  While standard persistent homology has been successful in extracting
information from metric datasets, its applicability to more general data, e.g.
directed networks, is hindered by its natural insensitivity to asymmetry. We
study a construction of homology of digraphs due to Grigor'yan, Lin, Muranov
and Yau, and extend this construction to the persistent framework. The result,
which we call persistent path homology, can provide information about the
digraph structure of a directed network at varying resolutions. Moreover, this
method encodes a rich level of detail about the asymmetric structure of the
input directed network. We test our method on both simulated and real-world
directed networks and conjecture some of its characteristics.
