Dynamic network models and graphon estimation

TL;DR

This paper develops adaptive estimation methods for dynamic network models, including stochastic block models and graphons, providing theoretical guarantees and simplifying mathematical analysis.

Contribution

It introduces a penalized least squares estimator for dynamic network connection probabilities, achieving oracle inequalities and minimax bounds, with adaptivity to unknown parameters.

Findings

Estimator satisfies oracle inequality.

Estimator attains minimax lower bounds.

Results are non-asymptotic and extendable.

Abstract

In the present paper we consider a dynamic stochastic network model. The objective is estimation of the tensor of connection probabilities $\Lambda$ when it is generated by a Dynamic Stochastic Block Model (DSBM) or a dynamic graphon. In particular, in the context of the DSBM, we derive a penalized least squares estimator $\widehat{\Lambda}$ of $\Lambda$ and show that $\widehat{\Lambda}$ satisfies an oracle inequality and also attains minimax lower bounds for the risk. We extend those results to estimation of $\Lambda$ when it is generated by a dynamic graphon function. The estimators constructed in the paper are adaptive to the unknown number of blocks in the context of the DSBM or to the smoothness of the graphon function. The technique relies on the vectorization of the model and leads to much simpler mathematical arguments than the ones used previously in the stationary set up. In addition, all results in the paper are non-asymptotic and allow a variety of extensions.