Power-law relations in random networks with communities

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 $\tau=\gamma-1$.