  The theory of path homology for digraphs was developed by Alexander
Grigor'yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. In this paper, we
generalize the path homology for digraphs. We prove that for any digraph $G$,
any $t\geq 0$, any $0\leq q\leq 2t$, and any $(2t+1)$-dimensional element
$\alpha$ in the differential algebra on the set of the vertices, we always have
an $(\alpha,q)$-path homology for $G$. In particular, if $t=0$, then the
$(\alpha,0)$-path homology gives the weighted path homology for vertex-weighted
digraphs.
