MCMC Sampling of Directed Flag Complexes with Fixed Undirected Graphs

TL;DR

This paper introduces a novel MCMC-based method to generate null models of directed graphs with fixed undirected structures, enabling statistical validation of topological features in connectome data.

Contribution

It presents a new sampling algorithm and implementation for directed graph null models that retain the undirected graph structure, advancing topological data analysis in neuroscience.

Findings

Betti numbers in connectomes are statistically significant outliers.

The method confirms the meaningfulness of topological features in microscale connectome analysis.

The approach provides a rigorous statistical framework for topological data analysis.

Abstract

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.