# Popularity Adjusted Block Models are Generalized Random Dot Product Graphs

arXiv: https://arxiv.org/abs/2109.04010 · HTML: https://mlsanity.com/paper/2109.04010/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 9 Sep 2021.

## Builds on (most cited first, 4 of 4)

- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Noisy Sparse Subspace Clustering](https://mlsanity.com/paper/1309.1233/) (2013)
- [Network Enhancement: a general method to denoise weighted biological networks](https://mlsanity.com/paper/1805.03327/) (2018)
- [Producing a Unified Graph Representation from Multiple Social Network Views](https://mlsanity.com/paper/1301.5809/) (2013)

## Built on by (most cited first, 4 of 4)

- [Signed Diverse Multiplex Networks: Clustering and Inference](https://mlsanity.com/paper/2402.10242/) (2024)
- [Strongly Consistent Community Detection in Popularity Adjusted Block Models](https://mlsanity.com/paper/2506.07224/) (2025)
- [Optimal Graph Clustering without Edge Density Signals](https://mlsanity.com/paper/2510.21669/) (2025)
- [Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embedding](https://mlsanity.com/paper/2603.11965/) (2026)

## Abstract (arXiv)

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.