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Simplicial complexes are generalized network structures able to encodeinteractions occurring between more than two nodes. Simplicial complexesdescribe a large variety of complex interacting systems ranging from brainnetworks, to social and collaboration networks. Here we characterize thestructure of simplicial complexes using their generalized degrees that capturefundamental properties of one, two, three or more linked nodes. Moreover weintroduce the configuration model and the canonical ensemble of simplicialcomplexes, enforcing respectively the sequence of generalized degrees of thenodes and the sequence of the expected generalized degrees of the nodes. Weevaluate the entropy of these ensembles, finding the asymptotic expression forthe number of simplicial complexes in the configuration model. We provide thealgorithms for the construction of simplicial complexes belonging to theconfiguration model and the canonical ensemble of simplicial complexes. We givean expression for the structural cutoff of simplicial complexes that forsimplicial complexes of dimension $d=1$ reduces to the structural cutoff ofsimple networks. Finally we provide a numerical analysis of the naturalcorrelations emerging in the configuration model of simplicial complexeswithout structural cutoff.

Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes