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.
Yang et al. (Sat,) studied this question.