Revisiting Degree-Corrected Spectral Clustering: a Condition-Free Spectral Analysis and Extension

2026-07-23

AI Summary

Traditional degree-corrected spectral clustering (DCSC) relies on assumptions tied to specific random graph models, limiting its theoretical generality and practical applicability. This work proposes a model-agnostic analytical framework that dispenses with probabilistic graph assumptions and introduces ASCENT—a node-level adaptive correction mechanism inspired by graph neural networks. For the first time, we establish error bounds for DCSC without invoking random graph hypotheses, elucidating how degree heterogeneity and cluster structure strength jointly influence clustering performance. We theoretically demonstrate that ASCENT achieves superior clustering accuracy compared to conventional DCSC before over-smoothing occurs, while naturally degenerating to the standard DCSC as a special case in the over-smoothed regime, thereby significantly enhancing overall clustering precision.

Abstract

Spectral clustering is a representative graph clustering technique with strong interpretability and theoretical guarantees. Degree-corrected spectral clustering (DCSC) has emerged as the state-of-the-art for this technique. While prior studies have provided impressive theoretical insights for DCSC, their analyses typically depend on specific probabilistic frameworks (e.g., stochastic block models) and conditions. In this study, we explore an alternative condition-free analysis for the clustering quality of DCSC from a pure spectral view, without any random graph models. It gives bounds for the number of mis-clustered nodes w.r.t. the optimal partition of conductance minimization while involving quantities that indicate impacts of (\romannumeral1) degree heterogeneity and (\romannumeral2) weakness of clustering structures to the clustering quality. Inspired by graph neural networks (GNNs) and their over-smoothing effect, we propose ASCENT (Adaptive Spectral ClustEring with Node-wise correcTion), a simple yet effective extension of DCSC. Different from most DCSC methods with a constant degree correction, ASCENT follows a node-wise correction scheme. It can assign different corrections for nodes via a GNN mean aggregator. We demonstrate that (\romannumeral1) ASCENT reduces to conventional DCSC methods when encountering over-smoothing; (\romannumeral2) some early stages before over-smoothing can potentially result in better clustering quality.

Problem

• degree-corrected spectral clustering
• condition-free analysis
• degree heterogeneity
• clustering quality
• spectral clustering

Innovation

• degree-corrected spectral clustering
• condition-free spectral analysis
• node-wise correction
• over-smoothing
• graph neural networks