Abstract
We present a systematic examination of real network datasets using maximum likelihood estimation for exponential random graph models as well as new procedures to evaluate how well the models fit the observed graphs. These procedures compare structural statistics of the observed graph with the corresponding statistics on graphs simulated from the fitted model. We apply this approach to the study of friendship relations among high school students from the National Longitudinal Study of Adolescent Health (AddHealth). The sizes of the networks we fit range from 71 to 2209 nodes. The larger networks represent more than an order of magnitude increase over the size of any network previously fit using maximum likelihood methods for models of this kind. We argue that several well-studied models in the networks literature do not fit these data well, and we demonstrate that the fit improves dramatically when the models include the recently-developed geometrically weighted edgewise shared partner (GWESP), geometrically weighted dyadic shared partner (GWDSP), and geometrically weighted degree (GWD) network statistics. We conclude that these models capture aspects of the social structure of adolescent friendship relations not represented by previous models.