Closely estimating the entropy of sparse graph models

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.