Randomising hypergraphs by permuting their incidence matrix*

Network theory has emerged as a powerful paradigm to explain phenomena where units interact in a highly non-trivial way. So far, however, research in the field has mainly focused on pairwise interactions, disregarding the possibility that more-than-two constituent units could interact at a time. Hypergraphs represent a class of mathematical objects that could serve the scope of describing this novel kind of many-bodies interactions. In this paper, we propose benchmark models for hypergraphs analysis that generalise the usual Erdos-Renyi and Configuration Model in the simplest possible way, i.e. by randomising the hypergraph incidence matrix while preserving the corresponding connectivity/topological constraints - whose definition is, now, adapted to the novel framework. After exploring the mathematical properties of the proposed benchmark models, we consider two different applications: first, we define a novel quantity, the hyperedge assortativity, whose expected value we theoretically derive for all the introduced null models and which we, then, use to detect deviations in the corresponding real-world hypergraphs; second, we define a principled procedure for testing the statistical significance of the number of hyperedges connecting any two nodes.