Spectral Graph Neural Networks

• Spectral Graph Neural Networks (SGNNs) are graph neural architectures that exploit the Laplacian eigenstructure to perform global, frequency-aware filtering of graph signals.
• They employ mathematical techniques like Chebyshev polynomials, rational filters, and wavelet frames for tasks such as node classification, link prediction, and signal denoising.
• Recent advancements integrate spatio-spectral models, automated design, and LLM-guided adaptations to address scalability, expressivity, and transferability challenges.

Spectral Graph Neural Networks (SGNNs) are a class of graph neural architectures that define convolutional and propagation mechanisms based on the eigenstructure of graph Laplacian operators. In contrast to spatial GNNs, which aggregate neighbor features using local message-passing, SGNNs leverage the graph Fourier basis to implement global, frequency-aware filtering of graph signals. This spectral approach enables sophisticated manipulation of graph-structured data, spanning node classification, link prediction, and signal denoising, and has continually evolved from polynomial filter design to highly expressive, adaptive, and scalable paradigms.

1. Mathematical Foundations and Classical Spectral GNN Models

A core element of SGNNs is the spectral decomposition of the (normalized) graph Laplacian $L = I - D^{-1/2} A D^{-1/2} = U \Lambda U^\top$, where $U$ contains orthonormal eigenvectors and $\Lambda$ is diagonal with Laplacian eigenvalues. The graph Fourier transform of a node-signal $x$ is $\hat{x} = U^\top x$, and the inverse is $x = U \hat{x}$ (Chen, 2020). In this framework, a spectral filter is a function $g(\cdot)$ such that filtering $x$ is expressed as $g(L)x = U\,g(\Lambda)U^\top x$.

The classical SGNN architectures can be summarized as follows:

Model	Filter Parameterization	Computational Complexity
Bruna et al.	Full diagonal $g_\theta(\Lambda)$	$O(n^3)$ (EVD) + $O(n^2)$ per layer
ChebNet	Chebyshev poly. $g_\theta(L) = \sum_k \theta_k T_k(\tilde L)$	$O(K|E|)$ per layer
GCN	First-order approx., 1-hop $g_\theta(L)x \sim \theta[I + D^{-1/2}AD^{-1/2}]x$	$O(|E|)$ per layer
CayleyNet	Rational filters in Laplacian	$O(r|E|) +$ sparse linear solves
Lanczos	Krylov-subspace adaptive filtering	$O(M|E| + M^3)$

Later models such as BernNet, JacobiConv, APPNP, and GPR-GNN introduce further flexibility, including orthogonal polynomial bases, rational and teleportation-based propagation, and learnable high-order filters (Wang et al., 2022, Chen, 2020).

2. Universality, Expressive Power, and Limitations

Recent analyses formalize the expressive power of linear spectral GNNs. If the Laplacian has no repeated eigenvalues and the input features project onto all eigenvectors, then a linear SGNN can approximate any target node-signal via an appropriate polynomial filter—universality is achieved [(Wang et al., 2022), Thm.1].

However, monotonic increases in polynomial order do not always yield strictly more expressive filters in practice, especially under spectral multiplicity. For graphs with simple spectra (distinct eigenvalues), the expressivity of SGNNs exceeds that of 1-WL GNNs (Hordan et al., 5 Jun 2025). Nevertheless, even with simple spectra, certain SGNN paradigms (EPNN) are incomplete: there exist non-isomorphic graph pairs that are always mapped to the same output due to rotational and sign-permutation ambiguities in the eigenbasis. Enhanced models incorporating spectrum-aware rotation-equivariant feature channels (equiEPNN) strictly increase expressivity, eliminating these ambiguities (Hordan et al., 5 Jun 2025).

3. Filter Design: Polynomial, Rational, Wavelet, and State Space Models

While early SGNNs focused on polynomial filters (monomial, Chebyshev, Bernstein, Jacobi bases), recent advances exploit rational functions and adaptive constructions:

• Polynomial Filters: ChebNet, BernNet, JacobiConv, GPR-GNN use polynomial functions of Laplacian eigenvalues for multi-scale, yet localized, filtering. JacobiConv leverages orthogonal Jacobi polynomials and demonstrates that basis selection aligned with the empirical spectrum improves optimization and accuracy (Wang et al., 2022).
• Rational/State-Space Models: GrassNet introduces a structured state-space model (SSM) to process the sorted spectrum as a sequence, enabling the learning of rational-function filters strictly richer than any polynomial family. The bidirectional SSM allows each spectral mode to adapt its response in the context of the entire eigenvalue set, leading to superior accuracy especially on graphs with highly degenerate or concentrated spectra (Zhao et al., 16 Aug 2024).
• Wavelet and Multiscale Filters: Several models (ASWT-SGNN, LGWNN) construct multi-scale spectral wavelet frames, approximating them with Chebyshev polynomials for fast implementation. Lifting-based adaptive wavelets enable localized, attention-parametrized filtering that remains permutation-invariant and highly sparse, while self-supervised loss functions (e.g., contrastive loss) tune the tradeoff between global and local aggregation (Liu et al., 2023, Xu et al., 2021).
• PDE-inspired Filtering: Hyperbolic-PDE GNNs interpret spectral propagation as simulating solutions to hyperbolic wave equations on graphs. The resulting class of filters covers both oscillatory and diffusive regimes and provides a physically grounded parameterization space for SGNN design (2505.23014).

4. Extensions: Spatio-Spectral, Directed and Signed Graphs, and Scaling

Purely spectral GNNs can be rigid, sensitive to basis, and expensive to scale. Key extensions address these challenges:

• Spatio-Spectral GNNs: Combining polynomial message passing (spatial domain) with spectral/global low-/high-pass filtering (spectral domain) yields architectures such as S²GNN. These hybrid models effectively overcome over-squashing, allow global mixing in one step, and attain strictly tighter approximation-theoretic error bounds than spatial-only MPGNNs. They also support free positional encodings via Laplacian eigenvectors, achieving expressivity beyond the 1-WL test (Geisler et al., 29 May 2024).
• Directed and Signed Graphs: Spectral theory for signed or directed graphs employs normalized signed Laplacians or Hermitian magnetic Laplacians. The spectral perspective enables principled filter design for tasks such as node classification and link sign prediction, with global frequency concepts retained for orientation and sign (Singh et al., 2022, Geisler et al., 29 May 2024).
• Scalability: Spectral GNNs are typically bottlenecked by the need for expensive eigendecomposition or high-order Laplacian powers. Laplacian sparsification approaches approximate arbitrary polynomial filters with a single sketch matrix, allowing O(n log n / ε²) time and memory scaling. This makes spectral propagation tractable for graphs with 10⁸+ nodes and enables end-to-end differentiable learning—even for high-dimensional input features—surpassing detached-layer or precompute baselines without accuracy loss (Ding et al., 8 Jan 2025).

5. Automated and Adaptive Spectral Mechanism Discovery

Manual selection of spectral filters remains suboptimal and labor-intensive, particularly across graphs of varying homophily, heterophily, and structural complexity. Several frameworks automate spectral architecture design:

• LLM-driven SGNNs: Recent methods inject global statistical priors (notably, homophily estimates) derived from LLMs into the filter design process, adaptively biasing polynomial filters towards low (homophilic) or high/band-pass (heterophilic) regimes. LLM-guided priors yield consistently improved accuracy across both settings, even in low-label regimes, with negligible compute overhead (Lu et al., 17 Jun 2025).
• AutoSGNN: Combines evolutionary search and LLM code generation to explore the full combinatorial space of feature fitting, Laplacian-based filtering, and aggregation, optimizing for dataset-specific node classification accuracy. The method consistently outperforms both hand-designed and neural architecture search baselines, and automatically rediscovers or invents dataset-optimal spectral mechanisms (Mo et al., 17 Dec 2024).

6. Practical Considerations, Applications, and Limitations

SGNNs achieve state-of-the-art results on node classification, community detection, molecular property prediction, and graph signal denoising (Chen, 2020, Stachenfeld et al., 2020). Empirical evidence shows that:

• On dense/homophilic graphs, simple low-pass filters or low-order polynomial filters are optimal and highly efficient.
• On sparse/heterophilic or spectrally degenerate graphs, high-order, band-pass, or rational filters are necessary; global, adaptive or SSM-based filters alleviate performance degradations.
• In scalability, Laplacian sparsification, partial spectral approximations, and polynomial frameworks with small computational footprints enable large-scale inference and training (Ding et al., 8 Jan 2025, Zhao et al., 16 Aug 2024).

However, open limitations include sensitivity to Laplacian multiplicities and basis-induced ambiguities, difficulties in transferring learned filters across graphs with different spectra, and the cost or instability of full-spectrum and rational computations (Wang et al., 2022, Hordan et al., 5 Jun 2025, Zhao et al., 16 Aug 2024). Ongoing research focuses on cross-graph transfer, basis-free filtering, surrogate training and evaluation, and the incorporation of domain-specific global statistical priors via automation and LLMs (Mo et al., 17 Dec 2024, Lu et al., 17 Jun 2025).

7. Outlook and Future Directions

Current frontiers in SGNN research include integrating hybrid (spatio-spectral), attention, and wavelet-based mechanisms; enhancing transferability and invariance across graph structures; extending to dynamic, directed, or signed graphs; and scaling to billion-edge real-world benchmarks (Geisler et al., 29 May 2024, Singh et al., 2022, Ding et al., 8 Jan 2025). The continued confluence of mathematical advances in spectral graph theory, system identification, deep learning, and automated mechanism discovery is driving the development of highly expressive, principled, and scalable SGNNs for increasingly complex graph domains.