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Constructing null models to test the significance of extracted information isa crucial step in data analysis. In this work, we provide a uniformlysampleable null model of directed graphs with the same (or similar) number ofsimplices in the flag complex, with the restriction of retaining the underlyingundirected graph. We describe an MCMC-based algorithm to sample from this nullmodel and statistically investigate the mixing behaviour. This is paired with ahigh-performance, Rust-based, publicly available implementation. The motivationcomes from topological data analysis of connectomes in neuroscience. Inparticular, we answer the fundamental question: are the high Betti numbersobserved in the investigated graphs evidence of an interesting topology, or arethey merely a byproduct of the high numbers of simplices? Indeed, by applyingour new tool on the connectome of C. Elegans and parts of the statisticalreconstructions of the Blue Brain Project, we find that the Betti numbersobserved are considerable statistical outliers with respect to this new nullmodel. We thus, for the first time, statistically confirm that topological dataanalysis in microscale connectome research is extracting statisticallymeaningful information.

MCMC Sampling of Directed Flag Complexes with Fixed Undirected Graphs