  We connect two random graph models, the Popularity Adjusted Block Model
(PABM) and the Generalized Random Dot Product Graph (GRDPG), by demonstrating
that the PABM is a special case of the GRDPG in which communities correspond to
mutually orthogonal subspaces of latent vectors. This insight allows us to
construct new algorithms for community detection and parameter estimation for
the PABM, as well as improve an existing algorithm that relies on Sparse
Subspace Clustering. Using established asymptotic properties of Adjacency
Spectral Embedding for the GRDPG, we derive asymptotic properties of these
algorithms. In particular, we demonstrate that the absolute number of community
detection errors tends to zero as the number of graph vertices tends to
infinity. Simulation experiments illustrate these properties.
