Complexes of Tournaments, Directionality Filtrations and Persistent Homology

Dejan Govc; Ran Levi; Jason P. Smith

doi:10.1007/s41468-021-00068-0

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 journal › Article › peer-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
Number of pages	25
Journal	Journal of applied and computational topology
Early online date	19 Apr 2021
DOIs	https://doi.org/10.1007/s41468-021-00068-0
Publication status	E-pub ahead of print - 19 Apr 2021

Keywords

Tournaments
Flag complex
Persistent homology
Digraph

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https://arxiv.org/abs/2003.00324Licence: Unspecified

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AU - Levi, Ran

AU - Smith, Jason P.

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AB - 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.

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KW - Flag complex

KW - Persistent homology

KW - Digraph

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M3 - Article

JO - Journal of applied and computational topology

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