  Many empirical networks are intrinsically polyadic, with interactions
occurring within groups of agents of arbitrary size. There are, however, few
flexible null models that can support statistical inference for such polyadic
networks. We define a class of null random hypergraphs that hold constant both
the node degree and edge dimension sequences, generalizing the classical dyadic
configuration model. We provide a Markov Chain Monte Carlo scheme for sampling
from these models, and discuss connections and distinctions between our
proposed models and previous approaches. We then illustrate these models
through a triplet of applications. We start with two classical network topics
-- triadic clustering and degree-assortativity. In each, we emphasize the
importance of randomizing over hypergraph space rather than projected graph
space, showing that this choice can dramatically alter statistical inference
and study findings. We then define and study the edge intersection profile of a
hypergraph as a measure of higher-order correlation between edges, and derive
asymptotic approximations under the stub-labeled null. Our experiments
emphasize the ability of explicit, statistically-grounded polyadic modeling to
significantly enhance the toolbox of network data science. We close with
suggestions for multiple avenues of future work.
