  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.
