Network Overload due to Massive Attacks

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 $p_f$ as function of the strength of the initial attack, measured by the fraction of nodes $p$, which survive the initial attack for different values of tolerance $\alpha$ in random regular and Erd\"os-Renyi graphs. We find the existence of first order phase transition line $p_t(\alpha)$ on a $p-\alpha$ plane, such that if $p <p_t$ the cascade of failures lead to a very small fraction of survived nodes $p_f$ and the giant component of the network disappears, while for $p>p_t$, $p_f$ is large and the giant component of the network is still present. Exactly at $p_t$ the function $p_f(p)$ undergoes a first order discontinuity. We find that the line $p_t(\alpha)$ ends at critical point $(p_c,\alpha_c)$ ,in which the cascading failures are replaced by a second order percolation transition. We analytically find the average betweenness of nodes with different degrees before and after the initial attack, investigate their roles in the cascading failures, and find a lower bound for $p_t(\alpha)$. We also study the difference between a localized and random attacks.