The β-Model-Maximum Likelihood, Cramer-Rao Bounds, and Hypothesis Testing

We study the maximum-likelihood estimator in a setting where the dependent variable is a random graph and covariates are available on a graph level. The model generalizes the well-known beta-model for random graphs by replacing the constant model parameters with regression functions. Cramer-Rao bounds are derived for special cases of the undirected beta-model, the directed beta-model, and the covariate-based beta-model. The corresponding maximum-likelihood estimators are compared with the bounds by means of simulations. Moreover, examples are given on how to use the presented maximum-likelihood estimators to test for directionality and significance. Finally, the applicability of the model is demonstrated using temporal social network data describing communication among healthcare workers.