Estimation and Clustering in Popularity Adjusted Stochastic Block Model

Abstract

The paper considers the Popularity Adjusted Block model (PABM) introduced by Sengupta and Chen (2018). 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 modeling 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 the community structure and provide non-asymptotic upper bounds for the estimation and the clustering errors. We use the Sparse Subspace Clustering (SSC) approach to partition the network into communities, the approach that, to the best of our knowledge, has not been used for clustering network data. The theory is supplemented by a simulation study. In addition, we show advantages of the PABM for modeling a butterfly similarity network and a human brain functional network.

Eigenvalues of Stochastic Blockmodel Graphs and Random Graphs with Low-Rank Edge Probability Matrices

“…This construction permits settings where d < K, recalling that d = p + q. Similar constructions show that formulations of degree-corrected stochastic blockmodel graphs (Karrer and Newman, 2011), mixed-membership stochastic blockmodel graphs (Airoldi et al, 2008), and popularity adjusted blockmodels (Noroozi et al, 2019;Sengupta and Chen, 2018) are also special cases of generalized random dot product graphs.…”

Sparse Popularity Adjusted Stochastic Block Model

“…Hence, even if sparsity is not specifically enforced (as it happens in [28] where the penalty depends on n and K only), one still obtains a sparse estimator P with the support ĴK = JK .…”

“…Recently, [28] and [30] studied the Popularity Adjusted Block Model (PABM) which generalizes both the SBM and the DCBM and allows to model the matrix of probabilities in a more flexible way. In order to understand the PABM, consider a rearranged version P (Z, K) of matrix P where its first n 1 rows correspond to nodes from class 1, the next n 2 rows correspond to nodes from class 2 and the last n K rows correspond to nodes from class K. Denote the (k, l)-th block of matrix P (Z, K) by P (k,l) (Z, K).…”