Spatial graphs are particular graphs for which the nodes are localized in space (e.g., public transport network, molecules, branching biological structures). In this work, we consider the problem of spatial graph reduction, that aims to find a smaller spatial graph (i.e., with less nodes) with the same overall structure as the initial one. In this context, performing the graph reduction while preserving the main topological features of the initial graph is particularly relevant, due to the additional spatial information. Thus, we propose a topological spatial graph coarsening approach based on a new framework that finds a trade-off between the graph reduction and the preservation of the topological characteristics. The coarsening is realized by collapsing short edges. In order to capture the topological information required to calibrate the reduction level, we adapt the construction of classical topological descriptors made for point clouds (the so-called persistent diagrams) to spatial graphs. This construction relies on the introduction of a new filtration called triangle-aware graph filtration. Our coarsening approach is parameter-free and we prove that it is equivariant under rotations, translations and scaling of the initial spatial graph. We evaluate the performances of our method on synthetic and real spatial graphs, and show that it significantly reduces the graph sizes while preserving the relevant topological information.

翻译：空间图是一种特殊类型的图，其节点在空间中具有定位（例如公共交通网络、分子、分支生物结构）。在本工作中，我们考虑空间图简化问题，其目标是找到一个更小的空间图（即具有更少的节点），且其整体结构与初始图相同。在此背景下，由于存在额外的空间信息，在执行图简化时保持初始图的主要拓扑特征尤为重要。因此，我们提出了一种基于新框架的拓扑空间图粗化方法，该方法在图简化与拓扑特征保持之间寻求平衡。粗化通过折叠短边来实现。为了捕获校准简化级别所需的拓扑信息，我们将为点云构建的经典拓扑描述符（即所谓的持久图）的构造方法适配到空间图。这一构造依赖于引入一种称为三角形感知图滤过的新滤过。我们的粗化方法是无参数的，并且我们证明了其在初始空间图的旋转、平移和缩放变换下具有等变性。我们在合成和真实空间图上评估了该方法的性能，结果表明，它在保持相关拓扑信息的同时，显著减小了图的规模。