Hyperbolic Curvature Graph Neural Network

Hyperbolic space is emerging as a promising learning space for representation learning, owning to its exponential growth volume. Compared with the flat Euclidean space, the curved hyperbolic space is far more ambient and embeddable, particularly for datasets with implicit tree-like architectures, such as hierarchies and power-law distributions. On the other hand, the structure of a real-world network is usually intricate, with some regions being tree-like, some being flat, and others being circular. Directly embedding heterogeneous structural networks into a homogeneous embedding space unavoidably brings inductive biases and distortions. Inspiringly, the discrete curvature can well describe the local structure of a node and its surroundings, which motivates us to investigate the information conveyed by the network topology explicitly in improving geometric learning. To this end, we explore the properties of the local discrete curvature of graph topology and the continuous global curvature of embedding space. Besides, a Hyperbolic Curvature-aware Graph Neural Network, HCGNN, is further proposed. In particular, HCGNN utilizes the discrete curvature to lead message passing of the surroundings and adaptively adjust the continuous curvature simultaneously. Extensive experiments on node classification and link prediction tasks show that the proposed method outperforms various competitive models by a large margin in both high and low hyperbolic graph data. Case studies further illustrate the efficacy of discrete curvature in finding local clusters and alleviating the distortion caused by hyperbolic geometry.