  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.
