  Graph convolutional neural networks (GCNs) embed nodes in a graph into
Euclidean space, which has been shown to incur a large distortion when
embedding real-world graphs with scale-free or hierarchical structure.
Hyperbolic geometry offers an exciting alternative, as it enables embeddings
with much smaller distortion. However, extending GCNs to hyperbolic geometry
presents several unique challenges because it is not clear how to define neural
network operations, such as feature transformation and aggregation, in
hyperbolic space. Furthermore, since input features are often Euclidean, it is
unclear how to transform the features into hyperbolic embeddings with the right
amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural
Network (HGCN), the first inductive hyperbolic GCN that leverages both the
expressiveness of GCNs and hyperbolic geometry to learn inductive node
representations for hierarchical and scale-free graphs. We derive GCN
operations in the hyperboloid model of hyperbolic space and map Euclidean input
features to embeddings in hyperbolic spaces with different trainable curvature
at each layer. Experiments demonstrate that HGCN learns embeddings that
preserve hierarchical structure, and leads to improved performance when
compared to Euclidean analogs, even with very low dimensional embeddings:
compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to
63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node
classification, also improving state-of-the art on the Pubmed dataset.
