  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.
