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Keywords: Community detection, popularity adjusted block model, stochastic block model, spectral clustering, sparse subspace clustering, adjacency spectral embedding, generalized random dot product graph
We connect two random graph models, the Popularity Adjusted Block Model (PABM) and the Generalized Random Dot Product Graph (GRDPG), demonstrating that a PABM is a GRDPG in which communities correspond to certain mutually orthogonal subspaces of latent positions. This insight leads to the construction of improved algorithms for community detection and parameter estimation with PABM. Using established asymptotic properties of Adjacency Spectral Embedding (ASE) for GRDPG, we derive asymptotic properties of these algorithms, including algorithms that rely on Sparse Subspace Clustering (SSC). We illustrate these properties via simulation.