Modeling information spread across networks with communities using a multitype branching process framework

The dynamics of information diffusion in complex networks is widely studied in an attempt to understand how individuals communicate and how information travels and reaches individuals through interactions. However, complex networks often present community structure, and tools to analyse information diffusion on networks with communities are needed. In this paper, we develop theoretical tools using multi-type branching processes to model and analyse simple contagion information spread on a broad class of networks with community structure. We show how, by using limited information about the network -- the degree distribution within and between communities -- we can calculate standard statistical characteristics of the dynamics of information diffusion, such as the extinction probability, hazard function, and cascade size distribution. These properties can be estimated not only for the entire network but also for each community separately. Furthermore, we estimate the probability of information spreading from one community to another where it is not currently spreading. We demonstrate the accuracy of our framework by applying it to two specific examples: the Stochastic Block Model and a log-normal network with community structure. We show how the initial seeding location affects the observed cascade size distribution on a heavy-tailed network and that our framework accurately captures this effect.