  Network theory has primarily focused on pairwise relationships, disregarding
many-body interactions: neglecting them, however, can lead to misleading
representations of complex systems. Hypergraphs represent an increasingly
popular alternative for describing polyadic interactions: our innovation lies
in leveraging the representation of hypergraphs based on the incidence matrix
for extending the entropy-based framework to higher-order structures. In
analogy with the Exponential Random Graphs, we name the members of this novel
class of models Exponential Random Hypergraphs. Here, we focus on two explicit
examples, i.e. the generalisations of the Erd\"os-R\'enyi Model and of the
Configuration Model. After discussing their asymptotic properties, we employ
them to analyse real-world configurations: more specifically, i) we extend the
definition of several network quantities to hypergraphs, ii) compute their
expected value under each null model and iii) compare it with the empirical
one, in order to detect deviations from random behaviours. Differently from
currently available techniques, ours is analytically tractable, scalable and
effective in singling out the structural patterns of real-world hypergraphs
differing significantly from those emerging as a consequence of simpler,
structural constraints.
