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.

An application of neighbourhoods in digraphs to the classification of binary dynamics

“…Another way of saying the same is that a topological operator on digraphs is a functor from the category of digraphs and digraph inclusions to the category of topological spaces and inclusions. The flag complex of G (ignoring orientation), the directed flag complex [15], and the flag tournaplex [8] are examples of such operators. Definition 1.2.…”

“…To emphasise which option is taken we decorated the parameter codes from Table 1 with a subscript "high" (referring to the difference between the two largest moduli) or "low" (referring to the smallest modulus of a nonzero eigenvalue). For example, Figures 7,8,9 have bls low as a parameter, indicating the lowest nonzero value in the Bauer Laplacian spectrum (that is, the minimal nonzero eigenvalue of the Bauer Laplacian matrix). Another variant of the standard concepts of spectra is what we call the reversed spectral gap.…”

“…Moreover, natural systems such as neuronal networks tend to be very noisy, in the sense that the emerging dynamics from the same stimulus may take a rather large variety of forms. Finally, it is a general working hypothesis in studying network dynamics that the network structure affects its function (see for instance [20]), and therefore instead of considering individual vertices in the network, it makes sense to examine ensembles of vertices and the way that they behave as dynamical sub-units.In previous studies we considered cliques in a directed graph, with various orientations of the connections between nodes, as basic units from which one could extract information about binary dynamics [18,8]. However, the results in these papers fell short of suggesting an efficient classifier of binary dynamics (see [8,).…”

“…In previous studies we considered cliques in a directed graph, with various orientations of the connections between nodes, as basic units from which one could extract information about binary dynamics [18,8]. However, the results in these papers fell short of suggesting an efficient classifier of binary dynamics (see [8,). Indeed, when we applied…”