Title: Correlations between thresholds and degrees: An analytic approach to model attacks and failure cascades,
Abstract: Two node variables determine the evolution of cascades in random networks: a
node's degree and threshold. Correlations between both fundamentally change the
robustness of a network, yet, they are disregarded in standard analytic methods
as local tree or heterogeneous mean field approximations because of the bad
tractability of order statistics. We show how they become tractable in the
thermodynamic limit of infinite network size. This enables the analytic
description of node attacks that are characterized by threshold allocations
based on node degree. Using two examples, we discuss possible implications of
irregular phase transitions and different speeds of cascade evolution for the
control of cascades.