A framework for studying the vulnerability and the recovery of networks and interdependent networks will be presented. In interdependent networks, such as infrastructures, when nodes in one network fail, they cause dependent nodes in other networks to also fail. This may happen recursively and can lead to a cascade of failures and to a sudden fragmentation of the system. I will present analytical solutions based on percolation theory, for the critical thresholds and the giant component of a network of n interdependent networks. I will show, that the general theory has many novel features that are not present in the classical network theory. When recovery of components is possible, global spontaneous failure and recovery of the networks, as well as hysteresis phenomena, occur. The theory suggests an optimal repair strategy for a system of systems.
I will also show that interdependent networks embedded in space are
significantly more vulnerable compared to non-embedded networks. In particular, small localized attacks of zero fraction but above a critical size may lead to cascading failures that dynamically propagate and yield catastrophic consequences.
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