Popularity Adjusted Block Models are Generalized Random Dot Product Graphs

TL;DR

This paper reveals that the Popularity Adjusted Block Model is a special case of the Generalized Random Dot Product Graph, enabling new algorithms for community detection with asymptotic accuracy improvements.

Contribution

It establishes a theoretical connection between PABM and GRDPG, leading to novel algorithms and improved community detection methods.

Findings

Community detection errors tend to zero as graph size increases.

New algorithms outperform existing methods in simulations.

Theoretical link enables asymptotic analysis of algorithms.

Abstract

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.