Disintegration of Different Types of Networks by Overload under Massive Attack

We discuss a network which has a fraction of its nodes fail initially , and the redistribution of the betweenness centrality of the remaining nodes leads to subsequent failures, as in the Motter and Lai model; the subsequent change of the betweenness can lead to a cascade of failures that might disintegrate the network. There is a threshold in the size of the initial attack that leads to disintegration. The transition switches from first order to second when the tolerance of the nodes increases for networks with a narrow distribution of the degrees of their nodes ( Erdös-Rényi, random regular, small-world.,In the case of broader distributions, like a power law with exponent smaller than 3, the destruction of the initial nodes tends to stabilize the network, the value of the threshold goes to zero and the transition remains second order for all tolerances . We present an analytic calculation of the behavior of the betweenness of the different nodes during the disintegration and extensive numerical simulations .We consider the influence of the localized nature of the initial attacks on the disintegration. We show that these type of networks are, surprisingly, much more resilient vis-à-vis localized attacks.