Full-Spectrum Graph Neural Network: Expressive and Scalable

2026-05-07

AI Summary

This work addresses the limited expressivity of conventional spectral graph neural networks (GNNs), which are constrained by the 1-dimensional Weisfeiler–Lehman test and thus struggle to capture higher-order structural patterns, particularly on heterophilic graphs. To overcome this limitation, the authors propose Full-Spectrum GNN (FSpecGNN), a novel framework that lifts graph signals into the node-pair domain and introduces bivariate spectral filters to enhance representational capacity. FSpecGNN is the first spectral GNN to achieve universal approximation on node-pair signals, thereby surpassing the expressive limits of classical spectral methods and matching the expressivity of Local 2-GNNs. Through a scalable design employing low-rank approximations and avoiding explicit node-pair computations, FSpecGNN achieves state-of-the-art performance on heterophilic graph benchmarks while enabling efficient learning on large-scale graphs.

Abstract

It is well established that spectral graph neural networks (GNNs) can universally approximate node signals; however, their expressive power remains bounded by the 1-dimensional Weisfeiler-Lehman test, which is mirrored in their lack of universality for higher-order signals. To go beyond this bound, we propose the Full-Spectrum GNN (FSpecGNN), a second-order generalization of classical spectral GNNs. FSpecGNN advances spectral filtering in two perspectives: (1) it lifts the signal from the node domain to the node-pair domain; and (2) it extends the univariate spectral filter over eigenvalues to a bivariate filter over eigenvalue pairs. We show that classical spectral GNNs arise as a diagonal special case of FSpecGNN, and prove that FSpecGNN can be at most as expressive as Local 2-GNN while universally approximating node-pair signals, the latter being particularly beneficial for heterophilic graph learning. Moreover, FSpecGNN admits scalable implementations that avoid explicit node-pair-level computations; combined with a low-rank approximation that reduces full-spectrum convolution to a combination of polynomial spectral filters, it enables learning on large graphs. Empirically, FSpecGNN validates the predicted expressivity and delivers strong performance on heterophilic benchmarks.

Problem

• spectral graph neural networks
• expressive power
• higher-order signals
• heterophilic graphs
• Weisfeiler-Lehman test

Innovation

• Full-Spectrum GNN
• spectral graph neural networks
• higher-order signals
• bivariate spectral filter
• heterophilic graphs