B. (2001). The statistical evaluation of social network dynamics. In Sociological Methodology—2001 (M. E. Sobel and M. P. Becker, eds.) 361–395. Blackwell, London.
  • Snijders, T. A. B. (2006). Statistical methods for network dynamics. In Proceedings of the XLIII Scientific Meeting, Italian Statistical Society, (S. R. Luchini et al. eds.) 281–296. CLEUP, Padova.
  • Snijders, T. A. B. and van Duijn, M. A. J. (1997). Simulation for statistical inference in dynamic network models. In Simulating Social Phenomena (R. Conte, R. Hegselmann and P. Terna, eds.) 493–512. Springer, Berlin.
  • Snijders, T. A. B., Pattison, P. E., Robins, G. L. and Handcock, M. S. (2006). New specifications for exponential random graph models. Sociological Methodology 36 99–153.
  • Snijders, T. A. B., Steglich, C. E. G., Schweinberger, M., Huisman, M. (2009). Manual for SIENA version 3.2. Dept. Statistics, Univ. Oxford. Univ. Groningen, ICS. Available at
  • Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation (with discussion). J. Amer. Statist. Assoc. 82 528–550.
  • van de Bunt, G. G., van Duijn, M. A. J. and Snijders, T. A. B. (1999). Friendship networks through time: An actor-oriented statistical network model. Computational and Mathematical Organization Theory 5 167–192.
  • van Duijn, M. A. J., Zeggelink, E. P. H., Huisman, M., Stokman, F. M. and Wasseur, F. W. (2003). Evolution of sociology freshmen into a friendship network. Journal of Mathematical Sociology 27 153–191.
  • Wasserman, S. (1979). A stochastic model for directed graphs with transition rates determined by reciprocity. Sociological Methodology—1980 (K. F. Schuessler, ed.) 392–412. Jossey-Bass, San Francisco.
  • Wasserman, S. (1980). Analyzing social networks as stochastic processes. J. Amer. Statist. Assoc. 75 280–294.
  • Wasserman, S. and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge Univ. Press.
  • Wasserman, S. and Pattison, P. (1996). Logit models and logistic regression for social networks: I. An introduction to Markov graphs and p*. Psychometrika 61 401–425.