Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore, merely combining individual spatial and network representations cannot reveal the underlying interaction mechanism of spatial networks. Besides, existing spatial network representation learning methods can only consider networks embedded in Euclidean space, and can not well exploit the rich geometric information carried by irregular and non-uniform non-Euclidean space. In order to address this issue, in this paper we propose a novel generic framework to learn the representation of spatial networks that are embedded in non-Euclidean manifold space. Specifically, a novel message-passing-based neural network is proposed to combine graph topology and spatial geometry, where spatial geometry is extracted as messages on the edges. We theoretically guarantee that the learned representations are provably invariant to important symmetries such as rotation or translation, and simultaneously maintain sufficient ability in distinguishing different geometric structures. The strength of our proposed method is demonstrated through extensive experiments on both synthetic and real-world datasets.

翻译：空间网络是指其图拓扑结构受嵌入空间约束的网络。理解空间与图属性的耦合关系对于从空间网络中提取强表征至关重要。因此，单纯结合独立的空间表征与网络表征无法揭示空间网络的底层交互机制。此外，现有空间网络表征学习方法仅能处理嵌入欧几里得空间中的网络，无法有效利用非均匀且不规则的 非欧几里得空间所承载的丰富几何信息。为解决这一问题，本文提出了一种新颖的通用框架，用于学习嵌入非欧几里得流形空间中的空间网络表征。具体而言，我们设计了一种基于消息传递的新型神经网络，通过将图拓扑与空间几何相结合，将空间几何提取为边上的消息。我们从理论上保证学习到的表征对旋转或平移等重要对称性具有可证明的不变性，同时保持足够的能力以区分不同的几何结构。通过在合成数据集与真实数据集上的大量实验，验证了所提方法的优势。