Maximum entropy distributions on graphs

TL;DR

This paper develops a statistical framework for maximum entropy distributions on weighted graphs with specified expected degrees, establishing existence, uniqueness, and consistency of estimators, and extending classical graph degree sequence criteria.

Contribution

It introduces a novel maximum likelihood estimation approach for weighted graph models with given expected degrees, including proofs of estimator properties and new criteria for weighted graph degree sequences.

Findings

Proved existence and uniqueness of the MLE for vertex potentials.

Established the consistency of the MLE from a single large graph sample.

Derived weighted graph analogues of the Erdos-Gallai criterion.

Abstract

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.