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. 2019 Jul 23;9:10692. doi: 10.1038/s41598-019-47119-2

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© The Author(s) 2019

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Abrupt functioning efficiency collapse and the ‘connected but inefficient’ network state. Step by step strong links removal simulation on a sub-network (N = 25) of the US Airports and three nodes removal on the C. Elegans real-world complex weighted network. The total flow TF and the weighted efficiency Eff readily decrease under the removal of links with higher weight and nodes with higher strength. (a) Initial sub-network of 25 nodes and 153 links drew from US Airports network. The weight of the links identifies the number of passengers flowing from the connected airports; the weights ranges from a minimum of 9 to maximum of 2.253.992 passengers per year. For the illustrative example we maintain only the strongest links with weight >10⁶ (26 strongest dark grey links) and <380.000 passengers per year (127 weakest soft grey links). (b) The sub-network after the removal of 16 links of highest weight (10% of the links). (c) The sub-network after the removal of 26 links of highest weight (17% of the links). (d) C. Elegans real-world complex weighted network representing nodes-neurons and the links connections number among them (link weights). Dark grey are the strongest and soft grey are the weakest links. The three red nodes of higher strength are outlined in the center where the others nodes are in the circle layout. (e) C. Elegans network after the removal of the first highest strength node. (f) Neuronal network of C. Elegans following the removal of the three main nodes.