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We introduce a semiparametric block model for graphs, where the within- and between-cluster edge probabilities are not constants within the blocks but are described by logistic type models, reminiscent of the 50-year-old Rasch model and the newly introduced α - β models. Our purpose is to give a partition of the vertices of an observed graph so that the induced subgraphs and bipartite graphs obey these models, where their strongly interlaced parameters give multiscale evaluation of the vertices at the same time. In this way, a profoundly heterogeneous version of the stochastic block model is built via mixtures of the above submodels, while the parameters are estimated with a special EM iteration.

Estimating Parameters of a Probabilistic Heterogeneous Block Model via the EM Algorithm