The Annals of Applied Statistics

Co-evolution of social networks and continuous actor attributes

Nynke M. D. Niezink and Tom A. B. Snijders

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Social networks and the attributes of the actors in these networks are not static; they may develop interdependently over time. The stochastic actor-oriented model allows for statistical inference on the mechanisms driving this co-evolution process. In earlier versions of this model, dynamic actor attributes are assumed to be measured on an ordinal categorical scale. We present an extension of the stochastic actor-oriented model that does away with this restriction using a stochastic differential equation to model the evolution of continuous actor attributes. We estimate the parameters by a procedure based on the method of moments. The proposed method is applied to study the dynamics of a friendship network among the students at an Australian high school. In particular, we model the relationship between friendship and obesity, focusing on body mass index as a continuous co-evolving attribute.

Article information

Ann. Appl. Stat., Volume 11, Number 4 (2017), 1948-1973.

Received: April 2016
Revised: February 2017
First available in Project Euclid: 28 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Social networks longitudinal data Markov model stochastic differential equations method of moments


Niezink, Nynke M. D.; Snijders, Tom A. B. Co-evolution of social networks and continuous actor attributes. Ann. Appl. Stat. 11 (2017), no. 4, 1948--1973. doi:10.1214/17-AOAS1037.

Export citation


  • Agneessens, F. and Wittek, R. (2008). Social capital and employee well-being: Disentangling intrapersonal and interpersonal selection and influence mechanisms. Revue Française de Sociologie 49 613–637.
  • Amati, V., Schönenberger, F. and Snijders, T. A. B. (2015). Estimation of stochastic actor-oriented models for the evolution of networks by generalized method of moments. J. SFdS 156 140–165.
  • Arnold, L. (1974). Stochastic Differential Equations: Theory and Applications. Wiley-Interscience, New York. Translated from the German.
  • Bergstrom, A. R. (1984). Continuous time stochastic models and issues of aggregation over time. In Handbook of Econometrics, Vol. 2 (Z. Griliches and M. D. Intriligator, eds.). Handbooks in Economics 2 1146–1212. North-Holland, Amsterdam.
  • Bergstrom, A. R. (1988). The history of continuous-time econometric models. Econometric Theory 4 365–383.
  • Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Polit. Econ. 81 637–654.
  • Christakis, N. A. and Fowler, J. H. (2007). The spread of obesity in a large social network over 32 years. N. Engl. J. Med. 357 370–379.
  • Cohen-Cole, E. and Fletcher, J. M. (2008). Is obesity contagious? Social networks vs. environmental factors in obesity epidemic. J. Health Econ. 27 1382–1387.
  • De la Haye, K., Robins, G., Mohr, P. and Wilson, C. (2011). Homophily and contagion as explanations for weight similarities among adolescent friends. J. Adolesc. Health 49 421–427.
  • Dijkstra, J. K., Lindenberg, S., Veenstra, R., Steglich, C., Isaacs, J., Card, N. A. and Hodges, E. V. E. (2010). Influence and selection processes in weapon carrying during adolescence: The roles of status, aggression, and vulnerability. Criminology 48 187–220.
  • Dijkstra, J. K., Gest, S. D., Lindenberg, S., Veenstra, R. and Cillessen, A. H. N. (2012). The emergence of weapon carrying in peer context. Testing three explanations: The role of aggression, victimization, and friends. J. Adolesc. Health 50 371–376.
  • Hamerle, A., Singer, H. and Nagl, W. (1993). Identification and estimation of continuous time dynamic systems with exogenous variables using panel data. Econometric Theory 9 283–295.
  • Holland, P. W. and Leinhardt, S. (1977/1978). A dynamic model for social networks. J. Math. Sociol. 5 5–20.
  • Hunter, D. R. (2007). Curved exponential family models for social networks. Soc. Netw. 29 216–230.
  • Koskinen, J. H. and Snijders, T. A. B. (2007). Bayesian inference for dynamic social network data. J. Statist. Plann. Inference 137 3930–3938.
  • Kushner, H. J. and Yin, G. G. (2003). Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed. Applications of Mathematics (New York): Stochastic Modelling and Applied Probability 35. Springer, New York.
  • Lehmann, E. L. (1999). Elements of Large-Sample Theory. Springer, New York.
  • McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In Frontiers in Economics (P. Zarembka, ed.) 105–142. Academic Press, New York.
  • Merton, R. C. (1990). Continuous-Time Finance. Basil Blackwell, Oxford.
  • Oud, J. H. L. and Jansen, R. A. R. G. (2000). Continuous time state space modeling of panel data by means of SEM. Psychometrika 65 199–215.
  • Oud, J. H. L., Folmer, H., Patuelli, R. and Nijkamp, P. (2012). Continuous-time modeling with spatial dependence. Geogr. Anal. 44 29–46.
  • Ripley, R. M., Snijders, T. A. B., Boda, Z., Vörös, A. and Preciado, P. (2017). Manual for RSiena (version March 2017). Dept. Statistics, and Nuffield College, Univ. Oxford, Oxford.
  • Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Stat. 22 400–407.
  • Schweinberger, M. and Snijders, T. A. B. (2007). Markov models for digraph panel data: Monte Carlo-based derivative estimation. Comput. Statist. Data Anal. 51 4465–4483.
  • Shalizi, C. R. and Thomas, A. C. (2011). Homophily and contagion are generically confounded in observational social network studies. Sociol. Methods Res. 40 211–239.
  • Singer, H. (1996). Continuous-time dynamic models for panel data. In Analysis of Change: Advanced Techniques in Panel Data Analysis (U. Engel and J. Reinecke, eds.) 113–133. de Gruyter, Berlin.
  • Snijders, T. A. B. (2001). The statistical evaluation of social network dynamics. In Sociological Methodology (M. Sobel and M. Becker, eds.) 361–395. Basil Blackwell, Boston.
  • Snijders, T. A. B. (2005). Models for longitudinal network data. In Models and Methods in Social Network Analysis (P. J. Carrington, J. Scott and S. S. Wasserman, eds.) 215–247. Cambridge Univ. Press, New York.
  • Snijders, T. A. B., Koskinen, J. and Schweinberger, M. (2010). Maximum likelihood estimation for social network dynamics. Ann. Appl. Stat. 4 567–588.
  • Snijders, T. A. B., Steglich, C. E. G. and Schweinberger, M. (2007). Modeling the co-evolution of networks and behavior. In Longitudinal Models in the Behavioral and Related Sciences (K. van Montfort, H. Oud and A. Satorra, eds.) 41–71. Cambridge Univ. Press, New York.
  • Steele, J. M. (2001). Stochastic Calculus and Financial Applications. Applications of Mathematics (New York): Stochastic Modelling and Applied Probability 45. Springer, New York.
  • Steglich, C. E. G., Snijders, T. A. B. and Pearson, M. (2010). Dynamic networks and behavior: Separating selection from influence. Sociol. Method. 40 329–392.
  • Voelkle, M. C., Oud, J. H. L., Davidov, E. and Schmidt, P. (2012). An SEM approach to continuous time modeling of panel data: Relating authoritarianism and anomia. Psychol. Methods 17 176–192.
  • Wasserman, S. S. (1980). Analyzing social networks as stochastic processes. J. Amer. Statist. Assoc. 75 280–294.
  • Wasserman, S. S. and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge Univ. Press, Cambridge.
  • Weerman, F. (2011). Delinquent peers in context: A longitudinal network analysis of selection and influence effects. Criminology 49 253–286.