Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Complexes of Tournaments, Directionality Filtrations and Persistent Homology

Version 1 : Received: 2 March 2020 / Approved: 4 March 2020 / Online: 4 March 2020 (11:40:12 CET)

How to cite: Levi, R.; Govc, D.; Smith, J. Complexes of Tournaments, Directionality Filtrations and Persistent Homology. Preprints 2020, 2020030068 (doi: 10.20944/preprints202003.0068.v1). Levi, R.; Govc, D.; Smith, J. Complexes of Tournaments, Directionality Filtrations and Persistent Homology. Preprints 2020, 2020030068 (doi: 10.20944/preprints202003.0068.v1).

## 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 G, we associate with it a flag tournaplex'' which is a tournaplex containing the directed flag complex of 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.

## Keywords

Tournaments; Semi-Simplicial Sets; Neural networks; Persistent homology

## Subject

MATHEMATICS & COMPUTER SCIENCE, Geometry & Topology

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