Tolerance of scale-free networks against attack-induced cascades

Scale-free networks can be disintegrated by attack on a single or a very few nodes through the process of cascading failures. By utilizing a prototype cascading model, we previously determined the critical value of the capacity parameter below which the network can become disintegrated due to attack on a single node. A fundamental question in network security, which has not been addressed previously but may be more important and of wider interest, is how to design networks of finite capacity that are safe against cascading breakdown. Here we derive an upper bound for the capacity parameter, above which the network is immune to cascading breakdown. Our theory also yields estimates for the maximally achievable network integrity via controlled removal of a small set of low-degree nodes. The theoretical results are confirmed numerically.