{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T21:43:57Z","timestamp":1784843037224,"version":"3.55.0"},"reference-count":44,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T00:00:00Z","timestamp":1733702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Correctly identifying influential nodes in a complex network and implementing targeted protection measures can significantly enhance the overall security of the network. Currently, indicators such as degree centrality, closeness centrality, betweenness centrality, H-index, and K-shell are commonly used to measure node influence. Although these indicators can identify critical nodes to some extent, they often consider node attributes from a narrow perspective and have certain limitations. Therefore, evaluating the importance of nodes using most existing indicators remains incomplete. In this paper, we propose the multi-attribute CRITIC-TOPSIS network decision indicator, or MCTNDI, which integrates closeness centrality, betweenness centrality, H-index, and network constraint coefficients to identify critical nodes in a network. This indicator combines information from multiple perspectives, including local neighborhood importance, network topological location, path centrality, and node mutual information, thereby solving the issue of the one-sided perspective of single indicators and providing a more comprehensive measure of node importance. Additionally, MCTNDI is validated through the analysis of several real-world networks, including the Contiguous USA network, Dolphins network, USAir97 network, and Tech-routers-rf network. The validation is conducted from four aspects: the results of simulated network attacks, the distribution of node importance, the monotonicity of rankings, and the similarity of indicators, illustrating MCTNDI\u2019s effectiveness in real networks.<\/jats:p>","DOI":"10.3390\/e26121075","type":"journal-article","created":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T11:16:48Z","timestamp":1733743008000},"page":"1075","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["A Multi-Attribute Decision-Making Approach for Critical Node Identification in Complex Networks"],"prefix":"10.3390","volume":"26","author":[{"given":"Xinyun","family":"Zhao","sequence":"first","affiliation":[{"name":"Electronic Engineering Institute, National University of Defense Technology, Hefei 230037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongheng","family":"Zhang","sequence":"additional","affiliation":[{"name":"Electronic Engineering Institute, National University of Defense Technology, Hefei 230037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qingying","family":"Zhai","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Hefei University of Technology, Hefei 230601, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinrui","family":"Zhang","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Hefei University of Technology, Hefei 230601, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lanlan","family":"Qi","sequence":"additional","affiliation":[{"name":"Electronic Engineering Institute, National University of Defense Technology, Hefei 230037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ullah, A., Wang, B., Sheng, J., Long, J., Khan, N., and Sun, Z. (2021). Identification of nodes influence based on global structure model in complex networks. Sci. Rep., 11.","DOI":"10.1038\/s41598-021-84684-x"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1109\/MCAS.2003.1228503","article-title":"Complex networks: Small-world, scale-free and beyond","volume":"3","author":"Wang","year":"2003","journal-title":"Circuits Syst. Mag. IEEE"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of Scaling in Random Networks","volume":"286","author":"Barabasi","year":"1999","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/S0010-4655(02)00201-1","article-title":"The Structure and Function of Complex Networks","volume":"147","author":"Newman","year":"2003","journal-title":"Comput. Phys. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"134943","DOI":"10.1016\/j.jclepro.2022.134943","article-title":"How to improve the effectiveness of Chinese green certificate market? A complex network and social influence analysis","volume":"380","author":"Zheng","year":"2022","journal-title":"J. Clean. Prod."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2020.12.003","article-title":"Percolation on complex networks: Theory and application","volume":"907","author":"Li","year":"2021","journal-title":"Phys. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Li, A., Hu, Q., Liu, J., and Pan, Y. (2016). Resistance and Security Index of Networks: Structural Information Perspective of Network Security. Sci. Rep., 6.","DOI":"10.1038\/srep26810"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/S0004-3702(02)00262-X","article-title":"Emergence of social conventions in complex networks","volume":"141","author":"Delgado","year":"2002","journal-title":"Artif. Intell."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"e1566","DOI":"10.1002\/wics.1566","article-title":"Community detection in complex networks: From statistical foundations to data science applications","volume":"14","author":"Dey","year":"2021","journal-title":"Wiley Interdiscip. Rev. Comput. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1126\/science.1215842","article-title":"Identifying Influential and Susceptible Members of Social Networks","volume":"337","author":"Aral","year":"2012","journal-title":"Science"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"122529","DOI":"10.1016\/j.apenergy.2023.122529","article-title":"Statistical analysis of the regional air quality index of Yangtze River Delta based on complex network theory","volume":"357","author":"Liu","year":"2024","journal-title":"Appl. Energy"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Lu, Y., Yang, G., and Hang, Z. (2022). Multi-Attribute Decision Making Method for Node Importance Metric in Complex Network. Appl. Sci., 12.","DOI":"10.3390\/app12041944"},{"key":"ref_13","first-page":"1","article-title":"Approximating Betweenness Centrality to Identify Key Nodes in a Weighted Urban Complex Transportation Network","volume":"2019","author":"Liu","year":"2019","journal-title":"J. Adv. Transp."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"109637","DOI":"10.1016\/j.chaos.2020.109637","article-title":"Identifying influential nodes in complex networks from global perspective","volume":"133","author":"Zhao","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Nie, T., Guo, Z., Zhao, K., and Lu, Z.M. (2015). New attack strategies for complex networks. Phys. A Stat. Mech. Its Appl., 424.","DOI":"10.1016\/j.physa.2015.01.004"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_17","first-page":"79","article-title":"Evaluation method for node importance based on node contraction in complex networks","volume":"26","author":"Tan","year":"2006","journal-title":"Xitong Gongcheng Lilun Yu Shijian\/Syst. Eng. Theory Pract."},{"key":"ref_18","first-page":"104","article-title":"Determination of the Hub Nodes in the Emergencies\u2019 Information Dissemination Supernetwork","volume":"25","author":"Peng","year":"2013","journal-title":"Manag. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"017101","DOI":"10.1103\/PhysRevE.67.017101","article-title":"Betweenness centrality correlation in social networks","volume":"67","author":"Goh","year":"2003","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"10168","DOI":"10.1038\/ncomms10168","article-title":"The H-index of a network node and its relation to degree and coreness","volume":"7","author":"Zhou","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1038\/nphys1746","article-title":"Identification of influential spreaders in complex networks","volume":"6","author":"Kitsak","year":"2010","journal-title":"Nat. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"342","DOI":"10.2307\/2393394","article-title":"Assessing the Political Landscape: Structure, Cognition, and Power in Organizations","volume":"35","author":"Krackhardt","year":"1990","journal-title":"Adm. Sci. Q."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5059","DOI":"10.1016\/j.aej.2021.09.031","article-title":"Interval number multi-attribute decision-making method based on TOPSIS","volume":"61","author":"Jiang","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"032138","DOI":"10.1088\/1742-6596\/1168\/3\/032138","article-title":"Research on Importance Evaluation Method of Power Communication Network Node Based on Node Damage Resistance","volume":"1168","author":"Wang","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.ins.2017.10.035","article-title":"Measure of Invulnerability for Command and Control Network based on Mission Link","volume":"426","author":"Yunming","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Qi, Q., Meng, Y., Zhao, X., and Liu, J. (2022). Resilience Assessment of an Urban Metro Complex Network: A Case Study of the Zhengzhou Metro. Sustainability, 14.","DOI":"10.3390\/su141811555"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0378-8733(78)90021-7","article-title":"Centrality in Social Networks\u2019 Conceptual Clarification","volume":"1","author":"Freeman","year":"1979","journal-title":"Soc. Netw."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Xiang, X., Zhu, C., Wang, Q., Zhu, X., and Zhou, Y. (2020). Identifying vital nodes in complex networks by adjacency information entropy. Sci. Rep., 10.","DOI":"10.1038\/s41598-020-59616-w"},{"key":"ref_29","first-page":"1777","article-title":"Identifying influential nodes in complex networks","volume":"391","author":"Chen","year":"2012","journal-title":"Fuel Energy Abstr."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.physa.2014.11.012","article-title":"Weighted k-shell decomposition for complex networks based on potential edge weights","volume":"420","author":"Wei","year":"2015","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"35","DOI":"10.2307\/3033543","article-title":"A Set of Measures of Centrality Based on Betweenness","volume":"40","author":"Freeman","year":"1977","journal-title":"Sociometry"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.tra.2019.07.015","article-title":"Impact of high-speed rail network development on airport traffic and traffic distribution: Evidence from China and Japan","volume":"127","author":"Liu","year":"2019","journal-title":"Transp. Res. Part A Policy Pract."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"101972","DOI":"10.1016\/j.jocs.2023.101972","article-title":"Identifying influential nodes based on resistance distance","volume":"67","author":"Li","year":"2023","journal-title":"J. Comput. Sci."},{"key":"ref_34","first-page":"107","article-title":"The Anatomy of a Large-Scale Hypertextual Web Search Engine","volume":"30","author":"Brin","year":"1998","journal-title":"Comput. Netw."},{"key":"ref_35","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_36","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/S0378-8733(00)00020-4","article-title":"Dynamical systems to define centrality in social networks","volume":"22","author":"Poulin","year":"2000","journal-title":"Soc. Netw."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1016\/0305-0548(94)00059-H","article-title":"Determining objective weights in multiple criteria problems: The CRITIC method","volume":"22","author":"Diakoulaki","year":"1995","journal-title":"Comput. OR"},{"key":"ref_38","first-page":"287","article-title":"Multiple Attribute Decision Making","volume":"404","author":"Hwang","year":"1981","journal-title":"Lect. Notes Econ. Math. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1086\/421787","article-title":"Structural Holes and Good Ideas","volume":"110","author":"Burt","year":"2004","journal-title":"Am. J. Sociol."},{"key":"ref_40","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_41","doi-asserted-by":"crossref","unstructured":"Chen, D., Gao, H., L\u00fc, L., and Zhou, T. (2013). Identifying Influential Nodes in Large-Scale Directed Networks: The Role of Clustering. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0077455"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10618-020-00716-6","article-title":"A survey of community detection methods in multilayer networks","volume":"35","author":"Huang","year":"2021","journal-title":"Data Min. Knowl. Discov."},{"key":"ref_43","first-page":"1","article-title":"Foundations and Modeling of Dynamic Networks Using Dynamic Graph Neural Networks: A Survey","volume":"9","author":"Skardinga","year":"2021","journal-title":"IEEE Access"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3580516","article-title":"Opinion Leaders for Information Diffusion Using Graph Neural Network in Online Social Networks","volume":"17","author":"Jain","year":"2023","journal-title":"ACM Trans. Web"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/12\/1075\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:50:44Z","timestamp":1760115044000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/12\/1075"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,9]]},"references-count":44,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["e26121075"],"URL":"https:\/\/doi.org\/10.3390\/e26121075","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,9]]}}}