Cascading Failures in Interdependent Lattice Networks: The Critical Role of the Length of Dependency Links

We study the cascading failures in a system composed of two interdependent square lattice networks $A$ and $B$ placed on the same Cartesian plane, where each node in network $A$ depends on a node in network $B$ randomly chosen within a certain distance $r$ from the corresponding node in network $A$ and vice versa. Our results suggest that percolation for small $r$ below $r_{\max }\approx 8$ (lattice units) is a second-order transition, and for larger $r$ is a first-order transition. For $r<r_{\max }$, the critical threshold increases linearly with $r$ from 0.593 at $r=0$ and reaches a maximum, 0.738 for $r=r_{\max }$, and then gradually decreases to 0.683 for $r=\infty$. Our analytical considerations are in good agreement with simulations. Our study suggests that interdependent infrastructures embedded in Euclidean space become most vulnerable when the distance between interdependent nodes is in the intermediate range, which is much smaller than the size of the system.

Figure 1

Two square lattices $A$ and $B$ where in each lattice every node has two types of links: connectivity links and dependency links. Every node is initially connected to its four nearest neighbors within the same lattice via connectivity links. Also, each node $A_i$ in lattice $A$ depends on one and only one node $B_j$ in lattice $B$ via a dependency link (and vice versa), with the only constraint that $|x_i-x_j|\le r$ and $|y_i-y_j|\le r$. If node $A_i$ fails, then node $B_j$ fails. If node $B_j$ fails, then node $A_i$ fails. Network $A$ is shifted vertically for clarity.

Figure 2

Giant component size $P_{\infty }$ as a function of step $i$ at the first-order transition regime at $p=0.6825$ for $r=L=1000$. The simulation results (solid lines) are in good agreement with the theoretical results (dots). The value of $p$ is close to the percolation threshold $p_c^{\mu }=0.6827$.

Figure 3

The giant component size $P_{\infty }$ as a function of remaining fraction of nodes $p$. The solid curve is for conventional percolation on a single square lattice, which also describes the limiting case of $r=0$. The solid curve is obtained by numerical simulations on $N=4000\times 4000$ lattice sites with periodic boundary conditions and averaged over 100 realizations. The dash curve represents the theoretical result $\mu (p)$ for two interdependent lattice networks with $r=L$ given by Eq. (2). The simulation results (dots) are for two interdependent lattice networks with $N=1000\times 1000$ and $r=L$. Inset: A schematic graphical solution of Eq. (2) is shown. The curves are $\sqrt{p{P}_{\infty }(x)}$ for different $p$ and the solution of Eq. (2) is given by the intersection of the solid curves and the straight line $y=x$. The critical $p=p_c^{\mu }$ corresponds to the case when the solid curve is tangential to the straight line $y=x$. Numerical solutions of Eqs. (3, 4) yield $x_c=0.641$, $P_{\infty }(x_c)=0.602$, and $p_c^{\mu }=0.6827$.

Figure 4

Three different typical behaviors of interdependent lattices near criticality. Pictures of stable mutual giant component at criticality of two interdependent lattices ($N=1000\times 1000$) after cascading failures initiated by a random removal of $1-p$ of the nodes for (a) $r=4$ and $p=0.680$ and for (b) $r=1000$ and $p=0.683$. The dynamics of a growing bubble (explained in the text) for $r=20$ is demonstrated by three snapshots, (c), (d), and (e), of the nonstable giant component of the interdependent lattices ($N=500\times 500$) during the cascading process initiated with $p=0.700$.

Figure 5

The fraction of nodes in the giant component as a function of nodes survived after the initial attack. We perform the simulations by gradually removing additional nodes. For $r=6$, the decrease of giant component occurs in multiple steps, characteristic of a second-order transition. For $r=8$ and $r=16$, the giant component may completely collapse by removal of even a single additional node, characteristic of a first-order transition.

Figure 6

The critical $p_c^{\mu }$ as a function of interdependent distance $r$. The change from second to first order transition occurs at $r_{\max }\approx 8$. The critical $p_c^{\mu }$ of mutual percolation linearly increases for $r<r_{\max }$ following the percolation threshold for flat interface and then gradually decreases to $p_c^{\mu }=0.683$ at $r=\infty$, which is in good agreement with the theoretical results. Inset: Diameter of the hole size $\xi$ as a function of $r$ on conventional percolation on a single lattice network. ${\xi }_h\approx r_{\max }=8$ at $p=0.744$ is in good agreement with the simulation.