  We introduce a persistent Hochschild homology framework for directed graphs.
Hochschild homology groups of (path algebras of) directed graphs vanish in
degree $i\geq 2$. To extend them to higher degrees, we introduce the notion of
connectivity digraphs and analyse two main examples; the first, arising from
Atkin's $q$-connectivity, and the second, here called $n$-path digraphs,
generalising the classical notion of line graphs. Based on a categorical
setting for persistent homology, we propose a stable pipeline for computing
persistent Hochschild homology groups. This pipeline is also amenable to other
homology theories; for this reason, we complement our work with a survey on
homology theories of digraphs.
