Complexes of tournaments, directionality filtrations and persistent homology

Abstract Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer “tournaplexes”, whose simplices In particular, given digraph $${\mathcal {G}}$$ G , associate with it “flag tournaplex” which is tournaplex containing directed flag complex but also geometric realisation cliques not directed. define several types filtrations on tournaplexes, and exploiting persistent homology, observe tournaplexes provide finer means distinguishing graph dynamics than complex. then demonstrate power these ideas by applying them data arising from Blue Brain Project’s digital reconstruction rat’s neocortex.