Hochschild homology, and a persistent approach via connectivity digraphs

TL;DR

This paper develops a persistent Hochschild homology framework for directed graphs, introducing connectivity digraphs to extend homology to higher degrees and proposing a stable computational pipeline.

Contribution

It introduces connectivity digraphs and a categorical pipeline for persistent Hochschild homology, extending the theory to higher degrees and generalizing classical graph notions.

Findings

Hochschild homology vanishes in degree i≥2 for path algebras of directed graphs

Connectivity digraphs enable extension of homology to higher degrees

A stable computational pipeline for persistent Hochschild homology is proposed

Abstract

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.