We consider a network which, under an initial attack, has a fraction of its nodes fail, and the redistribution of the betweenness centrality of the remaining nodes leads to subsequent nodes failing when it exceeds its initial value multiplied by a factor of tolerance . We study the possible disintegration of the network as a function of the size of the initial attack, and the nature of the transition through which the failure takes place, which is determined by the tolerance of the nodes, switching from first order to second when the tolerance increases. We consider different types of networks ( Erdös-Rényi networks, regular graphs, small-world networks, scale-free networks ) and their different behaviors. We present some analytical insights as well as numerical simulations .We also study the influence of the localized nature of the initial attacks on the process of disintegration.
