  We develop a maximum-likelihood based method for regression 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.
Cram\'er-Rao bounds are derived for the undirected $\beta$-model, the directed
$\beta$-model, and the generalized $\beta$-model. The corresponding maximum
likelihood estimators are compared to 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. Last, the applicability
of the model is demonstrated using dynamic social network data describing
communication among healthcare workers.
