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.

本文提出了第一个归纳式的超几何图卷积神经网络(HGCN)，它利用了超几何空间的表达能力和嵌入高度曲率的特点来学习分层和无标度图的归纳式节点表示，并说明了如何将欧几里得输入特征转换为具有不同可训练曲率的超几何嵌入。 实验证明，HGCN学习到的嵌入保留了分层结构，并且与欧几里得模型相比，即使具有非常低的维度嵌入，也能获得更好的性能：在链路预测中，ROC AUC误差降低最多为63.1％，在节点分类中，F1 score提高最多为47.5％，也改善了Pubmed数据集的最新技术水平。

双曲图卷积神经网络