Networks of dyadic relationships between entities have emerged as a dominant
paradigm for modeling complex systems. Many empirical "networks" -- such as
collaboration networks; co-occurrence networks; and communication networks --
are intrinsically polyadic, with multiple entities interacting simultaneously.
Historically, such polyadic data has been represented dyadically via a standard
projection operation. While convenient, this projection often has unintended
and uncontrolled impact on downstream analysis, especially null
hypothesis-testing. In this work, we develop a class of random null models for
polyadic data in the framework of hypergraphs, therefore circumventing the need
for projection. The null models we define are uniform on the space of
hypergraphs sharing common degree and edge dimension sequences, and thus
provide direct generalizations of the classical configuration model of network
science. We also derive Metropolis-Hastings algorithms in order to sample from
these spaces. We then apply the model to study two classical network topics --
clustering and assortativity -- as well as one contemporary, polyadic topic --
simplicial closure. In each application, we emphasize the importance of
randomizing over hypergraph space rather than projected graph space, showing
that this choice can dramatically alter directional study conclusions and
statistical findings. For example, we find that many of the social networks we
study are less clustered than would be expected at random, a finding in tension
with much conventional wisdom within network science. Our findings underscore
the importance of carefully choosing appropriate null spaces for polyadic
relational data, and demonstrate the utility of random hypergraphs in many
study contexts.