Multi-Community Spectral Clustering for Geometric Graphs

TL;DR

This paper introduces a spectral clustering method for the soft geometric block model in dense graphs, proving its consistency and analyzing the spectral properties of the adjacency matrix.

Contribution

The paper develops a spectral clustering algorithm for SGBM, proves its weak and strong consistency, and analyzes the adjacency matrix spectrum using novel techniques.

Findings

Algorithm achieves weak and strong consistency.

Eigenvector-based embedding effectively recovers communities.

Spectral analysis provides insights into adjacency matrix behavior.

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.