Physical Review Letters · 2012 · 283 citations · 23 references
EngineeringComputer ScienceComplex SystemsComputer NetworksFault-tolerant SystemsCritical PhenomenaGraph TheoryInterdependent NetworksNetwork ReliabilityNetwork TheoryPercolation TheoryCascading FailuresInterdependent Network PercolationNetwork CriticalityNetwork Percolation ThresholdsPercolation On LatticesPercolation On NetworksPercolation ThresholdsPercolation Transition
The study investigates cascading failures in two interdependent square lattice networks where each node depends on a counterpart within a distance r. The authors model two interdependent square lattice networks on the same plane, linking each node to a node in the other network chosen uniformly within a distance r. The analysis shows that percolation is second‑order for r < ≈8 and first‑order for larger r, with the critical threshold rising linearly to 0.738 at r ≈8 before falling to 0.683 at infinite distance, and that interdependent lattices are most vulnerable when the dependency distance lies in an intermediate range.
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 ${r}_{\mathrm{max}}\ensuremath{\approx}8$ (lattice units) is a second-order transition, and for larger $r$ is a first-order transition. For $r<{r}_{\mathrm{max}}$, the critical threshold increases linearly with $r$ from 0.593 at $r=0$ and reaches a maximum, 0.738 for $r={r}_{\mathrm{max}}$, and then gradually decreases to 0.683 for $r=\ensuremath{\infty}$. 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.
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