  We introduce an algorithm for estimating the entropy of pairwise,
probabilistic graph models by leveraging bridges between social communities and
an accurate entropy estimator on sparse samples. We propose using a measure of
investment from the sociological literature, Burt's structural constraint, as a
heuristic for identifying bridges that partition a graph into conditionally
independent components. We combine this heuristic with the
Nemenman-Shafee-Bialek entropy estimator to obtain a faster and more accurate
estimator. We demonstrate it on the pairwise maximum entropy, or Ising, models
of judicial voting, to improve na\"ive entropy estimates. We use our algorithm
to estimate the partition function closely, which we then apply to the problem
of model selection, where estimating the likelihood is difficult. This serves
as an improvement over existing methods that rely on point correlation
functions to test fit can be extended to other graph models with a
straightforward modification of the open-source implementation.
