  Probability models on graphs are becoming increasingly important in many
applications, but statistical tools for fitting such models are not yet well
developed. Here we propose a general method of moments approach that can be
used to fit a large class of probability models through empirical counts of
certain patterns in a graph. We establish some general asymptotic properties of
empirical graph moments and prove consistency of the estimates as the graph
size grows for all ranges of the average degree including $\Omega(1)$.
Additional results are obtained for the important special case of degree
distributions.
