HSMAD: Heterophily-Driven Spectral and Manifold Learning for Graph Anomaly Detection

Graph anomaly detection (GAD) is a fundamental task in graph learning. However, most existing methods rely on the homophily assumption, which posits that connected nodes tend to share the same labels. This assumption often fails in the presence of edge heterophily, leading to degraded performance. We first observe that down-weighting heterophilic edges, relative to the original or randomly weighted graphs, results in a more concentrated spectral energy distribution, thereby facilitating the learning of discriminative spectral embeddings. Moreover, existing methods typically embed graphs in Euclidean spaces, neglecting the importance of heterophily in manifold spaces. Motivated by these observations, we propose HSMAD, a novel framework for GAD. It consists of two key components: the Heterophily-Weighted Spectral Filtering module, which reconstructs the Laplacian using heterophily-based edge weighting for spectral filtering, and the Heterophily-Routed Manifold Update module, which routes neighborhood messages to the appropriate manifold for node feature updates, enabling curvature-adaptive representation learning. These spectral and geometric representations are jointly leveraged for anomaly detection. Extensive experiments on six real-world datasets show that HSMAD achieves state-of-the-art performance across the average F1-Macro, AUROC, AUPRC, and G-Mean. Specifically, the average F1-Macro score improves by 2.66% over the best-performing method.