Stochastic blockmodels provide a convenient representation of relations between communities of nodes in a network. However, they imply a notion of stochastic equivalence that is often unrealistic for real networks, and they comprise large number of parameters that can make them hardly interpretable.

We discuss two extensions of stochastic blockmodels, and a recently proposed variable selection approach based on penalized inference, which allows to infer a sparse reduced graph summarizing relations between communities.

We compare this approach with maximum likelihood estimation on two datasets on face-to-face interactions in a French primary school and on bill cosponsorships in the Italian Parliament.