Maximum entropy distributions on graphs

Inspired by the problem of sensory coding in neuroscience, we study the maximum entropy distribution on weighted graphs with a given expected degree sequence. This distribution on graphs is characterized by independent edge weights parameterized by vertex potentials at each node. Using the general theory of exponential family distributions, we prove the existence and uniqueness of the maximum likelihood estimator (MLE) of the vertex parameters. We also prove the consistency of the MLE from a single graph sample, extending the results of Chatterjee, Diaconis, and Sly for unweighted (binary) graphs. Interestingly, our extensions require an intricate study of the inverses of diagonally dominant positive matrices. Along the way, we derive analogues of the Erd\H{o}s-Gallai criterion of graphical sequences for weighted graphs.