# Maximum entropy distributions on graphs

arXiv: https://arxiv.org/abs/1301.3321 · HTML: https://mlsanity.com/paper/1301.3321/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2013.

## Built on by (most cited first, 4 of 4)

- [Distributed Differential Privacy in Multi-Armed Bandits](https://mlsanity.com/paper/2206.05772/) (2022)
- [Modeling Attrition in Recommender Systems with Departing Bandits](https://mlsanity.com/paper/2203.13423/) (2022)
- [Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph boldsymbolβ-Model](https://mlsanity.com/paper/2307.02818/) (2023)
- [Asymptotic in a class of network models with an increasing sub-Gamma degree sequence](https://mlsanity.com/paper/2111.01301/) (2021)

## Abstract (arXiv)

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.