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December 10, 2021

Joint Vanderbilt/MTSU Graph Theory and Combinatorics Seminar-Estimation and Clustering in Network Models

We study the Popularity Adjusted Block model (PABM) that generalizes popular graph generative models such as the Stochastic Block Model and the Degree Corrected Block Model. We argue that the main appeal of the PABM is the flexibility of the graph’s spectral properties, which makes the PABM an attractive choice for modeling networks that appear in biological sciences. We expand the theory of PABM to the case of an arbitrary number of communities that possibly grow with a number of nodes in the network and is not assumed to be known. We produce the estimators of the probability matrix and the community structure and provide non-asymptotic upper bounds for the estimation and the clustering errors. Since the majority of real-life networks are sparse, we study a sparse stochastic network-enabled with a block structure. The popular models SBM and the DCBM address sparsity by placing an upper bound on the maximum probability of connections between any pair of nodes. As a result, sparsity describes only the behavior of the network as a whole, without distinguishing between the block-dependent sparsity patterns. To the best of our knowledge, the recently introduced PABM is the only block model that allows to introduce a structural sparsity where some probabilities of connections are identically equal to zero while the rest of them remain above a certain threshold. The latter presents a more nuanced view of the network. In this presentation, I will talk about the theoretical progress made and present the simulation study and the real data examples.

This will be the last seminar of the calendar year.