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 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.

Computing Homotopy Types of Directed Flag Complexes

“…These are a natural generalisation of the usual concept of simplicial complex and are characterised by the property that the simplices are determined by their ordered lists of vertices, rather than their sets of vertices. Flag tournaplexes are a related, even more recent construction [33].…”

“…Such computations were taken even further when Flagser [46] was developed, a software package based on Ripser [6] but specialised for working with directed flag complexes. A variation called Tournser which works for tournaplexes [33] was also presented in [46], as well as Deltser for ∆-complexes [34, Section 2.1].…”

“…This graph has also been investigated topologically in [57], where its directed flag complex and homology with Z 2 coefficients were computed. The flag tournaplex has also been computed and the corresponding directionality distribution has been compared with the one for the rat neocortical column [33]. Remarkably, the two directionality distributions appear very similar, whereas they can be strikingly different for other graphs.…”

“…The proof consists of applying the above procedure and recording the resulting sequence of collapses and coning operations (which constitute an explicit recipe to construct the required homotopy equivalence); this relies entirely on basic homotopy theoretic principles, and hence requires the computer only for speed. The corresponding flag tournaplex is also briefly examined and torsion is found in the integral homology of certain stages of its local directionality filtration [33], however the exact homotopy type is not analysed in this case (see Theorem 4.2):…”

“…This appears to be the first time torsion has been found in a flag complex arising from a biological neuronal network. Finally, some of the procedures presented here were originally developed to answer certain questions arising from the study of tournaplexes [33], one of these being how to distinguish nonisomorphic regular tournaments from one another. One possible approach is to examine the simplex counts, the Betti numbers and the homotopy types of their directed flag complexes.…”