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. 2025 Aug 21;16:7809. doi: 10.1038/s41467-025-63001-4

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© The Author(s) 2025

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, 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 licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence 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 licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.

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Fig. 4 — a, c, e, g, i, k Joint distributions of node flow and degree for Nets 1–6, and the Black Swan nodes are emphasized with triangles outlined in black. b, d, f, h, j, l The critical percolation thresholds under four scenarios for Nets 1–6. “O” represents the origin network without any enhancement strategy, “C” denotes the CML-based strategy, “F” signifies the flow-based strategy, “D” corresponds to the degree-based strategy, and “Rd” represents the random-based strategy. a, c, e, g Nets 1–4 exhibit analogous characteristics in their joint flow-degree distributions. i, k Nets 5 and 6 also manifest comparable joint flow-degree distribution characteristics. b, d, f, h Within Nets 1–4, the flow-based strategy and degree-based strategy show parallel performance. j and l In Nets 5 and 6, the performance of the CML-based strategy aligns closely with that of the flow-based strategy.