  Many real-world networks display a community structure. We study two random
graph models that create a network with similar community structure as a given
network. One model preserves the exact community structure of the original
network, while the other model only preserves the set of communities and the
vertex degrees. These models show that community structure is an important
determinant of the behavior of percolation processes on networks, such as
information diffusion or virus spreading: the community structure can both
\textit{enforce} as well as \textit{inhibit} diffusion processes. Our models
further show that it is the mesoscopic set of communities that matters. The
exact internal structures of communities barely influence the behavior of
percolation processes across networks. This insensitivity is likely due to the
relative denseness of the communities.
