Advances in exponential random graph ( p*) models applied to a large social network

Abstract

Recent advances in statistical network analysis based on the family of exponential random graph (ERG) models have greatly improved our ability to conduct inference on dependence in large social networks [Snijders, T.A.B., 2002. Markov Chain Monte Carlo Estimation of Exponential Random Graph Models. Journal of Social Structure 3, 2; Pattison, P.E., Robins, G.L., 2002. Neighbourhood-based models for social networks. Sociological Methodology 32, 301–337; Handcock, M.S., 2002. Statistical models for social networks: degeneracy and inference. In: Breiger, R., Carley, K., Pattison, P. (Eds.), Dynamic Social Network Modeling and Analysis. National Academies Press, Washington, DC, pp. 229–240; Handcock, M.S., 2003. Assessing Degeneracy in Statistical Models of Social Networks. Center for Statistics and the Social Sciences, University of Washington, Working Paper No. 39; Snijders, T.A.B., Pattison, P., Robins, G.L., Handcock, M., in press. New specifications for exponential random graph models. Sociological Methodology; Hunter, D.R., Goodreau, S.M. Handcock, M.S., 2005. Goodness of Fit of Social Network Models. Center for Statistics and the Social Sciences, University of Washington, Working Paper No. 47; Goodreau, S.M., Hunter, D.R., Morris, M., 2005. Statistical Modeling of Social Networks: Practical Advances and Results. Center for Studies in Demography and Ecology, University of Washington, Working Paper No. 05-01, previous papers this issue]. This paper applies advances in both model parameterizations and computational algorithms to an examination of the structure observed in an adolescent friendship network of 1681 actors from the National Longitudinal Study of Adolescent Health (AddHealth). ERG models of social network structure are fit using the R package statnet, and their adequacy assessed through comparison of model predictions with the observed data for higher-order network statistics. For this friendship network, the commonly used model of Markov dependence leads to the problems of degeneracy discussed by Handcock [Handcock, M.S., 2002. Statistical models for social networks: degeneracy and inference. In: Breiger, R., Carley, K., Pattison, P. (Eds.), Dynamic Social Network Modeling and Analysis. National Academic Press, Washington, DC, pp. 229–240; Handcock, M.S., 2003. Assessing Degeneracy in Statistical Models of Social Networks. Center for Statistics and the Social Sciences, University of Washington, Working Paper No. 39]. On the other hand, model parameterizations introduced by Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handcock, M., in press. New specifications for exponential random graph models. Sociological Methodology] and Hunter and Handcock [Hunter, D.R., Handcock, M.S., in press. Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics] avoid degeneracy and provide reasonable fit to the data. Degree-only models did a poor job of capturing observed network structure; those that did best included terms both for heterogeneous mixing on exogenous attributes (grade and self-reported race) as well as endogenous clustering. Networks simulated from this model were largely consistent with the observed network on multiple higher-order network statistics, including the number of triangles, the size of the largest component, the overall reachability, the distribution of geodesic distances, the degree distribution, and the shared partner distribution. The ability to fit such models to large datasets and to make inference about the underling processes generating the network represents a major advance in the field of statistical network analysis.