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Last edited on
Oct 18, 2021 by JJ
.
Thesis:
Master in Mathematics

Author:
Konstantin Klumpp

Title:
Minimax estimation in multilayer stochastic block models

Supervisor:
Jan JOHANNES

Abstract:
This master thesis treats the estimation of the array of connection probabilities in a model for inhomogeneous random multiplex networks. We propose generalizations of several known results regarding the analogous estimation problem in the inhomoge- neous random graph model. Firstly, the upper bounds on the mean squared errors of two least squares estimators from the stochastic block model, established by Gao et al. (2016) and Klopp et al. (2017), are transferred to the multilayer setting. For this purpose, the single-layer stochastic block model is replaced by two multilayer generalizations, the strata multilayer stochastic block model and what we call the inhomogeneous strata multilayer stochastic block model. Our upper bounds are for mulated as oracle inequalities to allow the consideration of true parameters without block structure. Secondly, lower bounds on the minimax risk in both of the considered multilayer stochastic block models are established with constructions based on those in Gao et al. (2015). These show that the least squares estimators are minimax up to a multiplicative constant in both models. In all the mentioned results, we include parameters which determine the sparsity of the observed networks and the likelihood for missing observations. Furthermore, to facilitate our proofs, we introduce a general block model which not only encompasses both of the considered multilayer stochastic block models, but also other models involving block structure, which may prove useful in future research.

Reference:
C. Gao, Y. Lu and H.H. Zhou (2015). Rate-optimal graphon estimation. The Annals of Statistics, 43(6):2624–2652.
O. Klopp, A.B. Tsybakov and N. Verzelen (2017). Oracle inequalities for network models and sparse graphon estimation. The Annals of Statistics, 45(1):316–354.