Title: On the Dynamics of Cascading Failures in Interdependent Networks

Abstract: Cascading failures in interdependent networks have been investigated using percolation theory in recent years. Here, we study the dynamics of the cascading failures, the average and fluctuations of the number of cascading as a function of system size $N$ near criticality. The system we analyzed is a pair of fully interdependent Erd$\ddot{\textup{o}}$s-R$\acute{\textup{e}}$nyi (ER) networks. We show that when $p$ is close to $p_{c}$, the whole dynamical process of cascading failures can be divided into three time stages. The giant component sizes in the second time stage, presented by a plateau in the size of giant component, have large standard deviations, which cannot be well predicted by the mean-field theory. We also investigate the standard deviation of the total time $\textup{\textbf{std}}(\tau)$ using simulations. When $p=p_{c}$, our numerical simulations indicate that $\textup{\textbf{std}}(\tau)\sim N^{1/3}$, which increases faster than the mean, $<\tau>\sim N^{1/4}$, predicted by the mean-field theory. We also find the scaling behavior as a function of $N$ and $p$ of $<\tau>$ and $\textup{\textbf{std}}(\tau)$ for $p<p_{c}$.