A framework for cascade size calculations on random networks

We present a framework to calculate the cascade size evolution for a large class of cascade models on random network ensembles in the limit of infinite network size. Our method is exact and applies to network ensembles with almost arbitrary degree distribution, degree-degree correlations and, in case of threshold models, with arbitrary threshold distribution. With our approach, we shift the perspective from the known branching process approximations to the iterative update of suitable probability distributions. Such distributions are key to capture cascade dynamics that involve possibly continuous quantities and that depend on the cascade history, e.g. if load is accumulated over time. These distributions respect the Markovian nature of the studied random processes. Random variables capture the impact of nodes that have failed at any point in the past on their neighborhood. As a proof of concept, we provide two examples: (a) Constant load models that cover many of the analytically tractable cascade models, and, as a highlight, (b) a fiber bundle model that was not tractable by branching process approximations before. Our derivations cover the whole cascade dynamics, not only their steady state. This allows to include interventions in time or further model complexity in the analysis.