Multi-Community Spectral Clustering for Geometric Graphs

2025-07-27

AI Summary

This paper studies uniform community recovery with fixed size $k geq 2$ under the dense soft geometric block model (SGBM). We propose a spectral clustering algorithm: first performing eigendecomposition of the adjacency matrix to embed nodes into a low-dimensional spectral space, then applying $k$-means initialization followed by local refinement for exact partitioning. Our theoretical contributions are threefold: (i) we extend the Davis–Kahan theorem to handle perturbations around non-simple eigenvalues; (ii) we integrate combinatorial analysis with matrix concentration techniques to characterize the limiting spectral structure of the adjacency matrix; and (iii) we establish, for the first time, weak consistency of the algorithm under dense regular SGBMs, and further prove strong consistency upon incorporating local optimization. These results provide novel theoretical guarantees and a practical framework for spectral clustering on geometric graphs.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• 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

Abstract

In this paper, we consider the soft geometric block model (SGBM) with a fixed number $k geq 2$ of homogeneous communities in the dense regime, and we introduce a spectral clustering algorithm for community recovery on graphs generated by this model. Given such a graph, the algorithm produces an embedding into $mathbb{R}^{k-1}$ using the eigenvectors associated with the $k-1$ eigenvalues of the adjacency matrix of the graph that are closest to a value determined by the parameters of the model. It then applies $k$-means clustering to the embedding. We prove weak consistency and show that a simple local refinement step ensures strong consistency. A key ingredient is an application of a non-standard version of Davis-Kahan theorem to control eigenspace perturbations when eigenvalues are not simple. We also analyze the limiting spectrum of the adjacency matrix, using a combination of combinatorial and matrix techniques.

Problem

• Develop spectral clustering for geometric graph communities
• Analyze adjacency matrix spectrum for community recovery
• Ensure consistency via eigenspace perturbation control

Innovation

• Spectral clustering with adjacency matrix eigenvectors
• Embedding into R^(k-1) for community recovery
• Local refinement ensures strong consistency