Despite the recently exhibited importance of higher-order interactions for
various processes, few flexible (null) models are available. In particular,
most studies on hypergraphs focus on a small set of theoretical models. Here,
we introduce a class of models for random hypergraphs which displays a similar
level of flexibility of complex network models and where the main ingredient is
the probability that a node belongs to a hyperedge. When this probability is a
constant, we obtain a random hypergraph in the same spirit as the Erdos-Renyi
graph. This framework also allows us to introduce different ingredients such as
the preferential attachment for hypergraphs, or spatial random hypergraphs. In
particular, we show that for the Erdos-Renyi case there is a transition
threshold scaling as 1/EN​ where N is the number of nodes and E the
number of hyperedges. We also discuss a random geometric hypergraph which
displays a percolation transition for a threshold distance scaling as
rc∗​∼1/E​. For these various models, we provide results for the
most interesting measures, and also introduce new ones in the spatial case for
characterizing the geometrical properties of hyperedges. These different models
might serve as benchmarks useful for analyzing empirical data.Comment: 10 pages, 10 figure