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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. 1 — a In Scenario 1, under an intact network topology, this flow allocation scheme causes slight congestion, resulting in a declined flow delivery function. b Scenario 2 maintains the same network structure as Scenario 1 but adopts a different flow allocation scheme, and no congestion is observed. c After one edge is damaged in Scenario 1, although the network remains connected, capacity limitations under this flow allocation scheme lead to severe congestion, which significantly degrades the flow delivery function. d Scenario 2 experiences more severe damage, resulting in a disconnected network, yet no congestion occurs. This indicates a disparity between the structural resilience and dynamical resilience of the FW network. e–g During disturbances, the FW network experiences flow redistribution and disturbance propagation, ultimately leading to cascading failures. h Initial state of the FW network before hypergraph construction. i Small-scale cascading failures occur after certain nodes in the network are disturbed. j Coupled effects emerge from small-scale cascading failures. k The network is divided into several distinct clusters. l The emergence of the Black Swan node results in global-scale failures. m Matrix representation of the hypergraph. n Black swan node in the hypergraph causes network-wide impact. o Protecting the Black Swan node from failure, the hypergraph remains connected. p The removal of a key hyperedge leads to a percolation transition in the hypergraph. q The distribution of hypergraph cardinality. r Percolation transition process on the hypergraph. s Forms of hyper-motifs existing in the network post-percolation transition. Hyper-motifs composed of more hyperedges can be represented by hyper-motifs formed by pairwise combinations. Therefore, this study only considers hyper-motifs formed by pairwise combinations.