Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network1,2,3,4,5,6,7,8,9,10,11,12,13,14. Modern systems are coupled together15,16,17,18,19 and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures (‘concurrent malfunction’) is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations20. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.
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We thank R. Burk for discussions that led the authors to focus on the interesting new scientific principles governing the catastrophic collapse of coupled networks. We also thank V. Rosato for providing the Italy 2003 blackout data. S.V.B. thanks the Office of Academic Affairs of Yeshiva University for funding the Yeshiva University high-performance computer cluster and acknowledges the partial support of this research through the Dr. Bernard W. Gamson Computational Science Center at Yeshiva College. S.V.B., G.P. and H.E.S. thank the Office of Naval Research for support. S.H. thanks the European EPIWORK project and the Israel Science Foundation for financial support. We thank E. Leicht and R. de Souza for discussing their unpublished work with us.
Author Contributions S.V.B., R.P., G.P., H.E.S. and S.H. all participated equally in the conceptual design of the model, the theoretical analysis, the computer simulations and the writing of the paper.
This file contains Supplementary Data, Supplementary References and Supplementary Figures 1- 4 with legends.
About this article
Nature Physics (2019)