Spectral Graph Neural Networks are Incomplete on Graphs with a Simple Spectrum
This addresses a fundamental limitation in graph neural networks for researchers and practitioners, offering a provable enhancement to expressive power, though it is incremental in the context of spectral methods.
The paper tackles the limited expressive power of spectrally-enhanced graph neural networks (SGNNs) on graphs with distinct eigenvalues, proving their incompleteness and proposing a rotation equivariant method to improve expressivity, with empirical validation on MNIST Superpixel and ZINC datasets.
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.