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.

翻译：树状结构（包括层级结构和幂律分布）在推荐系统、生态系统、金融网络、社交网络等实际应用中普遍存在。近年来，利用双曲空间进行树状结构建模因其指数级增长的容量而备受关注。与平坦的欧几里得空间相比，弯曲的双曲空间为数据嵌入提供了更灵活的空间，尤其适用于具有隐式树状架构的数据集。然而，真实树状数据的复杂性构成重大挑战——这些数据往往展现出树状、平坦和环状区域的异质混合结构。将此类异质结构直接嵌入同质嵌入空间（即双曲空间）不可避免地会导致严重失真。为缓解上述缺陷，本研究致力于探索离散结构与连续学习空间之间的曲率关系，旨在将网络拓扑传递的信息编码至学习过程中，从而改进树状结构建模。为此，本文提出曲率感知双曲图卷积神经网络{κ}HGCN，该模型利用曲率引导消息传递并增强长距离传播。在节点分类与链接预测任务上的大量实验表明，所提方法持续大幅优于各类竞争模型，验证了其优越性。