Optimal Graph Clustering without Edge Density Signals

TL;DR

This paper introduces the Popularity-Adjusted Block Model (PABM), a new graph clustering framework that captures degree heterogeneity and enables cluster recovery even when traditional signals vanish, with implications for spectral clustering methods.

Contribution

The paper characterizes the optimal error rate for clustering under PABM and demonstrates its advantages over SBM and DCBM, especially in low signal scenarios.

Findings

Cluster recovery is possible in PABM without edge-density signals.

Spectral clustering with $k^2$ eigenvectors outperforms traditional methods.

PABM's adjacency matrix rank ranges between $k$ and $k^2$, affecting spectral embedding.

Abstract

This paper establishes the theoretical limits of graph clustering under the Popularity-Adjusted Block Model (PABM), addressing limitations of existing models. In contrast to the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra- and inter-cluster connections. Our main contribution is the characterization of the optimal error rate for clustering under PABM, which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recovery remains possible in PABM even when traditional edge-density signals vanish, provided intra- and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability independently of global edge densities. Finally, because PABM exhibits a richer structure, its expected adjacency matrix has rank between $k$ and $k^2$, where $k$ is the number of clusters. As a result, spectral embeddings based on the top $k$ eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating $k^2$ eigenvectors outperform traditional spectral approaches.