Variational Inference for Stochastic Block Models from Sampled Data

This paper deals with non-observed dyads during the sampling of a network and consecutive issues in the Stochastic Block Model (SBM) inference. We review sampling designs and recover Missing At Random (MAR) and Not Missing At Random (NMAR) conditions for SBM. We introduce several variants of the variational EM (VEM) algorithm for inferring the SBM under various sampling designs (MAR and NMAR). The sampling design must be taken into account only in the NMAR case. Model selection criteria based on Integrated Classification Likelihood (ICL) are derived for selecting both the number of blocks and the sampling design. We investigate the accuracy and the range of applicability of these algorithms with simulations. We finally explore two real-world networks from ethnology (seed circulation network) and biology (protein-protein interaction network), where the interpretations considerably depends on the sampling designs considered