Tractable and Consistent Random Graph Models

We define a general class of network formation models, Statistical Exponential Random Graph Models (SERGMs), that nest standard exponential random graph models (ERGMs) as a special case. We analyze conditions for practical and consistent estimation of the network formation parameters. This addresses two holes in the estimation of exponential random graph models. First, although it is known that due to the enormity of the space of possible networks Markov chain Monte Carlo methods of estimation face slow (exponential) mixing times for many specifications, we provide a first set of results identifying nontrivial specifications for which practical, accurate estimation is possible. Second, we provide consistency results showing when maximum likelihood and GMM (generalized method of moments) estimates of parameter converge to the true values in SERGMs. In particular, we show that a class of SERGMs that count subgraphs of various types are consistently estimated using direct and fast techniques. We also define a related class of network formation models, SUGMs, and show that they are also consistently and easily estimated when networks are sufficiently sparse. We illustrate the application of the models and techniques with data.