{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T02:51:14Z","timestamp":1768618274164,"version":"3.49.0"},"reference-count":53,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,30]],"date-time":"2019-12-30T00:00:00Z","timestamp":1577664000000},"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>Numerous problems in many fields can be solved effectively through the approach of modeling by complex network analysis. Finding key nodes is one of the most important and challenging problems in network analysis. In previous studies, methods have been proposed to identify key nodes. However, they rely mainly on a limited field of local information, lack large-scale access to global information, and are also usually NP-hard. In this paper, a novel entropy and mutual information-based centrality approach (EMI) is proposed, which attempts to capture a far wider range and a greater abundance of information for assessing how vital a node is. We have developed countermeasures to assess the influence of nodes: EMI is no longer confined to neighbor nodes, and both topological and digital network characteristics are taken into account. We employ mutual information to fix a flaw that exists in many methods. Experiments on real-world connected networks demonstrate the outstanding performance of the proposed approach in both correctness and efficiency as compared with previous approaches.<\/jats:p>","DOI":"10.3390\/e22010052","type":"journal-article","created":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T03:28:53Z","timestamp":1578022133000},"page":"52","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Key Node Ranking in Complex Networks: A Novel Entropy and Mutual Information-Based Approach"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9166-087X","authenticated-orcid":false,"given":"Yichuan","family":"Li","sequence":"first","affiliation":[{"name":"Department of Computer Science, Shantou University, Shantou 515063, China"},{"name":"Key Laboratory of Intelligent Manufacturing Technology, Ministry of Education, Shantou University, Shantou 515063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weihong","family":"Cai","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Shantou University, Shantou 515063, China"},{"name":"Key Laboratory of Intelligent Manufacturing Technology, Ministry of Education, Shantou University, Shantou 515063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yao","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Shantou University, Shantou 515063, China"},{"name":"Key Laboratory of Intelligent Manufacturing Technology, Ministry of Education, Shantou University, Shantou 515063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Shantou University, Shantou 515063, China"},{"name":"Key Laboratory of Intelligent Manufacturing Technology, Ministry of Education, Shantou University, Shantou 515063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective Dynamics of Small World Networks","volume":"393","author":"Duncan","year":"1998","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1142\/S0217984906010883","article-title":"Optimization of scale-free network for random failures","volume":"20","author":"Liu","year":"2006","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Richard, J., Ananthi, A., and Drew, P. (2010). The Probabilistic Niche Model Reveals the Niche Structure and Role of Body Size in a Complex Food Web. PLoS ONE, 5.","DOI":"10.1371\/journal.pone.0012092"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hobson, E.A., and Dedeo, S. (2015). Social Feedback and the Emergence of Rank in Animal Society. PLoS Comput. Biol., 11.","DOI":"10.1101\/018374"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/978-3-540-44485-5_14","article-title":"Attacks and Cascades in Complex Networks","volume":"650","author":"Lai","year":"2004","journal-title":"Lect. Notes Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"025103","DOI":"10.1103\/PhysRevE.69.025103","article-title":"Structural vulnerability of the North American power grid","volume":"69","author":"Albert","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"974","DOI":"10.1016\/j.physa.2012.10.016","article-title":"A stochastic SIR epidemic on scale-free network with community structure","volume":"392","author":"Zhang","year":"2013","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Du, J., Zhu, G., Hou, M., and Zhang, N. (2014, January 7\u201310). Power supply restoration algorithm of multi-contact distribution network with distributed intelligent feeder automation system. Proceedings of the IEEE PES Asia-Power & Energy Engineering Conference, Kowloon Shangri-La, Hong Kong, China.","DOI":"10.1109\/APPEEC.2014.7066118"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Richardson, M., and Domingos, P. (2002, January 23\u201325). Mining knowledge-sharing sites for viral marketing. Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD\u201902), New York, NY, USA.","DOI":"10.1145\/775056.775057"},{"key":"ref_11","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_12","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_13","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.physa.2005.12.020","article-title":"Residual closeness in networks","volume":"365","author":"Chavdar","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_14","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_15","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_16","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1007\/BF02476438","article-title":"Structural parameters of communication networks","volume":"15","author":"Shimbel","year":"1953","journal-title":"Bull. Math. Biophys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0378-8733(94)00248-9","article-title":"Eccentricity and centrality in networks","volume":"17","author":"Hage","year":"1995","journal-title":"Soc. Netw."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0378-8733(91)90017-N","article-title":"Centrality in valued graphs: A measure of betweenness based on network flow","volume":"13","author":"Freeman","year":"1991","journal-title":"Soc. Netw."},{"key":"ref_19","unstructured":"(2019, July 02). Wikimedia. Available online: http:\/\/commons.wikimedia.org\/w\/index.php?title=File:6_centrality_measure.Png&oldid=344191846."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ai, X.B. (2017). Node Importance Ranking of Complex Networks with Entropy Variation. Entropy, 19.","DOI":"10.3390\/e19070303"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Qiao, T., Shan, W., Yu, G., and Liu, C. (2018). A Novel Entropy-Based Centrality Approach for Identifying Vital Nodes in Weighted Networks. Entropy, 20.","DOI":"10.3390\/e20040261"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"020204","DOI":"10.7498\/aps.62.020204","article-title":"Key nodes in complex networks identified by multi-attribute decision-making method","volume":"62","author":"Hui","year":"2013","journal-title":"Acta Phys. Sin. (Chin.)"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Dubitzky, W., Wolkenhauer, O., Cho, K.H., and Yokota, H. (2013). Degree Centrality. Encyclopedia of Systems Biology, Springer.","DOI":"10.1007\/978-1-4419-9863-7"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Guetzkow, H., and Simon, H.A. (1955). The Impact of Certain Communication Nets Upon Organization and Performance in Task-Oriented Groups, INFORMS.","DOI":"10.1287\/mnsc.1.3-4.233"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/S0378-8733(00)00031-9","article-title":"Eigenvector centrality a node centrality","volume":"22","author":"Ruhnau","year":"2000","journal-title":"Soc. Netw."},{"key":"ref_26","unstructured":"Page, L., Brin, S., Motwani, R., and Winograd, T. (1999). The PageRank Citation Ranking: Bringing Order to the Web, Stanford InfoLab Publication Server. Technical Report SIDL-WP-1999-0120."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1186\/s40537-017-0076-5","article-title":"Identification of top-K nodes in large networks using Katz centrality","volume":"4","author":"Zhan","year":"2017","journal-title":"J. Big Data"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"218701","DOI":"10.1103\/PhysRevLett.105.218701","article-title":"Thresholds for Epidemic Spreading in Networks","volume":"105","author":"Castellano","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","unstructured":"Hoboken, N.J. (2006). Elements of Information Theory, Wiley-Interscience. [2nd ed.]."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1002\/j.1538-7305.1948.tb00917.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Steiner, T. (2013, January 13\u201317). A meteoroid on steroids: Ranking media items stemming from multiple social networks. Proceedings of the 22nd International Conference on World Wide Web, Rio de Janeiro, Brazil.","DOI":"10.1145\/2487788.2487798"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1109\/42.563664","article-title":"Multimodality Image Registration by Maximization of Mutual Information","volume":"16","author":"Maes","year":"1997","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_33","unstructured":"(2019, July 04). Wikimedia. Available online: http:\/\/en.wikipedia.org\/w\/index.php?title=Mutual_information&oldid=9165.19092png&oldid=344191846."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5490","DOI":"10.1016\/j.physa.2013.06.059","article-title":"A modified evidential methodology of identifying influential nodes in weighted networks","volume":"392","author":"Cai","year":"2013","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1016\/j.physa.2017.05.098","article-title":"Identification of influential users by neighbors in online social networks","volume":"486","author":"Sheikhahmadi","year":"2017","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_36","unstructured":"(2019, July 07). Wikimedia. Available online: https:\/\/en.wikipedia.org\/w\/index.php?title=Directed_graph&oldid=91.6427033."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/S0378-8733(97)00007-5","article-title":"Integration and radiality: Measuring the extent of an individual\u2019s connectedness and reachability in a network","volume":"20","author":"Valente","year":"1998","journal-title":"Soc. Netw."},{"key":"ref_38","unstructured":"Hwang, W., Cho, Y., Zhang, A., Cho, Y.R., and Hwang, W. (2006, January 20\u201323). Bridging Centrality: Identifying Bridging Nodes in Scale-free Networks. Proceedings of the Twelfth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD\u201906), Philadelphia, PA, USA."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0012-365X(78)90098-5","article-title":"Another characterization of the centroid of a tree","volume":"24","author":"Mitchell","year":"1978","journal-title":"Discret. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"W438","DOI":"10.1093\/nar\/gkn257","article-title":"Hubba: Hub objects analyzer\u2014A framework of interactome hubs identification for network biology","volume":"36","author":"Lin","year":"2008","journal-title":"Nucleic Acids Res."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Min, L., Zhang, H., Wang, J.X., and Pan, Y. (2012). A new essential protein discovery method based on the integration of protein-protein interaction and gene expression data. BMC Syst. Biol., 6.","DOI":"10.1186\/1752-0509-6-15"},{"key":"ref_42","unstructured":"(2019, July 26). KONECT. Available online: http:\/\/konect.uni-koblenz.de\/networks\/moreno_vdb."},{"key":"ref_43","unstructured":"(2019, July 29). KONECT. Available online: http:\/\/konect.uni-koblenz.de\/networks\/opsahl-usairport."},{"key":"ref_44","unstructured":"(2019, July 29). KONECT. Available online: http:\/\/konect.uni-koblenz.de\/networks\/maayan-faa."},{"key":"ref_45","unstructured":"(2019, July 29). KONECT. Available online: http:\/\/konect.uni-koblenz.de\/networks\/subelj_euroroad."},{"key":"ref_46","unstructured":"(2019, August 01). KONECT. Available online: http:\/\/konect.uni-koblenz.de\/networks\/tntp-ChicagoRegional."},{"key":"ref_47","unstructured":"(2019, August 01). KONECT. Available online: http:\/\/konect.uni-koblenz.de\/networks\/dolphins."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1093\/comnet\/cnv005","article-title":"Assortativity in complex networks","volume":"3","author":"Noldus","year":"2018","journal-title":"J. Complex Netw."},{"key":"ref_49","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. Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/S0951-8320(96)00072-5","article-title":"Probability, conditional probability and complementary cumulative distribution functions in performance assessment for radioactive waste disposal","volume":"54","author":"Helton","year":"1996","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1088\/0256-307X\/22\/2\/068","article-title":"Epidemic spread in weighted scale-free networks","volume":"22","author":"Yan","year":"2005","journal-title":"Chin. Phys. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Butte, A.J., and Kohane, I.S. (1999). Mutual information relevance networks: Functional genomic clustering using pairwise entropy measurements. Pac. Symp. Biocomput., 418\u2013429.","DOI":"10.1142\/9789814447331_0040"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"056103","DOI":"10.1103\/PhysRevE.71.056103","article-title":"Subgraph centrality in complex networks","volume":"71","author":"Estrada","year":"2005","journal-title":"Phys. Rev. E"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:46:55Z","timestamp":1760190415000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,30]]},"references-count":53,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010052"],"URL":"https:\/\/doi.org\/10.3390\/e22010052","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,30]]}}}