Hochschild homology, and a persistent approach via connectivity digraphs

Abstract

We introduce a persistent Hochschild homology framework for directed graphs. Hochschild homology groups of (path algebras of) directed graphs vanish in degree i≥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 graph. 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 directed graphs.