The prevalence of tree-like structures, encompassing hierarchical structures
and power law distributions, exists extensively in real-world applications,
including recommendation systems, ecosystems, financial networks, social
networks, etc. Recently, the exploitation of hyperbolic space for tree-likeness
modeling has garnered considerable attention owing to its exponential growth
volume. Compared to the flat Euclidean space, the curved hyperbolic space
provides a more amenable and embeddable room, especially for datasets
exhibiting implicit tree-like architectures. However, the intricate nature of
real-world tree-like data presents a considerable challenge, as it frequently
displays a heterogeneous composition of tree-like, flat, and circular regions.
The direct embedding of such heterogeneous structures into a homogeneous
embedding space (i.e., hyperbolic space) inevitably leads to heavy distortions.
To mitigate the aforementioned shortage, this study endeavors to explore the
curvature between discrete structure and continuous learning space, aiming at
encoding the message conveyed by the network topology in the learning process,
thereby improving tree-likeness modeling. To the end, a curvature-aware
hyperbolic graph convolutional neural network, {kappa}HGCN, is proposed, which
utilizes the curvature to guide message passing and improve long-range
propagation. Extensive experiments on node classification and link prediction
tasks verify the superiority of the proposal as it consistently outperforms
various competitive models by a large margin.