# Construction of and efficient sampling from the simplicial configuration model

## TL;DR

This paper introduces the simplicial configuration model as a null model for complex systems represented by simplicial complexes, along with an efficient MCMC sampler, enabling comparison with empirical data and revealing organizational structures.

## Contribution

It presents the first efficient, uniform MCMC sampler for the simplicial configuration model, facilitating principled null model comparisons for multi-node interaction data.

## Key findings

- The sampler is efficient and uniform.
- Application to real systems shows non-random organization.
- Rejection of null hypothesis in two systems.

## Abstract

Simplicial complexes are now a popular alternative to networks when it comes to describing the structure of complex systems, primarily because they encode multi-node interactions explicitly. With this new description comes the need for principled null models that allow for easy comparison with empirical data. We propose a natural candidate, the simplicial configuration model. The core of our contribution is an efficient and uniform Markov chain Monte Carlo sampler for this model. We demonstrate its usefulness in a short case study by investigating the topology of three real systems and their randomized counterparts (using their Betti numbers). For two out of three systems, the model allows us to reject the hypothesis that there is no organization beyond the local scale.