Maximum entropy distributions on graphs

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.

36 pages

References in corpus (8)

• Weak pairwise correlations imply strongly correlated network states in a neural population
• Statistical mechanics for natural flocks of birds
• Maximum entropy models for antibody diversity
• Statistical Inference for Valued-Edge Networks: Generalized Exponential Random Graph Models
• Ising models for networks of real neurons
• Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters
• The Entropic Discriminant
• Inverses of symmetric, diagonally dominant positive matrices and applications

Cited by in corpus (12)

• Clique topology reveals intrinsic geometric structure in neural correlations
• Asymptotics in directed exponential random graph models with an increasing bi-degree sequence
• The Entropic Discriminant
• Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters
• The $\beta$-model for Random Graphs --- Regression, Cramér-Rao Bounds, and Hypothesis Testing
• Inverses of symmetric, diagonally dominant positive matrices and applications
• A Unified Framework for Inference in Network Models with Degree Heterogeneity and Homophily
• Block-Approximated Exponential Random Graphs
• A Probit Network Model with Arbitrary Dependence
• A sparse $p_0$ model with covariates for directed networks
• Affiliation networks with an increasing degree sequence
• A network Poisson model for weighted directed networks with covariates