Popularity Adjusted Block Models are Generalized Random Dot Product Graphs
This work provides theoretical insights for network analysis, but it is incremental as it links existing models without introducing a fundamentally new approach.
The paper connects two random graph models, showing that the Popularity Adjusted Block Model (PABM) is a special case of the Generalized Random Dot Product Graph (GRDPG), enabling new algorithms for community detection and parameter estimation with asymptotic guarantees of zero errors as graph size increases.
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.