Title: A framework for cascade size calculations on random networks,
Abstract: 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.