This paper considers an important Graph Anomaly Detection (GAD) task, namely open-set GAD, which aims to detect anomalous nodes using a small number of labelled training normal and anomaly nodes (known as seen anomalies) that cannot illustrate all possible inference-time abnormalities. The availability of that labelled data provides crucial prior knowledge about abnormalities for GAD models, enabling substantially reduced detection errors. However, current methods tend to over-emphasise fitting the seen anomalies, leading to a weak generalisation ability to detect unseen anomalies, i.e., those that are not illustrated by the labelled anomaly nodes. Further, they were introduced to handle Euclidean data, failing to effectively capture important non-Euclidean features for GAD. In this work, we propose a novel open-set GAD approach, namely Normal Structure Regularisation (NSReg), to achieve generalised detection ability to unseen anomalies, while maintaining its effectiveness on detecting seen anomalies. The key idea in NSReg is to introduce a regularisation term that enforces the learning of compact, semantically-rich representations of normal nodes based on their structural relations to other nodes. When being optimised with supervised anomaly detection losses, the regularisation term helps incorporate strong normality into the modelling, empowering the joint learning of both seen abnormality and normality of the nodes, and thus, it effectively avoids the over emphasis on solely fitting the seen anomalies during training. Extensive empirical results on six real-world datasets demonstrate the superiority of our proposed NSReg for open-set GAD.

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