{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T09:54:50Z","timestamp":1768730090677,"version":"3.49.0"},"reference-count":69,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,11]],"date-time":"2024-03-11T00:00:00Z","timestamp":1710115200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["T2293771"],"award-info":[{"award-number":["T2293771"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022ZD0211400"],"award-info":[{"award-number":["2022ZD0211400"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"STI 2030\u2013Major Projects","award":["T2293771"],"award-info":[{"award-number":["T2293771"]}]},{"name":"STI 2030\u2013Major Projects","award":["2022ZD0211400"],"award-info":[{"award-number":["2022ZD0211400"]}]},{"name":"New Cornerstone Science Foundation through the XPLORER PRIZE","award":["T2293771"],"award-info":[{"award-number":["T2293771"]}]},{"name":"New Cornerstone Science Foundation through the XPLORER PRIZE","award":["2022ZD0211400"],"award-info":[{"award-number":["2022ZD0211400"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Diverse higher-order structures, foundational for supporting a network\u2019s \u201cmeta-functions\u201d, play a vital role in structure, functionality, and the emergence of complex dynamics. Nevertheless, the problem of dismantling them has been consistently overlooked. In this paper, we introduce the concept of dismantling higher-order structures, with the objective of disrupting not only network connectivity but also eradicating all higher-order structures in each branch, thereby ensuring thorough functional paralysis. Given the diversity and unknown specifics of higher-order structures, identifying and targeting them individually is not practical or even feasible. Fortunately, their close association with k-cores arises from their internal high connectivity. Thus, we transform higher-order structure measurement into measurements on k-cores with corresponding orders. Furthermore, we propose the Belief Propagation-guided Higher-order Dismantling (BPHD) algorithm, minimizing dismantling costs while achieving maximal disruption to connectivity and higher-order structures, ultimately converting the network into a forest. BPHD exhibits the explosive vulnerability of network higher-order structures, counterintuitively showcasing decreasing dismantling costs with increasing structural complexity. Our findings offer a novel approach for dismantling malignant networks, emphasizing the substantial challenges inherent in safeguarding against such malicious attacks.<\/jats:p>","DOI":"10.3390\/e26030248","type":"journal-article","created":{"date-parts":[[2024,3,11]],"date-time":"2024-03-11T06:05:30Z","timestamp":1710137130000},"page":"248","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Network Higher-Order Structure Dismantling"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-5930-0900","authenticated-orcid":false,"given":"Peng","family":"Peng","sequence":"first","affiliation":[{"name":"Institute of Fundamental and Frontier Studies, University of Electronic Science and Technology of China, Chengdu 611731, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9456-6819","authenticated-orcid":false,"given":"Tianlong","family":"Fan","sequence":"additional","affiliation":[{"name":"School of Cyber Science and Technology, University of Science and Technology of China, Hefei 230026, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2156-0432","authenticated-orcid":false,"given":"Linyuan","family":"L\u00fc","sequence":"additional","affiliation":[{"name":"Institute of Fundamental and Frontier Studies, University of Electronic Science and Technology of China, Chengdu 611731, China"},{"name":"School of Cyber Science and Technology, University of Science and Technology of China, Hefei 230026, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1038\/35019019","article-title":"Error and attack tolerance of complex networks","volume":"406","author":"Albert","year":"2000","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1038\/nature14604","article-title":"Influence Maximization in Complex Networks through Optimal Percolation","volume":"524","author":"Morone","year":"2015","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1016\/j.physa.2010.11.027","article-title":"Link prediction in complex networks: A survey","volume":"390","author":"Zhou","year":"2011","journal-title":"Physica A Stat. 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