  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.
