[Paper Review] Rayleigh Quotient Graph Neural Networks for Graph-level Anomaly Detection

Xiangyu Dong, Xingyi Zhang+1 more|arXiv (Cornell University)|Oct 4, 2023

Anomaly Detection Techniques and ApplicationsComputer Science3 citations

TL;DR

This paper proposes RQGNN, a novel spectral graph neural network for graph-level anomaly detection that leverages the Rayleigh Quotient to capture intrinsic spectral differences between anomalous and normal graphs. By combining explicit Rayleigh Quotient learning (RQL) and a Chebyshev Wavelet GNN with RQ-pooling (CWGNN-RQ), RQGNN achieves state-of-the-art performance, outperforming the best baseline by 6.74% in Macro-F1 and 1.44% in AUC across 10 real-world datasets.

ABSTRACT

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.

Motivation & Objective

• To address the lack of spectral analysis in existing graph-level anomaly detection methods, which often rely on spatial GNNs without capturing underlying spectral properties of anomalous graphs.
• To investigate whether spectral energy differences, particularly through the Rayleigh Quotient, can serve as a discriminative signal for identifying anomalous graphs.
• To develop a new GNN framework that explicitly encodes the Rayleigh Quotient and implicitly learns spectral representations to improve detection performance.
• To overcome limitations of traditional pooling methods in graph-level anomaly detection by introducing RQ-pooling, a spectral-aware pooling mechanism.
• To mitigate data imbalance in graph-level anomaly detection using a class-balanced focal loss during training.

Proposed method

• Proposes the Rayleigh Quotient Learning (RQL) component to explicitly compute and embed the Rayleigh Quotient of each graph as a learnable representation.
• Introduces Chebyshev Wavelet GNN with RQ-pooling (CWGNN-RQ), which uses Chebyshev polynomial approximations to model graph signals in the spectral domain while incorporating Rayleigh Quotient as node weights.
• Employs RQ-pooling as a novel pooling mechanism that assigns attention weights based on the Rayleigh Quotient, enabling spectral-aware graph-level representation learning.
• Combines the explicit RQL and implicit CWGNN-RQ representations via concatenation to form the final graph embedding.
• Utilizes a class-balanced focal loss to address long-tail data distribution in graph-level anomaly detection, improving model robustness on minority anomalous classes.
• Theoretical analysis proves that accumulated spectral energy in graphs is equivalent to the Rayleigh Quotient, justifying its use as a core signal for anomaly detection.

Experimental results

Research questions

• RQ1 Can the Rayleigh Quotient serve as a discriminative spectral feature to distinguish anomalous graphs from normal ones?
• RQ2 How do the spectral energy distributions of anomalous and normal graphs differ, and can these differences be quantified and leveraged for detection?
• RQ3 Does incorporating the Rayleigh Quotient explicitly via RQL and implicitly via RQ-pooling improve graph-level anomaly detection performance compared to spatial GNNs?
• RQ4 To what extent does RQ-pooling outperform standard pooling functions like average or sum pooling in capturing spectral anomalies?
• RQ5 Can the Rayleigh Quotient distribution in test sets be used as a proxy for model generalization and detection reliability?

Key findings

• The Rayleigh Quotient distribution of anomalous graphs is statistically distinct from that of normal graphs, with consistent patterns observed across multiple datasets and sample sizes.
• RQGNN achieves a 6.74% average improvement in Macro-F1 score and a 1.44% average improvement in AUC over the best baseline model across 10 real-world datasets.
• Abllation studies confirm that both the RQL and RQ-pooling components are essential: removing either reduces performance by 3.92% in Macro-F1 and 1.76% in AUC on average.
• The RQ-pooling mechanism significantly outperforms average pooling, demonstrating the importance of spectral-aware pooling in anomaly detection.
• Case studies show that correctly classified graphs in the test set maintain Rayleigh Quotient distributions similar to the training set, confirming that RQGNN learns the intrinsic spectral trends of anomalies.
• Hyperparameter analysis reveals that a hidden dimension of 64, width of 4, and depth of 6 yield optimal performance across multiple datasets, ensuring stable and high-quality results.