论文标题
接地持续的路径同源性：稳定的拓扑拓扑描述符，用于加权挖掘
Grounded persistent path homology: a stable, topological descriptor for weighted digraphs
论文作者
论文摘要
加权挖掘物用于对各种天然系统进行建模，并且可以在各种尺度上表现出有趣的结构。为了理解和比较这些系统，我们需要稳定，可解释的多尺度描述符。为此，我们提出了接地的持续路径同源性（GRPPH） - 一种新的功能性拓扑描述符，该描述符通过持久性条形码描述了边缘加权挖掘的结构。我们显示，该图可以选择电路基础，该电路基础可为条形码中的特征提供几何解释的代表。此外，我们表明条形码在瓶颈距离处是稳定的，它们都与数值和结构扰动均具有稳定性。 GRPPH源自一个灵活的框架，该框架是通过选择Digraph链复合物和过滤的选择。为了完整性，我们还调查了定向标志复合物在GRPPH中使用的路径同源性复合物。
Weighted digraphs are used to model a variety of natural systems and can exhibit interesting structure across a range of scales. In order to understand and compare these systems, we require stable, interpretable, multiscale descriptors. To this end, we propose grounded persistent path homology (GrPPH) - a new, functorial, topological descriptor that describes the structure of an edge-weighted digraph via a persistence barcode. We show there is a choice of circuit basis for the graph which yields geometrically interpretable representatives for the features in the barcode. Moreover, we show the barcode is stable, in bottleneck distance, to both numerical and structural perturbations. GrPPH arises from a flexible framework, parametrised by a choice of digraph chain complex and a choice of filtration; for completeness, we also investigate replacing the path homology complex, used in GrPPH, by the directed flag complex.
