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. 2017 Nov 8;7:15059. doi: 10.1038/s41598-017-14384-y

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© The Author(s) 2017

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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Order parameters for the “mass rule”, for a system of networks with internal distribution P _A(k) = P _B(k) = δ _k,5, supply distributions P _sA(k) = P _sB(k) = δ _k,2 and thresholds k _A ^* = k _B ^* = 1. All the components of size h = 1 are deleted (q(1) = 1), and all the components of size h ≥ 3 are preserved (q(3) = q(4) = ... = q(h _max) = 0 where h _max is the maximum value of h). The curves represent the case q(2) = 0.3 (●, ), for which there is a continuous transition, and q(2) = 0.85 (, ), which leads to an abrupt breakdown of the order parameter. The dashed lines represent the theoretical results and the symbols the stochastic simulations. (a) Network A, (b) Network B.

Inline graphic — Order parameters for the “mass rule”, for a system of networks with internal distribution P _A(k) = P _B(k) = δ _k,5, supply distributions P _sA(k) = P _sB(k) = δ _k,2 and thresholds k _A ^* = k _B ^* = 1. All the components of size h = 1 are deleted (q(1) = 1), and all the components of size h ≥ 3 are preserved (q(3) = q(4) = ... = q(h _max) = 0 where h _max is the maximum value of h). The curves represent the case q(2) = 0.3 (●, ), for which there is a continuous transition, and q(2) = 0.85 (, ), which leads to an abrupt breakdown of the order parameter. The dashed lines represent the theoretical results and the symbols the stochastic simulations. (a) Network A, (b) Network B.