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Although several local centrality methods have been proposed, their accuracy may be compromised if interactions between nodes and their neighbors are not fully considered. To address this issue, this paper proposes a key node identification method based on multilayer neighbor node gravity and information entropy (MNNGE). The method works as follows: First, the relative gravity of the nodes is calculated based on their weights. Second, the direct gravity of the nodes is calculated by considering the attributes of neighboring nodes, thus capturing interactions within local triangular structures. Finally, the centrality of the nodes is obtained by aggregating the relative and direct gravity of multilayer neighbor nodes using information entropy. To validate the effectiveness of the MNNGE method, we conducted experiments on various real-world network datasets, using evaluation metrics such as the susceptible-infected-recovered (SIR) model, Kendall \u03c4 correlation coefficient, Jaccard similarity coefficient, monotonicity, and complementary cumulative distribution function. Our results demonstrate that MNNGE can identify key nodes more accurately than other methods, without requiring parameter settings, and is suitable for large-scale complex networks.<\/jats:p>","DOI":"10.3390\/e26121041","type":"journal-article","created":{"date-parts":[[2024,12,2]],"date-time":"2024-12-02T05:43:43Z","timestamp":1733118223000},"page":"1041","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Key Node Identification Method Based on Multilayer Neighbor Node Gravity and Information Entropy"],"prefix":"10.3390","volume":"26","author":[{"given":"Lidong","family":"Fu","sequence":"first","affiliation":[{"name":"College of Computer Science and Technology, Xi\u2019an University of Science and Technology, Xi\u2019an 710064, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-1130-8667","authenticated-orcid":false,"given":"Xin","family":"Ma","sequence":"additional","affiliation":[{"name":"College of Computer Science and Technology, Xi\u2019an University of Science and Technology, Xi\u2019an 710064, China"}]},{"given":"Zengfa","family":"Dou","sequence":"additional","affiliation":[{"name":"School of Computer and Information Science, Qinhai Institute of Technology, Xining 810016, China"}]},{"given":"Yun","family":"Bai","sequence":"additional","affiliation":[{"name":"College of Computer Science and Technology, Xi\u2019an University of Science and Technology, Xi\u2019an 710064, China"}]},{"given":"Xi","family":"Zhao","sequence":"additional","affiliation":[{"name":"Xi\u2019an Big Data Service Center, Xi\u2019an 710064, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"112045","DOI":"10.1016\/j.knosys.2024.112045","article-title":"Mining node attributes for link prediction with a non-negative matrix factorization-based approach","volume":"299","author":"Zhao","year":"2024","journal-title":"Knowl.-Based Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"114849","DOI":"10.1016\/j.chaos.2024.114849","article-title":"High-quality community detection in complex networks based on node influence analysis","volume":"182","author":"Wang","year":"2024","journal-title":"Chaos Soliton. Fract."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"120927","DOI":"10.1016\/j.eswa.2023.120927","article-title":"Finding critical nodes in a complex network from information diffusion and Matthew effect aggregation","volume":"233","author":"Sun","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"120326","DOI":"10.1016\/j.eswa.2023.120326","article-title":"LSS: A locality-based structure system to evaluate the spreader\u2019s importance in social complex networks","volume":"228","author":"Ullah","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1160","DOI":"10.1016\/j.isci.2019.07.043","article-title":"An Algorithmic Information Calculus for Causal Discovery and Reprogramming Systems","volume":"19","author":"Zenil","year":"2019","journal-title":"iScience"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0378-8733(78)90021-7","article-title":"Centrality in social networks: Conceptual clarification","volume":"1","author":"Freeman","year":"1978","journal-title":"Soc. Netw."},{"key":"ref_7","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_8","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_9","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S0378-8733(01)00038-7","article-title":"Eigenvector-like measures of centrality for asymmetric relations","volume":"23","author":"Bonacich","year":"2001","journal-title":"Soc. Netw."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Liu, R., Gao, L., and Wu, J. (2009, January 17\u201319). Key nodes mining in transport networks based on PageRank algorithm. Proceedings of the 2009 Chinese Control and Decision Conference, Guilin, China.","DOI":"10.1109\/CCDC.2009.5192339"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"L\u00fc, L., Zhang, Y.C., Yeung, C.H., and Zhou, T. (2011). Leaders in social networks, the delicious case. PLoS ONE, 6.","DOI":"10.1371\/journal.pone.0021202"},{"key":"ref_12","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_13","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."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"112859","DOI":"10.1016\/j.eswa.2019.112859","article-title":"Weighted k-shell degree neighborhood: A new method for identifying the influential spreaders from a variety of complex network connectivity structures","volume":"139","author":"Namtirtha","year":"2020","journal-title":"Expert Syst. Appl."},{"key":"ref_15","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_16","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."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1016\/j.physleta.2013.02.039","article-title":"Ranking spreaders by decomposing complex networks","volume":"377","author":"Zeng","year":"2013","journal-title":"Phys. Lett. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.physa.2016.05.048","article-title":"Fast ranking influential nodes in complex networks using a k-shell iteration factor","volume":"461","author":"Wang","year":"2016","journal-title":"Phys. A Stat. Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"107198","DOI":"10.1016\/j.knosys.2021.107198","article-title":"An improved gravity model to identify influential nodes in complex networks based on k-shell method","volume":"227","author":"Yang","year":"2021","journal-title":"Knowl.-Based Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1016\/j.chaos.2017.09.010","article-title":"Influential nodes ranking in complex networks: An entropy-based approach","volume":"104","author":"Zareie","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"124229","DOI":"10.1016\/j.physa.2020.124229","article-title":"Identifying influential spreaders in complex networks based on improved k-shell method","volume":"554","author":"Wang","year":"2020","journal-title":"Phys. A Stat. Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6985650","DOI":"10.1155\/2023\/6985650","article-title":"An entropy-based gravity model for influential spreaders identification in complex networks","volume":"2023","author":"Liu","year":"2023","journal-title":"Complexity"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"129302","DOI":"10.1016\/j.physa.2023.129302","article-title":"Identifying influential nodes in complex networks: A k-orders entropy-based method","volume":"632","author":"Wu","year":"2023","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"083402","DOI":"10.1088\/1742-5468\/acdceb","article-title":"Identifying influential nodes in complex networks based on network embedding and local structure entropy","volume":"2023","author":"Lu","year":"2023","journal-title":"J. Stat. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"131719","DOI":"10.1109\/ACCESS.2019.2939804","article-title":"Identifying influential nodes in complex networks based on local neighbor contribution","volume":"7","author":"Dai","year":"2019","journal-title":"IEEE Access"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8403738","DOI":"10.1155\/2021\/8403738","article-title":"Identification of influential nodes via effective distance-based centrality mechanism in complex networks","volume":"2021","author":"Ullah","year":"2021","journal-title":"Complexity"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"105580","DOI":"10.1016\/j.knosys.2020.105580","article-title":"Finding influential nodes in social networks based on neighborhood correlation coefficient","volume":"194","author":"Zareie","year":"2020","journal-title":"Knowl.-Based Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"012308","DOI":"10.1103\/PhysRevE.96.012308","article-title":"Low-Algorithmic-Complexity Entropy-Deceiving Graphs","volume":"96","author":"Zenil","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zenil, H., Hern\u00e1ndez-Orozco, S., Kiani, N.A., Soler-Toscano, F., Rueda-Toicen, A., and Tegn\u00e9r, J. (2018). A decomposition method for global evaluation of Shannon entropy and local estimations of algorithmic complexity. Entropy, 20.","DOI":"10.3390\/e20080605"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3200","DOI":"10.1103\/PhysRevLett.86.3200","article-title":"Epidemic spreading in scale-free networks","volume":"86","author":"Vespignani","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1140\/epjb\/e20020122","article-title":"Epidemic outbreaks in complex heterogeneous networks","volume":"26","author":"Moreno","year":"2002","journal-title":"Eur. Phys. J. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1093\/biomet\/30.1-2.81","article-title":"A new measure of rank correlation","volume":"30","author":"Kendall","year":"1938","journal-title":"Biometrika"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"128353","DOI":"10.1016\/j.physa.2022.128353","article-title":"Identification of node centrality based on Laplacian energy of networks","volume":"609","author":"Zhao","year":"2023","journal-title":"Phys. A Stat. Mech."},{"key":"ref_34","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. 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