Abstract
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 enforce as well as 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.
Highlights
Many complex systems across the sciences can be modeled as networks of vertices joined in pairs by edges
We extract the communities of these networks with the Infomap community detection algorithm[45], and use these communities as input for the hierarchical configuration model (HCM) and HCM* model, to create networks with a similar community structure as the original networks
The size of the largest component of real-world networks can be well predicted using the analytical estimates of HCM, which only uses the joint distribution of community sizes and the number of AS Enron HEP PGP FB yeast
Summary
Many complex systems across the sciences can be modeled as networks of vertices joined in pairs by edges. The community structure of a network influences the way a cooperation process behaves on real-world networks[25], and using community structure improves the prediction of which messages will go viral across a network[26]. Several stylized random graph models with a community structure have shown that communities influence the process of an epidemic across a network[27,28,29,30,31,32,33,34], but the extent to which community structure affects epidemics on real-world networks is largely unexplained. We study two random graph models that generate networks with a similar community structure as any given network We find that these models capture the behavior of epidemics or percolation on real-world networks accurately, and that the mesoscopic community structure is vital for understanding epidemic spreading. Correspondence and requests for materials should be addressed to C.S.
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