The Annals of Statistics

Oracle inequalities for network models and sparse graphon estimation

Olga Klopp, Alexandre B. Tsybakov, and Nicolas Verzelen

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Inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models. We consider the problem of statistical estimation of the matrix of connection probabilities based on the observations of the adjacency matrix of the network. Taking the stochastic block model as an approximation, we construct estimators of network connection probabilities—the ordinary block constant least squares estimator, and its restricted version. We show that they satisfy oracle inequalities with respect to the block constant oracle. As a consequence, we derive optimal rates of estimation of the probability matrix. Our results cover the important setting of sparse networks. Another consequence consists in establishing upper bounds on the minimax risks for graphon estimation in the $L_{2}$ norm when the probability matrix is sampled according to a graphon model. These bounds include an additional term accounting for the “agnostic” error induced by the variability of the latent unobserved variables of the graphon model. In this setting, the optimal rates are influenced not only by the bias and variance components as in usual nonparametric problems but also include the third component, which is the agnostic error. The results shed light on the differences between estimation under the empirical loss (the probability matrix estimation) and under the integrated loss (the graphon estimation).

Article information

Ann. Statist., Volume 45, Number 1 (2017), 316-354.

Received: July 2015
Revised: February 2016
First available in Project Euclid: 21 February 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 60C05: Combinatorial probability

Inhomogeneous random graph networks oracle inequality sparse graphon sparsity stochastic block model


Klopp, Olga; Tsybakov, Alexandre B.; Verzelen, Nicolas. Oracle inequalities for network models and sparse graphon estimation. Ann. Statist. 45 (2017), no. 1, 316--354. doi:10.1214/16-AOS1454.

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