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 Ω(1). Additional results are obtained
for the important special case of degree distributions.