Using Laplacian Spectrum as Graph Feature Representation

TL;DR

This paper advocates using the Laplacian spectrum as a simple yet effective graph feature that preserves structural information, invariance, and consistency, serving as a strong baseline for graph analysis tasks.

Contribution

It demonstrates that the Laplacian spectrum satisfies key structural preservation properties and provides bounds related to graph isomorphism divergence, establishing it as a robust graph feature.

Findings

Laplacian spectrum is invariant under graph isomorphism.

Bounds relate spectrum divergence to graph isomorphism divergence.

Spectrum-based representation performs well in classification tasks.

Abstract

Graphs possess exotic features like variable size and absence of natural ordering of the nodes that make them difficult to analyze and compare. To circumvent this problem and learn on graphs, graph feature representation is required. A good graph representation must satisfy the preservation of structural information, with two particular key attributes: consistency under deformation and invariance under isomorphism. While state-of-the-art methods seek such properties with powerful graph neural-networks, we propose to leverage a simple graph feature: the graph Laplacian spectrum (GLS). We first remind and show that GLS satisfies the aforementioned key attributes, using a graph perturbation approach. In particular, we derive bounds for the distance between two GLS that are related to the \textit{divergence to isomorphism}, a standard computationally expensive graph divergence. We finally experiment GLS as graph representation through consistency tests and classification tasks, and show that it is a strong graph feature representation baseline.