# Complexes of Tournaments, Directionality Filtrations and Persistent Homology

Dejan Govc, Ran Levi* (Corresponding Author), Jason P. Smith

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

## Abstract

Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph $\mathcal{G}$, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of $\mathcal{G}$, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neocortex.
Original language English 25 Journal of applied and computational topology 19 Apr 2021 https://doi.org/10.1007/s41468-021-00068-0 E-pub ahead of print - 19 Apr 2021

## Keywords

• Tournaments
• Flag complex
• Persistent homology
• Digraph