Position: Spectral GNNs Are Neither Spectral Nor Superior for Node Classification

TL;DR

Spectral GNNs do not truly leverage spectral properties of graphs for node classification and their empirical success is mainly due to message-passing dynamics, not spectral filtering, challenging their purported advantages.

Contribution

This paper critically examines Spectral GNNs, revealing they do not effectively utilize spectral information and are essentially equivalent to message-passing neural networks in practice.

Findings

Spectral GNNs do not have true Fourier bases for graph signals.

Polynomial approximation in spectral GNNs is not theoretically justified.

Performance issues arise when implementing spectral models consistently with their design.

Abstract

Spectral Graph Neural Networks (Spectral GNNs) for node classification promise frequency-domain filtering on graphs, yet rest on flawed foundations. Recent work shows that graph Laplacian eigenvectors do not in general have the key properties of a true Fourier basis, but leaves the empirical success of Spectral GNNs unexplained. We identify two theoretical glitches: (1) commonly used "graph Fourier bases" are not classical Fourier bases for graph signals; (2) (n-1)-degree polynomials (n = number of nodes) can exactly interpolate any spectral response via a Vandermonde system, so the usual "polynomial approximation" narrative is not theoretically justified. The effectiveness of GCN is commonly attributed to spectral low-pass filtering, yet we prove that low- and high-pass behaviors arise solely from message-passing dynamics rather than Graph Fourier Transform-based spectral formulations. We then analyze two representative directed spectral models, MagNet and HoloNet. Their reported effectiveness is not spectral: it arises from implementation issues that reduce them to powerful MPNNs. When implemented consistently with the claimed spectral algorithms, performance becomes weak. This position paper argues that: for node classification, Spectral GNNs neither meaningfully capture the graph spectrum nor reliably improve performance; competitive results are better explained by their equivalence to MPNNs, sometimes aided by implementations inconsistent with their intended design.