标题： 持久路径图的稳定性 显示英文标题

标题： Stability of Persistent Path Diagrams

摘要： 在现实世界系统中，组件之间的关系和连接非常复杂。 真实系统通常被描述为网络，其中节点表示系统中的对象，边表示节点之间的关系或连接。 随着研究的深入，网络被赋予了更丰富的结构，例如有向边、边权重，甚至涉及多个节点的超边。 持久同调是一种用于分析数据的代数方法。 它通过跟踪不同尺度参数下拓扑特征的出生和死亡来帮助我们理解数据的内在结构和模式。原始的持久同调不适用于有向网络。 然而，在有向图上建立的路径同调解决了这个问题。 本文研究作为加权有向图或边加权路径复形表示的复杂网络及其持久路径同调。 我们使用有向图和路径复形的同伦理论，以及持久模的交错性质和瓶颈距离，证明了针对加权有向图或边加权路径复形的持久路径图的稳定性。 因此，持久路径同调具有实际应用价值。 显示英文摘要

摘要： In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections between nodes. With the deepening of research, networks have been endowed with richer structures, such as directed edges, edge weights, and even hyperedges involving multiple nodes. Persistent homology is an algebraic method for analyzing data. It helps us understand the intrinsic structure and patterns of data by tracking the death and birth of topological features at different scale parameters.The original persistent homology is not suitable for directed networks. However, the introduction of path homology established on digraphs solves this problem. This paper studies complex networks represented as weighted digraphs or edge-weighted path complexes and their persistent path homology. We use the homotopy theory of digraphs and path complexes, along with the interleaving property of persistent modules and bottleneck distance, to prove the stability of persistent path diagram with respect to weighted digraphs or edge-weighted path complexes. Therefore, persistent path homology has practical application value.