Network overload due to massive attacks

Abstract

We study the cascading failure of networks due to overload, using the betweenness centrality of a node as the measure of its load following the Motter and Lai model. We study the fraction of survived nodes at the end of the cascade p[subscript f] as a function of the strength of the initial attack, measured by the fraction of nodes p that survive the initial attack for different values of tolerance α in random regular and Erdös-Renyi graphs. We find the existence of a first-order phase-transition line p[subscript t](α) on a p-α plane, such that if p<p[subscript t], the cascade of failures leads to a very small fraction of survived nodes p[subscript f] and the giant component of the network disappears, while for p>p[subscript t], p[subscript f] is large and the giant component of the network is still present. Exactly at p[subscript t], the function p[subscript f](p) undergoes a first-order discontinuity. We find that the line p[subscript t](α) ends at a critical point (p[subscript c],α[subscript c]), in which the cascading failures are replaced by a second-order percolation transition. We find analytically the average betweenness of nodes with different degrees before and after the initial attack, we investigate their roles in the cascading failures, and we find a lower bound for p[subscript t](α). We also study the difference between localized and random attacks.