Pseudolikelihood Estimation for Social Networks

Abstract Interest in log-linear modeling for social-network data has grown steadily since Holland and Leinhardt (1981) proposed their p 1 model. That model was designed for a single binary relationship (directed graph) representing interactions between individuals. It assumed that interactions between pairs of individuals are mutually independent. Subsequent work has extended the model in various ways, including block-modeling and the case of dependence between pairs of individuals. In empirical work it would often be desirable to fit a wide variety of these models, as the differences in predictions or goodness of fit are likely to provide insights into the data. This has not been common practice, however, because estimation for some of the models has been difficult, and the maximum likelihood schemes developed for others involve different computer programs not always available in standard packages. The focus of this article is on a general estimation technique that maximizes the pseudolikelihood, the product of the probabilities of the binary variables, with each probability conditional on the rest of the data. The method is shown to be equivalent to a weighted least squares procedure and thus can be carried out with standard computer packages. In cases where true maximum likelihood estimation is available for comparison the two methods seem to work about equally well. The pseudolikelihood estimation is used in an example where the fits of a large number of different models are compared. Some of these models, such as various Markov block models, have not previously been proposed. In this example (as in others considered) it appears that the p 1-type models are overparameterized, and that much more parsimonious models give tolerable fits.