  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.
