Simplex Closing Probabilities in Directed Graphs

TL;DR

This paper introduces a novel method using almost-d-simplices and their closing probabilities to analyze the emergence of high-dimensional simplices in directed graphs, revealing insights into biological connectomes and artificial networks.

Contribution

The paper presents a new approach for studying simplicial structures in directed graphs, including a fast algorithm and the concept of almost-d-simplices, to distinguish origins of connectomes.

Findings

Biological connectomes show abundant high-dimensional simplices and Betti-numbers.

The method identifies mechanisms behind simplex emergence in biological and artificial networks.

The approach suggests a link between simplex signatures and excitatory subnetworks in the brain.

Abstract

Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brains 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 Erd\H{o}s-R\'enyi-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 optimised code for this new method is publicly available.