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Graph Neural Networks(GNNs) are powerful methods to analyze the non-Euclidean data. As a dominant type of GNN, Graph Convolutional Networks(GCNs) have wide applications. However, the analysis of the generalization error for GCNs with multilayer is limited. Based on the review of single-layer GCNs, this paper analyzes the generalization error of two-layers GCNs and extends the conclusion to the general GCNs models. Firstly, this paper examines two-layers GCNs and obtains the stability of the GCNs algorithm. And then, based on this algorithmic stability, the generalization stability of multilayer GCNs is obtained. This paper shows that the algorithmic stability of GCNs depends upon the graph filters and its product with node features as well as the training procedure. Furthermore, the generalization error gap of GCNs tends to be enlarged with more layers, which can interpret why GCNs with deeper layers have relatively poorer performance in test datasets.

Fraud detection is essential for e-commerce platforms to maintain a fair business environment. Many existing works propose manually designed methods such as label propagation and dense block mining rules on built user-item graphs to detect fraud behaviours, but they are always heuristic and thus have limited performance. Other learning-based methods can either handle only the fraud detection problem well in the transductive scenario when there is only structural information or require rich content features to obtain a good inductive ability. Considering that content features are not always available in practice and there are usually many fraudulent behaviours that belong to newly emerging users and items, how to learn effective inductive rules with structures only is still underexplored. In this paper, we propose a subgraph-based method named SubGNN for collective fraud detection. In SubGNN, first, we extract the subgraphs around the given edges (user behaviours) to be tested. Then, we remove nodes’ global IDs so that SubGNN is entity-independent. Finally, by learning knowledge reasoning rules on extracted heterogeneous subgraphs using our proposed relational graph isomorphism network (R-GIN), a powerful graph neural network (GNN) model, SubGNN can achieve precise fraud detection. Experiments are conducted on publicly available Amazon and Yelp datasets and a newly collected Taobao dataset. The results clearly show the advantages and prospects of our method. When using SubGNN to detect fraudulent transactions on Taobao, the precision is higher than 0.99 and more than 90 % of fraud samples are recalled.