  Constructing null models to test the significance of extracted information is
a crucial step in data analysis. In this work, we provide a uniformly
sampleable null model of directed graphs with the same (or similar) number of
simplices in the flag complex, with the restriction of retaining the underlying
undirected graph. We describe an MCMC-based algorithm to sample from this null
model and statistically investigate the mixing behaviour. This is paired with a
high-performance, Rust-based, publicly available implementation. The motivation
comes from topological data analysis of connectomes in neuroscience. In
particular, we answer the fundamental question: are the high Betti numbers
observed in the investigated graphs evidence of an interesting topology, or are
they merely a byproduct of the high numbers of simplices? Indeed, by applying
our new tool on the connectome of C. Elegans and parts of the statistical
reconstructions of the Blue Brain Project, we find that the Betti numbers
observed are considerable statistical outliers with respect to this new null
model. We thus, for the first time, statistically confirm that topological data
analysis in microscale connectome research is extracting statistically
meaningful information.
