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The extreme vulnerability of interdependent spatially embedded networks

Abstract

Recent studies show that in interdependent networks a very small failure in one network may lead to catastrophic consequences. Above a critical fraction of interdependent nodes, even a single node failure can invoke cascading failures that may abruptly fragment the system, whereas below this critical dependency a failure of a few nodes leads only to a small amount of damage to the system. So far, research has focused on interdependent random networks without space limitations. However, many real systems, such as power grids and the Internet, are not random but are spatially embedded. Here we analytically and numerically study the stability of interdependent spatially embedded networks modelled as lattice networks. Surprisingly, we find that in lattice systems, in contrast to non-embedded systems, there is no critical dependency and any small fraction of interdependent nodes leads to an abrupt collapse. We show that this extreme vulnerability of very weakly coupled lattices is a consequence of the critical exponent describing the percolation transition of a single lattice.

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Figure 1: A system of interdependent networks is characterized by the structure (dimension) of the single networks as well as by the coupling between the networks.
Figure 2: Schematic solution of the critical point of coupled lattices and coupled RR networks.
Figure 3: Percolation transition of interdependent lattices compared with interdependent random networks.
Figure 4: The size of the abrupt collapse in coupled lattices compared with coupled random networks.
Figure 5: Mutual percolation transition in spatially embedded real-world systems.

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Acknowledgements

We acknowledge the European EPIWORK and MULTIPLEX (EU-FET project 317532) projects, the Deutsche Forschungsgemeinschaft (DFG), the Israel Science Foundation, ONR and DTRA for financial support.

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Authors

Contributions

A.B., Y.B., S.V.B. and S.H. conceived and designed the research. Y.B. carried out the numerical simulations. A.B. developed the theory and wrote the paper with contributions from all other authors.

Corresponding author

Correspondence to Amir Bashan.

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The authors declare no competing financial interests.

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Bashan, A., Berezin, Y., Buldyrev, S. et al. The extreme vulnerability of interdependent spatially embedded networks. Nature Phys 9, 667–672 (2013). https://doi.org/10.1038/nphys2727

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