Journal articles
Year
2017
Abstract
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 L2 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).
KLOPP, O., TSYBAKOV, A. et VERZELEN, N. (2017). Oracle inequalities for network models and sparse graphon estimation. Annals of Statistics, 45(1), pp. 316-354.