Hyperbolic graph convolutional networks (GCNs) demonstrate powerful representation ability to model graphs with hierarchical structure. Existing hyperbolic GCNs resort to tangent spaces to realize graph convolution on hyperbolic manifolds, which is inferior because tangent space is only a local approximation of a manifold. In this paper, we propose a hyperbolic-to-hyperbolic graph convolutional network (H2H-GCN) that directly works on hyperbolic manifolds. Specifically, we developed a manifold-preserving graph convolution that consists of a hyperbolic feature transformation and a hyperbolic neighborhood aggregation. The hyperbolic feature transformation works as linear transformation on hyperbolic manifolds. It ensures the transformed node representations still lie on the hyperbolic manifold by imposing the orthogonal constraint on the transformation sub-matrix. The hyperbolic neighborhood aggregation updates each node representation via the Einstein midpoint. The H2H-GCN avoids the distortion caused by tangent space approximations and keeps the global hyperbolic structure. Extensive experiments show that the H2H-GCN achieves substantial improvements on the link prediction, node classification, and graph classification tasks.
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