Stability of Persistent Path Diagrams

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.