  We investigate the number of maximal cliques, i.e., cliques that are not
contained in any larger clique, in three network models: Erd\H{o}s-R\'enyi
random graphs, inhomogeneous random graphs (also called Chung-Lu graphs), and
geometric inhomogeneous random graphs. For sparse and not-too-dense
Erd\H{o}s-R\'enyi graphs, we give linear and polynomial upper bounds on the
number of maximal cliques. For the dense regime, we give super-polynomial and
even exponential lower bounds. Although (geometric) inhomogeneous random graphs
are sparse, we give super-polynomial lower bounds for these models. This comes
form the fact that these graphs have a power-law degree distribution, which
leads to a dense subgraph in which we find many maximal cliques. These lower
bounds seem to contradict previous empirical evidence that (geometric)
inhomogeneous random graphs have only few maximal cliques. We resolve this
contradiction by providing experiments indicating that, even for large
networks, the linear lower-order terms dominate, before the super-polynomial
asymptotic behavior kicks in only for networks of extreme size.
