Simplicial and homological approaches on directed networks

Applying topological data analysis to neural networks (in the sense of brains, not machine learning) has evolved into a field that has come to be known as topological neuroscience. For topological toolbox we first need to build a topological space on the directed graph representing our network. One convenient construction is the directed flag complex built out of directed cliques. However, when making topological measurements, like computing simplicial homology, the directionality information of the network is lost to some extent. I will show how to build new spaces out of directed cliques, that are more sensitive to the directionality, by employing so called q-connectivity of Atkin's. I will then outline another, algebraic, homology that can be computed for directed graphs, the Hochschild homology of the path algebra, and how that can be enhanced with the new spaces just built out of directed cliques. I will aim to give an accessible talk by introducing the possibly unfamiliar topological notions.