Spatio-Spectral Graph Neural Networks

TL;DR

Spatio-Spectral Graph Neural Networks (S2GNNs) tackle the fundamental limitations of traditional MPGNNs by fusing spatial message passing with spectrally parameterized filters that operate globally. This hybrid approach enables efficient propagation across long-range graph structure, eliminates over-squashing, and yields tighter approximation bounds than purely spatial methods; it also introduces permutation-equivariant spectral networks and stable, expressive positional encodings that surpass 1-WL expressivity. The framework extends naturally to directed graphs via the Magnetic Laplacian and demonstrates strong empirical performance on long-range peptide benchmarks, sequence modeling tasks, and large-scale datasets, while remaining scalable with partial eigendecompositions. The work broadens the GNN design space, offering practical, scalable, and expressive models for graph-structured data with long-range interactions.

Abstract

Spatial Message Passing Graph Neural Networks (MPGNNs) are widely used for learning on graph-structured data. However, key limitations of l-step MPGNNs are that their "receptive field" is typically limited to the l-hop neighborhood of a node and that information exchange between distant nodes is limited by over-squashing. Motivated by these limitations, we propose Spatio-Spectral Graph Neural Networks (S$^2$GNNs) -- a new modeling paradigm for Graph Neural Networks (GNNs) that synergistically combines spatially and spectrally parametrized graph filters. Parameterizing filters partially in the frequency domain enables global yet efficient information propagation. We show that S$^2$GNNs vanquish over-squashing and yield strictly tighter approximation-theoretic error bounds than MPGNNs. Further, rethinking graph convolutions at a fundamental level unlocks new design spaces. For example, S$^2$GNNs allow for free positional encodings that make them strictly more expressive than the 1-Weisfeiler-Lehman (WL) test. Moreover, to obtain general-purpose S$^2$GNNs, we propose spectrally parametrized filters for directed graphs. S$^2$GNNs outperform spatial MPGNNs, graph transformers, and graph rewirings, e.g., on the peptide long-range benchmark tasks, and are competitive with state-of-the-art sequence modeling. On a 40 GB GPU, S$^2$GNNs scale to millions of nodes.

Figures (28)

• Figure 1: S2GNN principle.
• Figure 2: S2GNN framework with adjacency matrix ${\bm{A}}$, node features ${\bm{X}}$, and Laplacian ${\bm{L}}$ (function of ${\bm{A}}$).
• Figure 3: Message-passing interpretation of ${\bm{V}} (\hat{g}_\vartheta(\boldsymbol{\lambda}) \odot [ {\bm{V}}^\top {\bm{X}}])$ (spectral filter): via the Fourier coefficients they may exchange information globally and allow intra- and inter-cluster message passing. Edge width/color denotes the magnitude/sign of ${\bm{V}}$.
• Figure 4: Further hierarchies (lowest eigenspaces).
• Figure 5: Spectral filters do not exhibit over-squashing on "Clique Path" graphs di_giovanni_over-squashing_2023.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 1: Stable ()
• Theorem 5
• Theorem 6
• Theorem 7: Adapted from di_giovanni_over-squashing_2023
• Theorem 8: Adapted from di_giovanni_over-squashing_2023
• Theorem 8