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Title: Cascading failures in coupled networks with both inner-dependency and inter-dependency links
Authors: Liu, RR
Li, M
Jia, CX
Wang, BH
Issue Date: 2016
Source: Scientific reports, 4 2016, v. 6, no. , p. 1-10
Abstract: We study the percolation in coupled networks with both inner-dependency and inter-dependency links, where the inner- and inter-dependency links represent the dependencies between nodes in the same or different networks, respectively. We find that when most of dependency links are inner- or inter-ones, the coupled networks system is fragile and makes a discontinuous percolation transition. However, when the numbers of two types of dependency links are close to each other, the system is robust and makes a continuous percolation transition. This indicates that the high density of dependency links could not always lead to a discontinuous percolation transition as the previous studies. More interestingly, although the robustness of the system can be optimized by adjusting the ratio of the two types of dependency links, there exists a critical average degree of the networks for coupled random networks, below which the crossover of the two types of percolation transitions disappears, and the system will always demonstrate a discontinuous percolation transition. We also develop an approach to analyze this model, which is agreement with the simulation results well.
Publisher: Nature Publishing Group
Journal: Scientific reports 
EISSN: 2045-2322
DOI: 10.1038/srep25294
Rights: This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit
© The Author(s) 2016
The following publication Liu, R.-R. et al. Cascading failures in coupled networks with both inner-dependency and inter-dependency links. Sci. Rep. 6, 25294 (2016) is available at
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