  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.
