Many real-world networks display a community structure. We study two randomgraph models that create a network with similar community structure as a givennetwork. One model preserves the exact community structure of the originalnetwork, while the other model only preserves the set of communities and thevertex degrees. These models show that community structure is an importantdeterminant of the behavior of percolation processes on networks, such asinformation diffusion or virus spreading: the community structure can both\textit{enforce} as well as \textit{inhibit} diffusion processes. Our modelsfurther show that it is the mesoscopic set of communities that matters. Theexact internal structures of communities barely influence the behavior ofpercolation processes across networks. This insensitivity is likely due to therelative denseness of the communities.
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