Simplex closing probabilities in directed graphs

Florian Fedor Fridolin Unger*, Jonathan Krebs, Michael Günther Müller

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brain's sparse adjacency graph, the so called connectome. These biological connectomes show an abundance of both high-dimensional directed simplices and Betti-numbers in all viable dimensions – in contrast to Erdős–Rényi-graphs of comparable size and density. An analysis of synthetically trained connectomes reveals similar findings, raising questions about the graphs comparability and the nature of origin of the simplices.

We present a new method capable of delivering insight into the emergence of simplices and thus simplicial abundance. Our approach allows to easily distinguish simplex-rich connectomes of different origin. The method relies on the novel concept of an almost-d-simplex, that is, a simplex missing exactly one edge, and consequently the almost-d-simplex closing probability by dimension. We also describe a fast algorithm to identify almost-d-simplices in a given graph. Applying this method to biological and artificial data allows us to identify a mechanism responsible for simplex emergence, and suggests this mechanism is responsible for the simplex signature of the excitatory subnetwork of a statistical reconstruction of the mouse primary visual cortex. Our highly optimized code for this new method is publicly available.
Original languageEnglish
Article number101941
JournalComputational Geometry
Publication statusPublished - Feb 2023


  • topology
  • flag complexes
  • connectome
  • Motifs
  • Flag complex
  • Simplicial complex
  • Simplex closing
  • Connectome

ASJC Scopus subject areas

  • Computational Mathematics
  • Control and Optimization
  • Geometry and Topology
  • Computer Science Applications
  • Computational Theory and Mathematics


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