{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T16:53:24Z","timestamp":1787072004081,"version":"build-2736575974"},"reference-count":44,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T00:00:00Z","timestamp":1787011200000},"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":["62502529"],"award-info":[{"award-number":["62502529"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["62502529"],"award-info":[{"award-number":["62502529"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Accurately identifying critical nodes in complex networks and applying targeted protection strategies significantly enhances network security. Traditional importance metrics rely on a single topological feature and cannot fully capture node influence. Existing multi-attribute fusion methods integrate multiple structural sources but typically use fixed weights or predefined rules, failing to adaptively adjust attribute contributions based on local and global network characteristics, which limits their generalization across diverse networks. To address this, we propose the CRITIC-based Objective Weighting and Multi-Metric Fusion Method (COWMF). COWMF first builds a Graph Attention Network with Virtual Global\u2013Local Integration (GAT-VGL), taking four low-complexity topological metrics, degree centrality (DC), H-index, degree and neighborhood information centrality (DNC), and k-shell, as input. Through a learnable attention mechanism, GAT-VGL adaptively aggregates multi-hop neighborhood information and explicitly incorporates global structural information via a virtual node to achieve whole-graph topological awareness, generating a global influence score with good discriminative power and high computational efficiency. This score is then integrated with DC and DNC into an improved CRITIC-based objective weighting fusion scheme, enabling adaptive synergy among local connectivity, semi-local radiation, and global structure. Experiments on six real-world networks of varying types and scales show that COWMF demonstrates relatively stable and competitive performance in both simulated attack and susceptible-infected-recovered (SIR) spreading simulations, two complementary experiments, demonstrating satisfactory disruptive capability and propagation influence. Its importance scores exhibit high monotonicity across all networks, with good discriminative power.<\/jats:p>","DOI":"10.3390\/e28080927","type":"journal-article","created":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T16:12:11Z","timestamp":1787069531000},"page":"927","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Objective Multi-Metric Fusion for Critical Node Identification via CRITIC and Global\u2013Local Context Modeling"],"prefix":"10.3390","volume":"28","author":[{"given":"Pengcheng","family":"Cai","sequence":"first","affiliation":[{"name":"College of Electronic Engineering of NUDT, Hefei 230037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Canjv","family":"Lu","sequence":"additional","affiliation":[{"name":"College of Electronic Engineering of NUDT, Hefei 230037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ying","family":"Huang","sequence":"additional","affiliation":[{"name":"College of Electronic Engineering of NUDT, Hefei 230037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Xie","sequence":"additional","affiliation":[{"name":"College of Electronic Engineering of NUDT, Hefei 230037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/j.eswa.2023.122171","article-title":"An improved gravity centrality for finding important nodes in multi-layer networks based on multi-PageRank","volume":"238","author":"Lv","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"116151","DOI":"10.1016\/j.chaos.2025.116151","article-title":"Finding influential nodes via graph embedding and hybrid centrality in complex networks","volume":"194","author":"Ullah","year":"2025","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"16622","DOI":"10.1109\/TITS.2022.3163756","article-title":"Traffic node importance evaluation based on clustering in represented transportation networks","volume":"23","author":"Huang","year":"2022","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1093\/comnet\/cnu016","article-title":"Multilayer networks","volume":"2","author":"Arenas","year":"2014","journal-title":"J. Complex Netw."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2119","DOI":"10.1109\/TNSM.2022.3151552","article-title":"Software defined multicast for large-scale multi-layer leo satellite networks","volume":"19","author":"Hu","year":"2022","journal-title":"IEEE Trans. Netw. Serv. Manag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1677","DOI":"10.1007\/s00500-025-10471-9","article-title":"k-InfNode: Ranking top-k influential nodes in complex networks with random walk","volume":"29","author":"Hasan","year":"2025","journal-title":"Soft Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"121154","DOI":"10.1016\/j.eswa.2023.121154","article-title":"Cagm: A communicability-based adaptive gravity model for influential nodes identification in complex networks","volume":"235","author":"Xu","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"129123","DOI":"10.1016\/j.neucom.2024.129123","article-title":"Adaptive opinion dynamics over community networks when agents cannot express opinions freely","volume":"618","author":"Peng","year":"2025","journal-title":"Neurocomputing"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1186\/s40537-025-01159-w","article-title":"Identifying influential nodes in weighted complex networks by considering the importance of shortest paths","volume":"12","author":"Wang","year":"2025","journal-title":"J. Big Data"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"108015","DOI":"10.1016\/j.epsr.2022.108015","article-title":"Importance evaluation of power network nodes based on community division and characteristics of coupled network","volume":"209","author":"Liu","year":"2022","journal-title":"Electr. Power Syst. Res."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zhu, D., Wang, H., Wang, R., Duan, J., and Bai, J. (2022). Identification of key nodes in a power grid based on modified PageRank algorithm. Energies, 15.","DOI":"10.3390\/en15030797"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"130256","DOI":"10.1016\/j.physa.2024.130256","article-title":"The collaborative role of K-Shell and PageRank for identifying influential nodes in complex networks","volume":"658","author":"Esfandiari","year":"2025","journal-title":"Phys. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1007\/s10796-022-10242-z","article-title":"Fake news on social media: The impact on society","volume":"26","author":"Olan","year":"2024","journal-title":"Inf. Syst. Front."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"110177","DOI":"10.1016\/j.engappai.2025.110177","article-title":"A novel semi-local centrality to identify influential nodes in complex networks by integrating multidimensional factors","volume":"145","author":"Zhang","year":"2025","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"115843","DOI":"10.1016\/j.chaos.2024.115843","article-title":"Development of a multidimensional centrality metric for ranking nodes in complex networks","volume":"191","author":"Meng","year":"2025","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"119928","DOI":"10.1016\/j.eswa.2023.119928","article-title":"Vital spreaders identification synthesizing cross entropy and information entropy with Kshell method","volume":"224","author":"Tong","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"110163","DOI":"10.1016\/j.knosys.2022.110163","article-title":"Ranking influential spreaders based on both node k-shell and structural hole","volume":"260","author":"Zhao","year":"2023","journal-title":"Knowl.-Based Syst."},{"key":"ref_18","first-page":"238","article-title":"Centrality in social networks: Conceptual clarification","volume":"1","author":"Freeman","year":"2002","journal-title":"Soc. Netw."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"130518","DOI":"10.1016\/j.physa.2025.130518","article-title":"Distance-parameterized h-index gravity model for influential node identification in complex networks","volume":"666","author":"Yu","year":"2025","journal-title":"Phys. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.socnet.2004.11.009","article-title":"A measure of betweenness centrality based on random walks","volume":"27","author":"Newman","year":"2005","journal-title":"Soc. Netw."},{"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":"25","DOI":"10.1007\/s11235-024-01240-4","article-title":"Identifying influential nodes in complex networks: A semi-local centrality measure based on augmented graph and average shortest path theory","volume":"88","author":"Pan","year":"2025","journal-title":"Telecommun. Syst."},{"key":"ref_23","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. Oper. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1080\/0952813X.2023.2217831","article-title":"Community based influencer node identification using hybrid optimisation algorithm in social networks","volume":"37","author":"Devi","year":"2025","journal-title":"J. Exp. Theor. Artif. Intell."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1007\/BF02289527","article-title":"The Centrality Index of a Graph","volume":"31","author":"Sabidussi","year":"1966","journal-title":"Psychometrika"},{"key":"ref_26","unstructured":"Miorandi, D., and De Pellegrini, F. (2010). K-shell decomposition for dynamic complex networks. Proceedings of the 8th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks, Avignon, France, 31 May\u20134 June 2010, IEEE."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1750014","DOI":"10.1142\/S0129183117500140","article-title":"Identifying the most influential spreaders in complex networks by an Extended Local K-Shell Sum","volume":"28","author":"Yang","year":"2017","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1170","DOI":"10.1086\/228631","article-title":"Power and centrality: A family of measures","volume":"92","author":"Bonacich","year":"1987","journal-title":"Am. J. Sociol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s10462-024-10725-2","article-title":"A new semi-local centrality for identifying influential nodes based on local average shortest path with extended neighborhood","volume":"57","author":"Xiao","year":"2024","journal-title":"Artif. Intell. Rev."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Zhao, N., Yang, S., Wang, H., Zhou, X., Luo, T., and Wang, J. (2024). A Novel Method to Identify Key Nodes in Complex Networks Based on Degree and Neighborhood Information. Appl. Sci., 14.","DOI":"10.3390\/app14020521"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Fu, L., Ma, X., Dou, Z., Bai, Y., and Zhao, X. (2024). Key Node Identification Method Based on Multilayer Neighbor Node Gravity and Information Entropy. Entropy, 26.","DOI":"10.3390\/e26121041"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"10215","DOI":"10.1038\/s41598-026-40209-y","article-title":"Identifying influential nodes through hierarchical k-shell and extended neighborhood integration","volume":"16","author":"Wang","year":"2026","journal-title":"Sci. Rep."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"93711","DOI":"10.1109\/ACCESS.2024.3424834","article-title":"Quantifying node influence in networks: Isolating-betweenness centrality for improved ranking","volume":"12","author":"Chiranjeevi","year":"2024","journal-title":"IEEE Access"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Esfandiari, S. (May, January 28). A Multi-Attribute Model to Identify Influential Nodes in Complex Networks: Addressing the K-Shell Limitations. Proceedings of the Companion Proceedings of the ACM on Web Conference 2025, Sydney, Australia.","DOI":"10.1145\/3701716.3735086"},{"key":"ref_35","unstructured":"Veli\u010dkovi\u0107, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., and Bengio, Y. (2017). Graph attention networks. arXiv."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Qu, D.-Z., and Ma, Y.-F. (2025, January 7\u20139). AMON-Net: Integrating Graph Attention and Modularity Refinement for Community Detection in Complex Networks. Proceedings of the 2025 International Conference on Artificial Intelligence, Computer, Data Sciences and Applications (ACDSA), Antalya, Turkiye.","DOI":"10.1109\/ACDSA65407.2025.11166445"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Yuan, L., Jiang, Y., Wang, Q., Hu, C., Li, G., and Zheng, C.H. (2026). DeepSGE: Predicting spatial gene expression using residual network with efficient channel attention and dynamic graph attention network. BMC Genom., 27.","DOI":"10.1186\/s12864-026-12568-3"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Wei, Z.-L., An, H.-Y., Yao, Y., Su, W.-C., Li, G., Sun, B.-F., and Wang, M.-J.-S. (2025). FSTGAT: Financial Spatio-Temporal Graph Attention Network for Non-Stationary Financial Systems and Its Application in Stock Price Prediction. Symmetry, 17.","DOI":"10.3390\/sym17081344"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2550037","DOI":"10.1142\/S0129065725500376","article-title":"Enhanced graph attention network by integrating transformer for epileptic EEG identification","volume":"35","author":"Xie","year":"2025","journal-title":"Int. J. Neural Syst."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Chang, H., Rong, Y., Xu, T., Huang, W., Sojoudi, S., Huang, J., and Zhu, W. (2021). Spectral graph attention network with fast eigen-approximation. Proceedings of the 30th ACM International Conference on Information & Knowledge Management, Virtual Event, Australia, Association for Computing Machinery.","DOI":"10.1145\/3459637.3482187"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1142\/S1793005726500134","article-title":"A Stochastic SIR Epidemic with Nonlinear Power Functions and Logistic Growth","volume":"22","author":"Aznague","year":"2026","journal-title":"New Math. Nat. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1007\/s11538-026-01656-w","article-title":"Identifiability in epidemic models with prior immunity and under-reporting","volume":"88","author":"Favero","year":"2026","journal-title":"Bull. Math. Biol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"4531","DOI":"10.1109\/TCYB.2022.3207878","article-title":"A Learning Convolutional Neural Network Approach for Network Robustness Prediction","volume":"53","author":"Lou","year":"2023","journal-title":"IEEE Trans. Cybern."},{"key":"ref_44","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."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/927\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T16:13:47Z","timestamp":1787069627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/927"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,18]]},"references-count":44,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["e28080927"],"URL":"https:\/\/doi.org\/10.3390\/e28080927","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,18]]}}}