Distributions of Betweenness in Cascades of Overload Failure in Random Regular Networks

We study the Motter and Lai [1] model of cascading failures of a network by overload based on the betweenness centrality of the nodes, for the case of a random regular network. ~We study numerically by several means~the disintegration of the network as a function of the fraction~$p$~of the nodes that survive an initial random attack: the size of the final giant component, the number of cascade stages, and the distribution of the betweenness of the nodes for different stages of the cascade. ~We find that the nature of the transition through which the network disintegrates changes from first order to second order as the tolerance increases. After this large drop, in which a substantial part of the network disintegrates, we find that the size of the final~giant component does not decrease monotonically when increasing~the size of the initial attack~\textit{(1-p)}, but rather presents a series of maxima and minima as a function of~$p$. [1] Cascade-based attacks on complex networks, Phys. Rev. \textbf{E 66}, 065102(R) (2002)