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. 2019 Jul 12;9:10133. doi: 10.1038/s41598-019-46507-y

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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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The relative difference of the outbreak size Δr as a function of the fraction of initial spreaders f₀. The Degree (a), Betweenness (b), Closeness (c) and Eigenvector (d) centrality measures derived from the Overlapping Modular centrality are compared to the standard counterpart designed for networks with no community structure. Real-world networks with medium community structure (Princeton, Caltech and Georgetown networks) are used. The estimated values of their mixing coefficient is equal respectively to 0.22, 0.26 and 0.3.