  We study maximum likelihood estimation for the statistical model for
undirected random graphs, known as the $\beta$-model, in which the degree
sequences are minimal sufficient statistics. We derive necessary and sufficient
conditions, based on the polytope of degree sequences, for the existence of the
maximum likelihood estimator (MLE) of the model parameters. We characterize in
a combinatorial fashion sample points leading to a nonexistent MLE, and
nonestimability of the probability parameters under a nonexistent MLE. We
formulate conditions that guarantee that the MLE exists with probability
tending to one as the number of nodes increases.
