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 $i\ge 2$. 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.

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• Uncovering the overlapping community structure of complex networks in nature and society
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• First Betti number of the path homology of random directed graphs
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Cited by in corpus (2)

• Grounded persistent path homology: a stable, topological descriptor for weighted digraphs
• Towards a Quantitative Theory of Digraph-Based Complexes and its Applications in Brain Network Analysis