Abstract
An analytical approach to determining the mean avalanche size in a broad class of dynamical models on random networks is introduced. Previous results on percolation transitions and epidemic sizes are shown to be special cases of the method. The time-dependence of cascades and extensions to networks with community structure or degree-degree correlations are discussed. Analytical results for the rate of spread of innovations in a modular network and for the size of k cores in networks with degree-degree correlations are confirmed with numerical simulations.
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