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Two node variables determine the evolution of cascades in random networks: anode's degree and threshold. Correlations between both fundamentally change therobustness of a network, yet, they are disregarded in standard analytic methodsas local tree or heterogeneous mean field approximations because of the badtractability of order statistics. We show how they become tractable in thethermodynamic limit of infinite network size. This enables the analyticdescription of node attacks that are characterized by threshold allocationsbased on node degree. Using two examples, we discuss possible implications ofirregular phase transitions and different speeds of cascade evolution for thecontrol of cascades.

Correlations between thresholds and degrees: An analytic approach to model attacks and failure cascades