  Inspired by applications to theories of coding and communication in networks
of nervous tissue, we study maximum entropy distributions on weighted graphs
with a given expected degree sequence. These distributions are characterized by
independent edge weights parameterized by a shared vector of vertex potentials.
Using the general theory of exponential family distributions, we derive the
existence and uniqueness of the maximum likelihood estimator (MLE) of the
vertex parameters. We also prove consistency of the MLE from a single sample in
the limit of large graphs, extending results of Chatterjee, Diaconis, and Sly
in the unweighted case (the "beta-model" in statistics). Interestingly, our
proofs require tight estimates on the norms of inverses of symmetric,
diagonally dominant positive matrices. Along the way, we derive analogues of
the Erdos-Gallai criterion of graphical degree sequences for weighted graphs.
