  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.
