  Exponential random graph models have attracted significant research attention
over the past decades. These models are maximum-entropy ensembles under the
constraints that the expected values of a set of graph observables are equal to
given values. Here we extend these maximum-entropy ensembles to random
simplicial complexes, which are more adequate and versatile constructions to
model complex systems in many applications. We show that many random simplicial
complex models considered in the literature can be casted as maximum-entropy
ensembles under certain constraints. We introduce and analyze the most general
random simplicial complex ensemble $\mathbf{\Delta}$ with statistically
independent simplices. Our analysis is simplified by the observation that any
distribution $\mathbb{P}(O)$ on any collection of objects $\mathcal{O}=\{O\}$,
including graphs and simplicial complexes, is maximum-entropy under the
constraint that the expected value of $-\ln \mathbb{P}(O)$ is equal to the
entropy of the distribution. With the help of this observation, we prove that
ensemble $\mathbf{\Delta}$ is maximum-entropy under two types of constraints
that fix the expected numbers of simplices and their boundaries.
