Spectral graph clustering with inhomogeneous latent geometry

2026-08-11

AI Summary

This work addresses the challenge that conventional spectral clustering struggles to accurately recover communities under non-uniform latent geometric structures, as its dominant eigenvectors are often distorted by geometric interference. Building upon a block-wise latent space model, the authors analyze the spectrum of the adjacency matrix and its associated limiting integral operator, revealing that deeper eigenvectors—beyond the leading ones—encode meaningful community structure. Leveraging this insight, they propose DBSPEC, a density-based spectral clustering algorithm that does not rely on the uniform torus assumption and requires only coarse localization of informative eigenvalues. The method exhibits robustness in settings with poor eigenvalue separation. Theoretical predictions align closely with empirical observations, and DBSPEC demonstrates substantially improved community recovery performance in complex geometric settings.

Abstract

We study spectral clustering in the presence of a confounding latent geometry. The leading eigenvectors may then be dominated by the latent geometry rather than by the communities. Nevertheless, we show in a block latent-space model that communities can be recovered from eigenvectors deeper in the spectrum. We analyze the spectral properties of the adjacency matrix through a limiting integral operator and use its structure to develop DBSPEC, a density-based spectral clustering algorithm that requires only approximate localization of the informative eigenvalue and is robust to poor eigenvalue separation. Crucially, this approach handles general latent geometries, overcoming restrictions to homogeneous toroidal models in prior works. Our theoretical predictions for the location of the informative eigenvalue notably align with observations in real-world experiments.

Problem

• spectral clustering
• latent geometry
• community detection
• inhomogeneous structure
• eigenvector localization

Innovation

• spectral clustering
• latent geometry
• inhomogeneous
• density-based
• eigenvector localization