Simplex closing probabilities in directed graphs

Recent work in mathematical neuroscience has calculated the directed graph homology of simplicial complex given by brains sparse adjacency graph, so called connectome. These biological connectomes show an abundance both high-dimensional simplices and Betti-numbers all viable dimensions - contrast to Erd\H{o}s-R\'enyi-graphs comparable size density. An analysis synthetically trained reveals similar findings, raising questions about graphs comparability nature origin simplices. We present a new method capable delivering insight into emergence thus abundance. Our approach allows easily distinguish simplex-rich different origin. The relies on novel concept almost-d-simplex, that is, simplex missing exactly one edge, consequently almost-d-simplex closing probability dimension. also describe fast algorithm identify almost-d-simplices graph. Applying this artificial data us mechanism responsible for emergence, suggests is signature excitatory subnetwork statistical reconstruction mouse primary visual cortex. highly optimised code publicly available.