Spectral Graph Neural Networks are Incomplete on Graphs with a Simple Spectrum

TL;DR

This paper investigates the limitations of spectral graph neural networks (SGNNs) on graphs with simple spectra, introduces an expressivity hierarchy, and proposes a method to enhance their power, validated through experiments on image and molecular datasets.

Contribution

It introduces an expressivity hierarchy for SGNNs based on eigenvalue multiplicity and proposes a rotation-equivariant method to improve their expressivity on simple spectrum graphs.

Findings

Many SGNNs are incomplete on graphs with distinct eigenvalues.

The proposed method improves SGNNs' expressivity on simple spectrum graphs.

Empirical results confirm theoretical improvements on MNIST Superpixel and ZINC datasets.

Abstract

Spectral features are widely incorporated within Graph Neural Networks (GNNs) to improve their expressive power, or their ability to distinguish among non-isomorphic graphs. One popular example is the usage of graph Laplacian eigenvectors for positional encoding in MPNNs and Graph Transformers. The expressive power of such Spectrally-enhanced GNNs (SGNNs) is usually evaluated via the k-WL graph isomorphism test hierarchy and homomorphism counting. Yet, these frameworks align poorly with the graph spectra, yielding limited insight into SGNNs' expressive power. We leverage a well-studied paradigm of classifying graphs by their largest eigenvalue multiplicity to introduce an expressivity hierarchy for SGNNs. We then prove that many SGNNs are incomplete even on graphs with distinct eigenvalues. To mitigate this deficiency, we adapt rotation equivariant neural networks to the graph spectra setting to propose a method to provably improve SGNNs' expressivity on simple spectrum graphs. We empirically verify our theoretical claims via an image classification experiment on the MNIST Superpixel dataset and eigenvector canonicalization on graphs from ZINC.