  Most random graph models are locally tree-like - do not contain short cycles
- rendering them unfit for modeling networks with a community structure. We
introduce the hierarchical configuration model (HCM), a generalization of the
configuration model that includes community structures, while properties such
as the size of the giant component, and the size of the giant percolating
cluster under bond percolation can still be derived analytically. Viewing
real-world networks as realizations of the HCM, we observe two previously
unobserved power-law relations: between the number of edges inside a community
and the community sizes, and between the number of edges going out of a
community and the community sizes. We also relate the power-law exponent $\tau$
of the degree distribution with the power-law exponent of the community size
distribution $\gamma$. In the special case of extremely dense communities
(e.g., complete graphs), this relation takes the simple form $\tau=\gamma-1$.
