Hyperbolic Graph Convolutional Neural Networks

TL;DR

This paper introduces Hyperbolic Graph Convolutional Neural Networks (HGCN), which embed graph nodes in hyperbolic space to better capture hierarchical structures, outperforming Euclidean GCNs in link prediction and node classification tasks.

Contribution

The paper develops the first inductive hyperbolic GCN, deriving operations in hyperbolic space and enabling Euclidean features to be mapped into hyperbolic embeddings with trainable curvature.

Findings

HGCN achieves up to 63.1% error reduction in ROC AUC for link prediction.

HGCN improves F1 score by up to 47.5% in node classification.

HGCN outperforms state-of-the-art Euclidean GCNs on multiple benchmarks.

Abstract

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.