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.
Highlights
The idea of this article arose from considering topological objects associated to directed graphs, or digraphs
This leads to the central idea of this article, namely to the construction of a complex built of all possible cliques in the digraph, while preserving partial information that is determined by edge orientation
By analogy to the flag complex of a graph and the directed flag complex of a digraph, we introduce the flag tournaplex associated to a digraph
Summary
We start by defining the basic objects of study and some of their properties. All digraphs considered in this article are assume to be finite and simple. By simple we mean loopfree, namely edges of the form (v, v) are not allowed for any vertex v, and if v and w are two distinct vertices a digraph may contain the reciprocal edges (v, w) and (w, v), but two edges in the same orientation between v and w are not allowed. If G is any digraph, by its underlying undirected graph we mean the graph G on the same vertices and edges, where edge orientation is ignored (i.e. any directed edge or pair of reciprocal edges is replaced by a single undirected edge). For any set of k vertices in σ , the induced subgraph is a k-tournament Such sub-tournaments will be referred to as the faces of σ. We will sometimes refer to (n + 1)-tournaments as n-dimensional
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