{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T21:54:42Z","timestamp":1779918882746,"version":"3.53.1"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T00:00:00Z","timestamp":1739923200000},"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":["71971150"],"award-info":[{"award-number":["71971150"]}],"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":["Xq16B05"],"award-info":[{"award-number":["Xq16B05"]}],"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":["SXYPY202313"],"award-info":[{"award-number":["SXYPY202313"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Project of Research Center for System Sciences and Enterprise Development","award":["71971150"],"award-info":[{"award-number":["71971150"]}]},{"name":"Project of Research Center for System Sciences and Enterprise Development","award":["Xq16B05"],"award-info":[{"award-number":["Xq16B05"]}]},{"name":"Project of Research Center for System Sciences and Enterprise Development","award":["SXYPY202313"],"award-info":[{"award-number":["SXYPY202313"]}]},{"name":"Fundamental Research Funds for the Central Universities of China","award":["71971150"],"award-info":[{"award-number":["71971150"]}]},{"name":"Fundamental Research Funds for the Central Universities of China","award":["Xq16B05"],"award-info":[{"award-number":["Xq16B05"]}]},{"name":"Fundamental Research Funds for the Central Universities of China","award":["SXYPY202313"],"award-info":[{"award-number":["SXYPY202313"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Systems"],"abstract":"<jats:p>This study addresses the critical challenge of identifying key nodes in complex networks, an essential task for optimizing network stability, efficiency, and resilience. Traditional approaches often rely on single-dimensional metrics, which may fail to fully capture the multifaceted roles that nodes play in maintaining network functionality. To overcome this limitation, a novel framework was proposed, integrating the C3 dimensions (Cohesion, Connectivity, Conciseness) with the TOPSIS method and Pareto dominated set (PDS) to enable a comprehensive, multi-dimensional evaluation of node importance. The method introduces an optimized parameter \u03b1 for relative conciseness, validated through area-under-curve (AUC) minimization, ensuring adaptability across diverse networks. Scalability analysis demonstrates its feasibility for large-scale systems, with computational complexity managed through approximation algorithms. The method was applied to eight real-world networks including transportation systems, power grids, and social networks. Results demonstrated that the PDS-based STS method outperformed traditional centrality measures, such as degree centrality and betweenness centrality, particularly during the early stages of network degradation. The framework effectively identified critical nodes in highly connected systems, with the conciseness metric proving instrumental in highlighting irreplaceable nodes whose removal would severely disrupt network functionality. This study concludes that the \u201cC3-TOPSIS-Pareto\u201d based model provides a more accurate and robust approach for critical node identification, offering a reliable tool for enhancing network design and resilience, particularly in systems with complex interdependencies. Limitations in fragmented networks are discussed, with future directions proposed for dynamic weight adaptation and functional dynamics integration.<\/jats:p>","DOI":"10.3390\/systems13020138","type":"journal-article","created":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T11:49:27Z","timestamp":1739965767000},"page":"138","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A \u201cC3-TOPSIS-Pareto\u201d Based Model for Identifying Critical Nodes in Complex Networks"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1782-2804","authenticated-orcid":false,"given":"Ziqiang","family":"Zeng","sequence":"first","affiliation":[{"name":"Uncertain Decision Making Laboratory, Business School, Sichuan University, Chengdu 610065, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Weiye","family":"Zhang","sequence":"additional","affiliation":[{"name":"Uncertain Decision Making Laboratory, Business School, Sichuan University, Chengdu 610065, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongling","family":"Jin","sequence":"additional","affiliation":[{"name":"Uncertain Decision Making Laboratory, Business School, Sichuan University, Chengdu 610065, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.inffus.2021.02.001","article-title":"The fractal dimension of complex networks: A review","volume":"73","author":"Wen","year":"2021","journal-title":"Inf. Fusion"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"eabm8310","DOI":"10.1126\/sciadv.abm8310","article-title":"Network structural origin of instabilities in large complex systems","volume":"8","author":"Duan","year":"2022","journal-title":"Sci. Adv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1038\/nature15737","article-title":"Influence maximization in complex networks through optimal percolation","volume":"527","author":"Morone","year":"2015","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1038\/s42256-020-0177-2","article-title":"Finding key players in complex networks through deep reinforcement learning","volume":"2","author":"Fan","year":"2020","journal-title":"Nat. Mach. Intell."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"115348","DOI":"10.1016\/j.chaos.2024.115348","article-title":"Strategic node identification in complex network dynamics","volume":"187","author":"Nikougoftar","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1080\/0022250X.1972.9989806","article-title":"Factoring and weighting approaches to status scores and clique identification","volume":"2","author":"Bonacich","year":"1972","journal-title":"J. Math. Sociol."},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.physa.2013.10.047","article-title":"Identifying and ranking influential spreaders in complex networks by neighborhood coreness","volume":"395","author":"Bae","year":"2014","journal-title":"Phys. A-Stat. Mech. Its Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"126170","DOI":"10.1016\/j.physa.2021.126170","article-title":"Identification of critical nodes in multimodal transportation network","volume":"580","author":"Wang","year":"2021","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1007\/BF02289527","article-title":"Centrality index of a graph","volume":"31","author":"Sabidussi","year":"1966","journal-title":"Psychometrika"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1016\/j.socnet.2007.04.002","article-title":"Some unique properties of eigenvector centrality","volume":"29","author":"Bonacich","year":"2007","journal-title":"Soc. Netw."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.ins.2011.12.027","article-title":"Laplacian centrality: A new centrality measure for weighted networks","volume":"194","author":"Qi","year":"2012","journal-title":"Inf. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"19","DOI":"10.4236\/sn.2013.21003","article-title":"Terrorist networks, network energy and node removal: A new measure of centrality based on Laplacian energy","volume":"2","author":"Qi","year":"2013","journal-title":"Soc. Netw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1016\/j.physa.2019.04.205","article-title":"General link prediction with influential node identification","volume":"523","author":"Wu","year":"2019","journal-title":"Phys. A-Stat. Mech. Its Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Sadhu, S., Bhuiya, A., and Dutta, A. (2023, January 26\u201329). DSGCN: A Degree Strength Graph Convolution Network for Identifying Influential Nodes in Complex Networks. Proceedings of the 2023 IEEE\/WIC International Conference on Web Intelligence and Intelligent Agent Technology (WI-IAT), Venice, Italy.","DOI":"10.1109\/WI-IAT59888.2023.00053"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Yu, Y., Zhou, B., Chen, L., Gao, T., and Liu, J. (2022). Identifying Important Nodes in Complex Networks Based on Node Propagation Entropy. Entropy, 24.","DOI":"10.3390\/e24020275"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Jiang, Y., Liu, L., and Shu, J. (2023, January 24\u201326). Overview of Key Node Evaluation in Complex Networks. Proceedings of the 2023 3rd International Conference on Intelligent Communications and Computing (ICC), Nanchang, China.","DOI":"10.1109\/ICC59986.2023.10421031"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zeng, Z., Sun, Y., and Zhang, X. (2024). Entropy-based node importance identification method for public transportation infrastructure coupled networks: A case study of Chengdu. Entropy, 26.","DOI":"10.3390\/e26020159"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1016\/j.physa.2011.09.017","article-title":"Identifying influential nodes in complex networks","volume":"391","author":"Chen","year":"2012","journal-title":"Phys. A-Stat. Mech. Its Appl."},{"key":"ref_20","first-page":"9376","article-title":"Identifying the influential nodes in complex social networks using centrality-based approach","volume":"34","author":"Ishfaq","year":"2022","journal-title":"J. King Saud Univ.-Comput. Inf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Boccaletti, S., Bianconi, G., Criado, R., del Genio, C.I., G\u00f3mez-Garde\u00f1es, J., Romance, M., Sendi\u00f1a-Nadal, I., Wang, Z., and Zanin, M. (2014). The structure and dynamics of multilayer networks. arXiv.","DOI":"10.1016\/j.physrep.2014.07.001"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2350009","DOI":"10.1142\/S0219525923500091","article-title":"Identifying vital nodes in complex network by considering multiplex influences","volume":"26","author":"Ren","year":"2023","journal-title":"Adv. Complex Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"115778","DOI":"10.1016\/j.eswa.2021.115778","article-title":"Identifying vital nodes from local and global perspectives in complex networks","volume":"186","author":"Ullah","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.06.007","article-title":"Vital nodes identification in complex networks","volume":"650","author":"Lu","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"028902","DOI":"10.1088\/1674-1056\/ad9734","article-title":"Critical station identification of metro networks based on the integrated topological-functional algorithm: A case study of Chengdu","volume":"34","author":"Zeng","year":"2025","journal-title":"Chin. Phys. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6868","DOI":"10.1038\/ncomms7868","article-title":"Ranking in interconnected multilayer networks reveals versatile nodes","volume":"6","author":"Omodei","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1850216","DOI":"10.1142\/S0217984918502160","article-title":"A dynamic weighted TOPSIS method for identifying influential nodes in complex networks","volume":"32","author":"Yang","year":"2018","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Li, Y., Cai, W., Li, Y., and Du, X. (2020). Key Node Ranking in Complex Networks: A Novel Entropy and Mutual Information-Based Approach. Entropy, 22.","DOI":"10.3390\/e22010052"},{"key":"ref_29","unstructured":"Qiao, L., Wu, M., and Zhao, M. (2021, January 10\u201313). Identification of Key Nodes in Complex Networks. Proceedings of the 2021 7th International Conference on Computer and Communications (ICCC), Chengdu, China."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"01N02","DOI":"10.1142\/S0219525924500048","article-title":"IMine: A customizable framework for influence mining in complex networks","volume":"27","author":"Hussain","year":"2024","journal-title":"Adv. Complex Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.ins.2022.10.070","article-title":"A novel method to identify influential nodes in complex networks based on gravity centrality","volume":"618","author":"Zhang","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1177\/01655515241238401","article-title":"Assessing journals through a three-dimensional framework based on article citation, author and institution influence","volume":"50","author":"Zeng","year":"2024","journal-title":"J. Inf. Sci."}],"container-title":["Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-8954\/13\/2\/138\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:38:14Z","timestamp":1760027894000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-8954\/13\/2\/138"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,19]]},"references-count":32,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["systems13020138"],"URL":"https:\/\/doi.org\/10.3390\/systems13020138","relation":{},"ISSN":["2079-8954"],"issn-type":[{"value":"2079-8954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,19]]}}}