  Simplicial complexes are generalized network structures able to encode
interactions occurring between more than two nodes. Simplicial complexes
describe a large variety of complex interacting systems ranging from brain
networks, to social and collaboration networks. Here we characterize the
structure of simplicial complexes using their generalized degrees that capture
fundamental properties of one, two, three or more linked nodes. Moreover we
introduce the configuration model and the canonical ensemble of simplicial
complexes, enforcing respectively the sequence of generalized degrees of the
nodes and the sequence of the expected generalized degrees of the nodes. We
evaluate the entropy of these ensembles, finding the asymptotic expression for
the number of simplicial complexes in the configuration model. We provide the
algorithms for the construction of simplicial complexes belonging to the
configuration model and the canonical ensemble of simplicial complexes. We give
an expression for the structural cutoff of simplicial complexes that for
simplicial complexes of dimension $d=1$ reduces to the structural cutoff of
simple networks. Finally we provide a numerical analysis of the natural
correlations emerging in the configuration model of simplicial complexes
without structural cutoff.
