Cascade of failures in coupled network systems with multiple support-dependent relations
(1011.0234)Abstract
We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependent relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support node in the other network. If both networks A and B are Erd\H{o}s-R\'enyi networks, A and B, with (i) sizes $NA$ and $NB$, (ii) average degrees $a$ and $b$, and (iii) $c{AB}_0NB$ support links from network A to B and $c{BA}_0NB$ support links from network B to A, we find that under random attack with removal of fractions $(1-RA)NA$ and $(1-RB)NB$ nodes respectively, the percolating giant components of both networks at the end of the cascading failures, $\muA_\infty$ and $\muB_\infty$, are given by the percolation laws $\muA_\infty = RA [1-\exp{({-c{BA}0\muB\infty})}] [1-\exp{({-a\muA_\infty})}]$ and $\muB_\infty = RB [1-\exp{({-c{AB}0\muA\infty})}] [1-\exp{({-b\muB_\infty})}]$. In the limit of $c{BA}_0 \to \infty$ and $c{AB}_0 \to \infty$, both networks become independent, and the giant components are equivalent to a random attack on a single Erd\H{o}s-R\'enyi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.
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