Higher-Order Spectral Clustering for Geometric Graphs

TL;DR

This paper introduces higher-order spectral clustering, a novel method for effectively clustering geometric graphs, demonstrating its theoretical consistency and practical efficiency through numerical experiments.

Contribution

It generalizes classical spectral clustering by using higher-order eigenvalues and proves its weak and strong consistency for Soft Geometric Block Models.

Findings

Proves weak consistency of the method for a broad class of geometric graphs.

Shows strong consistency with a small algorithm adjustment.

Demonstrates effectiveness in numerical experiments on modest-sized graphs.

Abstract

The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak consistency of this algorithm for a wide class of geometric graphs which we call Soft Geometric Block Model. A small adjustment of the algorithm provides strong consistency. We also show that our method is effective in numerical experiments even for graphs of modest size.