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. 2022 Jun 2;13:3088. doi: 10.1038/s41467-022-30705-w

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© The Author(s) 2022

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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Fig. 7 — a Heatmap of Pearson correlation between local dimension and node infectiousness for small-world graph (n = 100, average degree k = 10, probability of rewiring p = 0.015). The black line is the average proportion of infected nodes given a single-seed node for a given infection probability β. The transition from low to high proportion of infected nodes indicates the critical point while the dashed line is the maximum correlation for each β. b We vary the probability of rewiring edges p of small world graphs and display i the diffusion time that maximises the correlation between local dimension and infectiousness for varying β, and ii the associated correlation coefficient. The correlation is near one close to criticality and above 0.8 for a large range of β. We repeat the analysis in a for c a Delaunay grid graph (n = 400) and d the European powergrid network to observe a similar linear relationships between scale and infection probability β prior to criticality.