  We consider a multi-parameter model for randomly constructing simplicial
complexes. This model interpolates between random clique complexes and
Linial-Meshulam random $k$-dimensional complexes, two models that have been
extensively studied. While these models asymptotically exhibit nontrivial
cohomology in only one or two dimensions, we show that in this generalized
setting nontrivial cohomology can occur in several dimensions simultaneously.
We establish upper and lower thresholds for the appearance of nontrivial
cohomology in a particular dimension, and in some instances characterize the
behavior at criticality.
