Lorentzian Graph Convolutional Networks

TL;DR

This paper introduces Lorentzian GCNs, a novel hyperbolic graph neural network that rigorously adheres to hyperbolic geometry, improving representation accuracy for scale-free and hierarchical graphs.

Contribution

It proposes Lorentzian graph operations and a centroid-based neighborhood aggregation, ensuring hyperbolic geometric correctness and enhancing GCN performance.

Findings

LGCN outperforms state-of-the-art hyperbolic GCNs on six datasets.

LGCN exhibits lower distortion in representing tree-like graphs.

Replacing existing hyperbolic GCN operations with proposed ones improves performance.

Abstract

Graph convolutional networks (GCNs) have received considerable research attention recently. Most GCNs learn the node representations in Euclidean geometry, but that could have a high distortion in the case of embedding graphs with scale-free or hierarchical structure. Recently, some GCNs are proposed to deal with this problem in non-Euclidean geometry, e.g., hyperbolic geometry. Although hyperbolic GCNs achieve promising performance, existing hyperbolic graph operations actually cannot rigorously follow the hyperbolic geometry, which may limit the ability of hyperbolic geometry and thus hurt the performance of hyperbolic GCNs. In this paper, we propose a novel hyperbolic GCN named Lorentzian graph convolutional network (LGCN), which rigorously guarantees the learned node features follow the hyperbolic geometry. Specifically, we rebuild the graph operations of hyperbolic GCNs with Lorentzian version, e.g., the feature transformation and non-linear activation. Also, an elegant neighborhood aggregation method is designed based on the centroid of Lorentzian distance. Moreover, we prove some proposed graph operations are equivalent in different types of hyperbolic geometry, which fundamentally indicates their correctness. Experiments on six datasets show that LGCN performs better than the state-of-the-art methods. LGCN has lower distortion to learn the representation of tree-likeness graphs compared with existing hyperbolic GCNs. We also find that the performance of some hyperbolic GCNs can be improved by simply replacing the graph operations with those we defined in this paper.