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Abstract The paper considers the Popularity Adjusted Block model (PABM) introduced by Sengupta and Chen (Journal of the Royal Statistical Society Series B, 2018, 80, 365–386). We argue that the main appeal of the PABM is the flexibility of the spectral properties of the graph which makes the PABM an attractive choice for modelling networks that appear in biological sciences. We expand the theory of PABM to the case of an arbitrary number of communities which possibly grows with a number of nodes in the network and is not assumed to be known. We produce estimators of the probability matrix and of the community structure and, in addition, provide non-asymptotic upper bounds for the estimation and the clustering errors. We use the Sparse Subspace Clustering (SSC) approach for partitioning the network into communities, the approach that, to the best of our knowledge, has not been used for the clustering network data. The theory is supplemented by a simulation study. In addition, we show advantages of the PABM for modelling a butterfly similarity network and a human brain functional network. 

The paper considers a Mixture Multilayer Stochastic Block Model (MMLSBM), where layers can be partitioned into groups of similar networks, and networks in each group are equipped with a distinct Stochastic Block Model. The goal is to partition the multilayer network into clusters of similar layers, and to identify communities in those layers. Jing et al. (2020) introduced the MMLSBM and developed a clustering methodology, TWIST, based on regularized tensor decomposition. The present paper proposes a different technique, an alternating minimization algorithm (ALMA), that aims at simultaneous recovery of the layer partition, together with estimation of the matrices of connection probabilities of the distinct layers. Compared to TWIST, ALMA achieves higher accuracy, both theoretically and numerically. 

The paper considers a Mixture Multilayer Stochastic Block Model (MMLSBM), where layers can be partitioned into groups of similar networks, and networks in each group are equipped with a distinct Stochastic Block Model. The goal is to partition the multilayer network into clusters of similar layers, and to identify communities in those layers. Jing et al. (2020) introduced the MMLSBM and developed a clustering methodology, TWIST, based on regularized tensor decomposition. The present paper proposes a different technique, an alternating minimization algorithm (ALMA), that aims at simultaneous recovery of the layer partition, together with estimation of the matrices of connection probabilities of the distinct layers. Compared to TWIST, ALMA achieves higher accuracy, both theoretically and numerically.