An Improved and Generalised Analysis for Spectral Clustering

2025-11-28

AI Summary

Spectral clustering lacks rigorous theoretical analysis for graphs with hierarchical or directed structures. Method: We propose a general performance criterion based on spectral gaps: accurate recovery of multiscale and directionally coherent clusters is guaranteed when the smallest eigenvalues of a Hermitian matrix representation form well-separated groups from the rest of the spectrum. Our approach transcends traditional Laplacian-based frameworks by extending spectral clustering theory to arbitrary Hermitian matrix representations—including Hermitianized formulations for directed graphs—and integrates symmetric graph representations with spectral graph theory via eigenvalue decomposition and spectral gap analysis. Contribution/Results: The resulting theory yields verifiable, interpretable guarantees. Experiments demonstrate that it accurately predicts clustering performance on synthetic benchmarks and real-world ecological networks (e.g., trophic level inference), significantly enhancing the interpretability and applicability of spectral clustering on complex-structured graphs.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

We revisit the theoretical performances of Spectral Clustering, a classical algorithm for graph partitioning that relies on the eigenvectors of a matrix representation of the graph. Informally, we show that Spectral Clustering works well as long as the smallest eigenvalues appear in groups well separated from the rest of the matrix representation's spectrum. This arises, for example, whenever there exists a hierarchy of clusters at different scales, a regime not captured by previous analyses. Our results are very general and can be applied beyond the traditional graph Laplacian. In particular, we study Hermitian representations of digraphs and show Spectral Clustering can recover partitions where edges between clusters are oriented mostly in the same direction. This has applications in, for example, the analysis of trophic levels in ecological networks. We demonstrate that our results accurately predict the performances of Spectral Clustering on synthetic and real-world data sets.

Problem

• Analyzing theoretical performance of spectral clustering algorithm
• Extending analysis to hierarchical clusters and digraph representations
• Validating improved spectral clustering on synthetic and real datasets

Innovation

• Generalised spectral clustering for hierarchical clusters
• Extended analysis to Hermitian digraph representations
• Validated on synthetic and real-world datasets