  In recent years, A. Grigor'yan, Y. Lin, Y. Muranov and S.T. Yau [6, 7, 8, 9]
constructed a path homology theory for digraphs. Later, S. Chowdhury and F.
Memoli [3] studied the persistent path homology for directed networks. In this
paper, we generalize the path homology theory for digraphs and construct a
weighted path homology for weighted digraphs. We study the persistent weighted
path homology for weighted digraphs and detect the effects of the weights on
the persistent weighted path homology. We prove a persistent version of a
Kunneth-type formula for joins of weighted digraphs.
