RSCF-PM: Relation-Specific Curvature Fields on Product Manifolds for Fraud Detection in Multi-Relational Social Networks

Graph-based fraud detection in multi-relational social networks must capture heterogeneous relation semantics and diverse fraud patterns while preserving geometric consistency and remaining scalable. Existing methods often either force all relations into a shared Euclidean or single-curvature space, or fuse relation-wise embeddings after mapping them to tangent coordinates, which weakens curvature-dependent metric information. We propose Relation-Specific Curvature Fields on Product Manifolds (RSCF-PM), a geometry-consistent framework that learns relation-specific curvature and represents each node as a tuple on a Riemannian product manifold. Each relation is encoded in its own hyperbolic space, and cross-relation fusion is performed directly through the product metric rather than Euclidean concatenation. On top of this representation, we introduce a multi-prototype classifier to model multiple fraud modes within each class. To support large-scale training, we adopt tangent-space aggregation as an efficient approximation to the Fréchet mean. Experiments on four public fraud detection benchmarks, including the 5.78M-node T-Social network, show that RSCF-PM achieves the best results on T-Social, FDCompCN, and YelpChi, while remaining highly competitive on Amazon, with up to 4.96% AUC improvement over strong baselines. Ablation and efficiency studies further confirm the complementary value of each component and the practical scalability of the framework.