# Rayleigh Quotient Graph Neural Networks for Graph-level Anomaly Detection

arXiv: https://arxiv.org/abs/2310.02861 · HTML: https://mlsanity.com/paper/2310.02861/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 41 references that are in the map.
- First posted on arXiv 4 Oct 2023.

## Builds on (most cited first, 26 of 26)

- [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)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Class-Balanced Loss Based on Effective Number of Samples](https://mlsanity.com/paper/1901.05555/) (2019)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Rethinking Graph Neural Networks for Anomaly Detection](https://mlsanity.com/paper/2205.15508/) (2022)
- [Understanding Attention and Generalization in Graph Neural Networks](https://mlsanity.com/paper/1905.02850/) (2019)
- [Revisiting Heterophily For Graph Neural Networks](https://mlsanity.com/paper/2210.07606/) (2022)
- [Rumor Detection on Social Media with Bi-Directional Graph Convolutional Networks](https://mlsanity.com/paper/2001.06362/) (2020)
- [G-Mixup: Graph Data Augmentation for Graph Classification](https://mlsanity.com/paper/2202.07179/) (2022)
- [Accurate Learning of Graph Representations with Graph Multiset Pooling](https://mlsanity.com/paper/2102.11533/) (2021)
- [Causal Attention for Interpretable and Generalizable Graph Classification](https://mlsanity.com/paper/2112.15089/) (2021)
- [Specformer: Spectral Graph Neural Networks Meet Transformers](https://mlsanity.com/paper/2303.01028/) (2023)
- [Deep Graph-level Anomaly Detection by Glocal Knowledge Distillation](https://mlsanity.com/paper/2112.10063/) (2021)
- [Structural Entropy Guided Graph Hierarchical Pooling](https://mlsanity.com/paper/2206.13510/) (2022)
- [Spam Review Detection with Graph Convolutional Networks](https://mlsanity.com/paper/1908.10679/) (2019)
- [On Using Classification Datasets to Evaluate Graph-Level Outlier Detection: Peculiar Observations and New Insights](https://mlsanity.com/paper/2012.12931/) (2020)
- [Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities](https://mlsanity.com/paper/2204.07321/) (2022)
- [Raising the Bar in Graph-level Anomaly Detection](https://mlsanity.com/paper/2205.13845/) (2022)
- [High-Order Pooling for Graph Neural Networks with Tensor Decomposition](https://mlsanity.com/paper/2205.11691/) (2022)
- [FAITH: Few-Shot Graph Classification with Hierarchical Task Graphs](https://mlsanity.com/paper/2205.02435/) (2022)
- [Label-invariant Augmentation for Semi-Supervised Graph Classification](https://mlsanity.com/paper/2205.09802/) (2022)
- [Graph-level Anomaly Detection via Hierarchical Memory Networks](https://mlsanity.com/paper/2307.00755/) (2023)
- [Total Variation Graph Neural Networks](https://mlsanity.com/paper/2211.06218/) (2022)

## Built on by (most cited first, 19 of 19)

- [Deep Graph Anomaly Detection: A Survey and New Perspectives](https://mlsanity.com/paper/2409.09957/) (2024)
- [Unifying Unsupervised Graph-Level Anomaly Detection and Out-of-Distribution Detection: A Benchmark](https://mlsanity.com/paper/2406.15523/) (2024)
- [UniGAD: Unifying Multi-level Graph Anomaly Detection](https://mlsanity.com/paper/2411.06427/) (2024)
- [SmoothGNN: Smoothing-aware GNN for Unsupervised Node Anomaly Detection](https://mlsanity.com/paper/2405.17525/) (2024)
- [DiffGAD: A Diffusion-based Unsupervised Graph Anomaly Detector](https://mlsanity.com/paper/2410.06549/) (2024)
- [Decoupled Graph Energy-based Model for Node Out-of-Distribution Detection on Heterophilic Graphs](https://mlsanity.com/paper/2502.17912/) (2025)
- [TINED: GNNs-to-MLPs by Teacher Injection and Dirichlet Energy Distillation](https://mlsanity.com/paper/2412.11180/) (2024)
- [Rumor Detection on Social Media with Reinforcement Learning-based Key Propagation Graph Generator](https://mlsanity.com/paper/2405.13094/) (2024)
- [Rethinking Cancer Gene Identification through Graph Anomaly Analysis](https://mlsanity.com/paper/2412.17240/) (2024)
- [Chi-Square Wavelet Graph Neural Networks for Heterogeneous Graph Anomaly Detection](https://mlsanity.com/paper/2505.18934/) (2025)
- [When Personalization Tricks Detectors: The Feature-Inversion Trap in Machine-Generated Text Detection](https://mlsanity.com/paper/2510.12476/) (2025)
- [GLADformer: A Mixed Perspective for Graph-level Anomaly Detection](https://mlsanity.com/paper/2406.00734/) (2024)
- [GLADMamba: Unsupervised Graph-Level Anomaly Detection Powered by Selective State Space Model](https://mlsanity.com/paper/2503.17903/) (2025)
- [Towards Anomaly-Aware Pre-Training and Fine-Tuning for Graph Anomaly Detection](https://mlsanity.com/paper/2504.14250/) (2025)
- [JANUS: A Dual-Constraint Generative Framework for Stealthy Node Injection Attacks](https://mlsanity.com/paper/2509.13266/) (2025)
- [FracAug: Fractional Augmentation boost Graph-level Anomaly Detection under Limited Supervision](https://mlsanity.com/paper/2509.20978/) (2025)
- [Smoothness Errors in Dynamics Models and How to Avoid Them](https://mlsanity.com/paper/2602.05352/) (2026)
- [Interpretable Graph-Level Anomaly Detection via Contrast with Normal Prototypes](https://mlsanity.com/paper/2602.10708/) (2026)
- [Cone Rayleigh Levels: Finite Perturbations and Certified Control](https://mlsanity.com/paper/2608.27122/) (2026)

## Abstract (arXiv)

Graph-level anomaly detection has gained significant attention as it finds applications in various domains, such as cancer diagnosis and enzyme prediction. However, existing methods fail to capture the spectral properties of graph anomalies, resulting in unexplainable framework design and unsatisfying performance. In this paper, we re-investigate the spectral differences between anomalous and normal graphs. Our main observation shows a significant disparity in the accumulated spectral energy between these two classes. Moreover, we prove that the accumulated spectral energy of the graph signal can be represented by its Rayleigh Quotient, indicating that the Rayleigh Quotient is a driving factor behind the anomalous properties of graphs. Motivated by this, we propose Rayleigh Quotient Graph Neural Network (RQGNN), the first spectral GNN that explores the inherent spectral features of anomalous graphs for graph-level anomaly detection. Specifically, we introduce a novel framework with two components: the Rayleigh Quotient learning component (RQL) and Chebyshev Wavelet GNN with RQ-pooling (CWGNN-RQ). RQL explicitly captures the Rayleigh Quotient of graphs and CWGNN-RQ implicitly explores the spectral space of graphs. Extensive experiments on 10 real-world datasets show that RQGNN outperforms the best rival by 6.74% in Macro-F1 score and 1.44% in AUC, demonstrating the effectiveness of our framework. Our code is available at https://github.com/xydong127/RQGNN.