{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T21:58:07Z","timestamp":1761429487286,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T00:00:00Z","timestamp":1703116800000},"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":["62071240","BK20231142"],"award-info":[{"award-number":["62071240","BK20231142"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Jiangsu Province","award":["62071240","BK20231142"],"award-info":[{"award-number":["62071240","BK20231142"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Malicious attacks can cause significant damage to the structure and functionality of complex networks. Previous research has pointed out that the ability of networks to withstand malicious attacks becomes weaker when networks are coupled. However, traditional research on improving the robustness of networks has focused on individual low-order or higher-order networks, lacking studies on coupled networks with higher-order and low-order networks. This paper proposes a method for optimizing the robustness of coupled networks with higher-order and low-order based on a simulated annealing algorithm to address this issue. Without altering the network\u2019s degree distribution, the method rewires the edges, taking the robustness of low-order and higher-order networks as joint optimization objectives. Making minimal changes to the network, the method effectively enhances the robustness of coupled networks. Experiments were conducted on Erd\u0151s\u2013R\u00e9nyi random networks (ER), scale-free networks (BA), and small-world networks (SW). Finally, validation was performed on various real networks. The results indicate that this method can effectively enhance the robustness of coupled networks with higher-order and low-order.<\/jats:p>","DOI":"10.3390\/e26010008","type":"journal-article","created":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T05:36:12Z","timestamp":1703136972000},"page":"8","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Improving Robustness of High-Low-Order Coupled Networks against Malicious Attacks Based on a Simulated Annealing Algorithm"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4458-5843","authenticated-orcid":false,"given":"Chengjun","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Computer Science, Nanjing University of Information Science and Technology, Nanjing 210044, China"},{"name":"Wuxi Institute of Technology, Nanjing University of lnformation Science & Technology, Wuxi 214000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yifan","family":"Xie","sequence":"additional","affiliation":[{"name":"School of Computer Science, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4448-2617","authenticated-orcid":false,"given":"Yadang","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Computer Science, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4786-4036","authenticated-orcid":false,"given":"Wenbin","family":"Yu","sequence":"additional","affiliation":[{"name":"Wuxi Institute of Technology, Nanjing University of lnformation Science & Technology, Wuxi 214000, China"},{"name":"School of Software, Nanjing University of Information Science and Technology, Nanjing 210044, China"},{"name":"Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CI-CAEET), Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gaofeng","family":"Xiang","sequence":"additional","affiliation":[{"name":"School of Computer Science, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peijun","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Computer Science, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7395-0873","authenticated-orcid":false,"given":"Yi","family":"Lei","sequence":"additional","affiliation":[{"name":"School of Computer Science, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7308","DOI":"10.1038\/s41467-022-34634-6","article-title":"Geometrical congruence, greedy navigability and myopic transfer in complex networks and brain connectomes","volume":"13","author":"Cannistraci","year":"2022","journal-title":"Nat. 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