## The Annals of Statistics

### Dynamic network models and graphon estimation

Marianna Pensky

#### Abstract

In the present paper, we consider a dynamic stochastic network model. The objective is estimation of the tensor of connection probabilities $\mathbf{{\Lambda}}$ 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 $\widehat{\boldsymbol{\Lambda}}$ of $\mathbf{{\Lambda}}$ and show that $\widehat{\boldsymbol{\Lambda}}$ satisfies an oracle inequality and also attains minimax lower bounds for the risk. We extend those results to estimation of $\mathbf{{\Lambda}}$ 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 nonasymptotic and allow a variety of extensions.

#### Article information

Source
Ann. Statist., Volume 47, Number 4 (2019), 2378-2403.

Dates
Revised: March 2018
First available in Project Euclid: 21 May 2019

https://projecteuclid.org/euclid.aos/1558425649

Digital Object Identifier
doi:10.1214/18-AOS1751

Mathematical Reviews number (MathSciNet)
MR3953455

Zentralblatt MATH identifier
07082290

#### Citation

Pensky, Marianna. Dynamic network models and graphon estimation. Ann. Statist. 47 (2019), no. 4, 2378--2403. doi:10.1214/18-AOS1751. https://projecteuclid.org/euclid.aos/1558425649

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#### Supplemental materials

• Supplementary material. The supplement contains proofs of all statements in the paper.