# A Survey on Spectral Graph Neural Networks

arXiv: https://arxiv.org/abs/2302.05631 · HTML: https://mlsanity.com/paper/2302.05631/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 63 references that are in the map.
- First posted on arXiv 11 Feb 2023.

## Builds on (most cited first, 30 of 51)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [GNNExplainer: Generating Explanations for Graph Neural Networks](https://mlsanity.com/paper/1903.03894/) (2019)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Rethinking Graph Transformers with Spectral Attention](https://mlsanity.com/paper/2106.03893/) (2021)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Adversarial Attacks on Neural Networks for Graph Data](https://mlsanity.com/paper/1805.07984/) (2018)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [Spectral Temporal Graph Neural Network for Multivariate Time-series Forecasting](https://mlsanity.com/paper/2103.07719/) (2021)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [Network Embedding as Matrix Factorization: Unifying DeepWalk, LINE, PTE, and node2vec](https://mlsanity.com/paper/1710.02971/) (2017)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Graph Neural Networks with convolutional ARMA filters](https://mlsanity.com/paper/1901.01343/) (2019)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [How Powerful are Spectral Graph Neural Networks](https://mlsanity.com/paper/2205.11172/) (2022)
- [Graph Wavelet Neural Network](https://mlsanity.com/paper/1904.07785/) (2019)

## Built on by (most cited first, 26 of 26)

- [Generalized Graph Prompt: Toward a Unification of Pre-Training and Downstream Tasks on Graphs](https://mlsanity.com/paper/2311.15317/) (2023)
- [Graph Transformers: A Survey](https://mlsanity.com/paper/2407.09777/) (2024)
- [How Universal Polynomial Bases Enhance Spectral Graph Neural Networks: Heterophily, Over-smoothing, and Over-squashing](https://mlsanity.com/paper/2405.12474/) (2024)
- [Graph Condensation: A Survey](https://mlsanity.com/paper/2401.11720/) (2024)
- [Graph Neural Network-based EEG Classification: A Survey](https://mlsanity.com/paper/2310.02152/) (2023)
- [Spectral Invariant Learning for Dynamic Graphs under Distribution Shifts](https://mlsanity.com/paper/2403.05026/) (2024)
- [Spatio-Spectral Graph Neural Networks](https://mlsanity.com/paper/2405.19121/) (2024)
- [FedSSP: Federated Graph Learning with Spectral Knowledge and Personalized Preference](https://mlsanity.com/paper/2410.20105/) (2024)
- [Exploring Causal Learning through Graph Neural Networks: An In-depth Review](https://mlsanity.com/paper/2311.14994/) (2023)
- [Beyond Sequence: Impact of Geometric Context for RNA Property Prediction](https://mlsanity.com/paper/2410.11933/) (2024)
- [Shape-aware Graph Spectral Learning](https://mlsanity.com/paper/2310.10064/) (2023)
- [Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains](https://mlsanity.com/paper/2410.04655/) (2024)
- [ERGNN: Spectral Graph Neural Network With Explicitly-Optimized Rational Graph Filters](https://mlsanity.com/paper/2412.19106/) (2024)
- [Beyond Grid Data: Exploring Graph Neural Networks for Earth Observation](https://mlsanity.com/paper/2411.03223/) (2024)
- [Graph Neural Networks Are More Than Filters: Revisiting and Benchmarking from A Spectral Perspective](https://mlsanity.com/paper/2412.07188/) (2024)
- [Graph Persistence goes Spectral](https://mlsanity.com/paper/2506.06571/) (2025)
- [Learnable quantum spectral filters for hybrid graph neural networks](https://mlsanity.com/paper/2507.05640/) (2025)
- [Coefficient Decomposition for Spectral Graph Convolution](https://mlsanity.com/paper/2405.03296/) (2024)
- [YOSO: You-Only-Sample-Once via Compressed Sensing for Graph Neural Network Training](https://mlsanity.com/paper/2411.05693/) (2024)
- [Rethinking Link Prediction for Directed Graphs](https://mlsanity.com/paper/2502.05724/) (2025)
- [Interpretable graph-based models on multimodal biomedical data integration: A technical review and benchmarking](https://mlsanity.com/paper/2505.01696/) (2025)
- [Spectral Manifold Harmonization for Graph Imbalanced Regression](https://mlsanity.com/paper/2507.01132/) (2025)
- [A Graph Laplacian Eigenvector-based Pre-training Method for Graph Neural Networks](https://mlsanity.com/paper/2509.02803/) (2025)
- [Learning from Heterophilic Graphs: A Spectral Theory Perspective on the Impact of Self-Loops and Parallel Edges](https://mlsanity.com/paper/2509.13139/) (2025)
- [Grassmanian Interpolation of Low-Pass Graph Filters: Theory and Applications](https://mlsanity.com/paper/2510.23235/) (2025)
- [Multi-Head Spectral-Adaptive Graph Anomaly Detection](https://mlsanity.com/paper/2512.22291/) (2025)

## Abstract (arXiv)

Graph neural networks (GNNs) have attracted considerable attention from the research community. It is well established that GNNs are usually roughly divided into spatial and spectral methods. Despite that spectral GNNs play an important role in both graph signal processing and graph representation learning, existing studies are biased toward spatial approaches, and there is no comprehensive review on spectral GNNs so far. In this paper, we summarize the recent development of spectral GNNs, including model, theory, and application. Specifically, we first discuss the connection between spatial GNNs and spectral GNNs, which shows that spectral GNNs can capture global information and have better expressiveness and interpretability. Next, we categorize existing spectral GNNs according to the spectrum information they use, ie, eigenvalues or eigenvectors. In addition, we review major theoretical results and applications of spectral GNNs, followed by a quantitative experiment to benchmark some popular spectral GNNs. Finally, we conclude the paper with some future directions.