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### Directed networks through simplicial paths and Hochschild homology

## Abstract

Directed graphs are a model for various phenomena in the

sciences. In topological data analysis particularly the advent of

applying topological tools to networks of brain neurons has spawned

interest in constructing topological spaces out of digraphs, developing

computational tools for obtaining topological information, and using

these to understand networks. At the end of the day, (homological)

computations of the spaces reveal something about the geometric

realisation, thereby losing the directionality information.

However, digraphs can also be associated with path algebras. We can now

consider applying Hochschild homology to extract information, hopefully

obtaining something more refined in terms of the combinatorics of the

directed edges and paths in the digraph. Unfortunately, Hochschild

homology tends to vanish beyond degree 1. We can overcome this by

considering different higher paths of simplices, and thus introduce

Hochschild homology of digraphs in higher degrees. Moreover, this

procedure gives an implementable persistence pipeline for network

analysis. This is a joint work with Luigi Caputi.