Rayleigh Quotient Graph Neural Networks for Graph-level Anomaly Detection

Graph-level anomaly detection has gained significant attention as it finds many applications in various domains, such as cancer diagnosis and enzyme prediction. However, existing methods fail to capture the underlying properties of graph anomalies, resulting in unexplainable framework design and unsatisfying performance. In this paper, we take a step back and 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 for graph-level anomaly detection, providing a new perspective on exploring the inherent spectral features of anomalous graphs. Specifically, we introduce a novel framework that consists of 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.