{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T00:50:33Z","timestamp":1784595033410,"version":"3.55.0"},"reference-count":65,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,15]],"date-time":"2020-12-15T00:00:00Z","timestamp":1607990400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001866","name":"Fonds National de la Recherche Luxembourg","doi-asserted-by":"publisher","award":["11601404"],"award-info":[{"award-number":["11601404"]}],"id":[{"id":"10.13039\/501100001866","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Information entropy metrics have been applied to a wide range of problems that were abstracted as complex networks. This growing body of research is scattered in multiple disciplines, which makes it difficult to identify available metrics and understand the context in which they are applicable. In this work, a narrative literature review of information entropy metrics for complex networks is conducted following the PRISMA guidelines. Existing entropy metrics are classified according to three different criteria: whether the metric provides a property of the graph or a graph component (such as the nodes), the chosen probability distribution, and the types of complex networks to which the metrics are applicable. Consequently, this work identifies the areas in need for further development aiming to guide future research efforts.<\/jats:p>","DOI":"10.3390\/e22121417","type":"journal-article","created":{"date-parts":[[2020,12,15]],"date-time":"2020-12-15T09:12:57Z","timestamp":1608023577000},"page":"1417","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":69,"title":["A Survey of Information Entropy Metrics for Complex Networks"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8350-4890","authenticated-orcid":false,"given":"Yamila M.","family":"Omar","sequence":"first","affiliation":[{"name":"Faculty of Science, Communication and Medicine, University of Luxembourg, L-1359 Luxembourg, Luxembourg"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3507-1397","authenticated-orcid":false,"given":"Peter","family":"Plapper","sequence":"additional","affiliation":[{"name":"Faculty of Science, Communication and Medicine, University of Luxembourg, L-1359 Luxembourg, Luxembourg"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell. Syst. Tech. J."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","unstructured":"Freeman, L.C. (1977). A set of measures of centrality based on betweenness. Sociometry, 35\u201341.","DOI":"10.2307\/3033543"},{"key":"ref_4","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_5","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_6","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_7","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"027105","DOI":"10.1103\/PhysRevE.75.027105","article-title":"Generalizations of the clustering coefficient to weighted complex networks","volume":"75","author":"Onnela","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"026107","DOI":"10.1103\/PhysRevE.76.026107","article-title":"Clustering in complex directed networks","volume":"76","author":"Fagiolo","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.amc.2007.12.010","article-title":"Information processing in complex networks: Graph entropy and information functionals","volume":"201","author":"Dehmer","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zenil, H., Kiani, N., and Tegn\u00e9r, J. (2018). A review of graph and network complexity from an algorithmic information perspective. Entropy, 20.","DOI":"10.3390\/e20080551"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.socnet.2004.11.008","article-title":"Centrality and network flow","volume":"27","author":"Borgatti","year":"2005","journal-title":"Soc. Netw."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Moher, D., Liberati, A., Tetzlaff, J., Altman, D.G., and Prisma Group (2009). Preferred reporting items for systematic reviews and meta-analyses: The PRISMA statement. PLoS Med., 6.","DOI":"10.1371\/journal.pmed.1000097"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.socnet.2006.10.001","article-title":"Entropy as a measure of centrality in networks characterized by path-transfer flow","volume":"29","author":"Tutzauer","year":"2007","journal-title":"Soc. Netw."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hussain, D.A., and Ortiz-Arroyo, D. (2008, January 3\u20135). Locating key actors in social networks using bayes\u2019 posterior probability framework. Proceedings of the European Conference on Intelligence and Security Informatics, Esbjerg, Denmark.","DOI":"10.1007\/978-3-540-89900-6_6"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ortiz-Arroyo, D., and Hussain, D.A. (2008, January 3\u20135). An information theory approach to identify sets of key players. Proceedings of the European Conference on Intelligence and Security Informatics, Esbjerg, Denmark.","DOI":"10.1007\/978-3-540-89900-6_5"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1080\/03637750903074727","article-title":"Entropy-Based Centralization and its Sampling Distribution in Directed Communication Networks","volume":"76","author":"Tutzauer","year":"2009","journal-title":"Commun. Monogr."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.ins.2010.08.041","article-title":"A history of graph entropy measures","volume":"181","author":"Dehmer","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"046117","DOI":"10.1103\/PhysRevE.83.046117","article-title":"Centrality measures and thermodynamic formalism for complex networks","volume":"83","author":"Delvenne","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Sun, R., Mu, A.l., Li, L., and Zhong, M. (2012, January 13). Evaluation of node importance based on topological potential in weighted complex networks. Proceedings of the Fourth International Conference on Machine Vision (ICMV 2011): Machine Vision, Image Processing, and Pattern Analysis, Singapore.","DOI":"10.1117\/12.920232"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Serin, E., and Balcisoy, S. (2012, January 26\u201329). Entropy based sensitivity analysis and visualization of social networks. Proceedings of the 2012 IEEE\/ACM International Conference on Advances in Social Networks Analysis and Mining, Istanbul, Turkey.","DOI":"10.1109\/ASONAM.2012.189"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Dehmer, M., and Sivakumar, L. (2012). Recent developments in quantitative graph theory: Information inequalities for networks. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0031395"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Fewell, J.H., Armbruster, D., Ingraham, J., Petersen, A., and Waters, J.S. (2012). Basketball teams as strategic networks. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0047445"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Chellappan, V., and Sivalingam, K.M. (2013, January 10\u201312). Application of entropy of centrality measures to routing in tactical wireless networks. Proceedings of the 2013 19th IEEE Workshop on Local & Metropolitan Area Networks (LANMAN), Bussels, Belgium.","DOI":"10.1109\/LANMAN.2013.6528278"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Chellappan, V., Sivalingam, K.M., and Krithivasan, K. (2014, January 21\u201323). An entropy maximization problem in shortest path routing networks. Proceedings of the 2014 IEEE 20th International Workshop on Local Metropolitan Area Networks(LANMAN), Reno, NV, USA.","DOI":"10.1109\/LANMAN.2014.7028625"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.laa.2013.11.009","article-title":"Walk entropies in graphs","volume":"443","author":"Estrada","year":"2014","journal-title":"Linear Algebra Its Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.laa.2013.12.014","article-title":"A note on walk entropies in graphs","volume":"445","author":"Benzi","year":"2014","journal-title":"Linear Algebra Its Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5416","DOI":"10.3390\/e16105416","article-title":"A note on distance-based graph entropies","volume":"16","author":"Chen","year":"2014","journal-title":"Entropy"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.socnet.2014.10.002","article-title":"On efficient use of entropy centrality for social network analysis and community detection","volume":"40","author":"Nikolaev","year":"2015","journal-title":"Soc. Netw."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.chaos.2014.12.021","article-title":"Ranking nodes according to their path-complexity","volume":"73","author":"Caravelli","year":"2015","journal-title":"Chaos Solitons Fractals"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"8217","DOI":"10.3390\/e17127871","article-title":"Some new properties for degree-based graph entropies","volume":"17","author":"Lu","year":"2015","journal-title":"Entropy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.physa.2016.02.009","article-title":"Using mapping entropy to identify node centrality in complex networks","volume":"453","author":"Nie","year":"2016","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Gialampoukidis, I., Kalpakis, G., Tsikrika, T., Vrochidis, S., and Kompatsiaris, I. (2016, January 17\u201319). Key player identification in terrorism-related social media networks using centrality measures. Proceedings of the 2016 European Intelligence and Security Informatics Conference (EISIC), Uppsala, Sweden.","DOI":"10.1109\/EISIC.2016.029"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.comnet.2016.04.015","article-title":"A centrality entropy maximization problem in shortest path routing networks","volume":"104","author":"Chellappan","year":"2016","journal-title":"Comput. Networks"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1016\/j.physa.2016.08.019","article-title":"Link influence entropy","volume":"465","author":"Singh","year":"2017","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Weber, C.M., Hasenauer, R.P., and Mayande, N.V. (2017, January 9\u201313). Quantifying nescience: A decision aid for practicing managers. Proceedings of the 2017 Portland International Conference on Management of Engineering and technology (PICMET), Portland, OR, USA.","DOI":"10.23919\/PICMET.2017.8125453"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1016\/j.ejor.2016.06.052","article-title":"Information diffusion, cluster formation and entropy-based network dynamics in equity and commodity markets","volume":"256","author":"Bekiros","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Wang, Q., Zeng, G., and Tu, X. (2017). Information technology project portfolio implementation process optimization based on complex network theory and entropy. Entropy, 19.","DOI":"10.3390\/e19060287"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Ai, X. (2017). Node importance ranking of complex networks with entropy variation. Entropy, 19.","DOI":"10.3390\/e19070303"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"013310","DOI":"10.1103\/PhysRevE.96.013310","article-title":"Eigenvector centrality for geometric and topological characterization of porous media","volume":"96","author":"Negre","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-017-09475-9","article-title":"Analysis and evaluation of the entropy indices of a static network structure","volume":"7","author":"Cai","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"042304","DOI":"10.1103\/PhysRevE.96.042304","article-title":"Mapping and discrimination of networks in the complexity-entropy plane","volume":"96","author":"Wiedermann","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_43","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_44","doi-asserted-by":"crossref","unstructured":"Qiao, T., Shan, W., and Zhou, C. (2017). How to identify the most powerful node in complex networks? A novel entropy centrality approach. Entropy, 19.","DOI":"10.3390\/e19110614"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"7390","DOI":"10.1109\/ACCESS.2018.2794324","article-title":"Identifying influential nodes based on community structure to speed up the dissemination of information in complex network","volume":"6","author":"Tulu","year":"2018","journal-title":"IEEE Access"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Oggier, F., Phetsouvanh, S., and Datta, A. (2018, January 28\u201331). Entropic Centrality for Non-Atomic Flow Networks. Proceedings of the 2018 International Symposium on Information Theory and Its Applications (ISITA), Singapore.","DOI":"10.23919\/ISITA.2018.8664236"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Stella, M., and De Domenico, M. (2018). Distance entropy cartography characterises centrality in complex networks. Entropy, 20.","DOI":"10.3390\/e20040268"},{"key":"ref_48","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_49","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1007\/s10955-018-2076-z","article-title":"Tackling information asymmetry in networks: A new entropy-based ranking index","volume":"173","author":"Barucca","year":"2018","journal-title":"J. Stat. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Ma, W., Zhang, Z., and Xiong, C. (2018, January 9\u201311). A transportation network stability analysis method based on betweenness centrality entropy maximization. Proceedings of the 2018 Chinese Control And Decision Conference (CCDC), Shenyang, China.","DOI":"10.1109\/CCDC.2018.8407591"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1109\/TITS.2018.2817282","article-title":"Discovery of Critical Nodes in Road Networks Through Mining From Vehicle Trajectories","volume":"20","author":"Xu","year":"2019","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.future.2018.11.023","article-title":"Influential node ranking in social networks based on neighborhood diversity","volume":"94","author":"Zareie","year":"2019","journal-title":"Future Gener. Comput. Syst."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Zarghami, S.A., Gunawan, I., and Schultmann, F. (2019). Entropy of centrality values for topological vulnerability analysis of water distribution networks. Built Environ. Proj. Asset Manag.","DOI":"10.1108\/BEPAM-02-2019-0014"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"e220","DOI":"10.7717\/peerj-cs.220","article-title":"A split-and-transfer flow based entropic centrality","volume":"5","author":"Oggier","year":"2019","journal-title":"PeerJ Comput. Sci."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"136012","DOI":"10.1109\/ACCESS.2019.2942843","article-title":"Predicting Essential Proteins Based on Second-Order Neighborhood Information and Information Entropy","volume":"7","author":"Zhao","year":"2019","journal-title":"IEEE Access"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Wang, L., Dai, W., Luo, G., and Zhao, Y. (2019). A Novel Approach to Support Failure Mode, Effects, and Criticality Analysis Based on Complex Networks. Entropy, 21.","DOI":"10.3390\/e21121230"},{"key":"ref_57","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_58","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.ins.2019.10.003","article-title":"Identification of influencers in complex networks by local information dimensionality","volume":"512","author":"Wen","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Guo, C., Yang, L., Chen, X., Chen, D., Gao, H., and Ma, J. (2020). Influential Nodes Identification in Complex Networks via Information Entropy. Entropy, 22.","DOI":"10.3390\/e22020242"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"123659","DOI":"10.1016\/j.physa.2019.123659","article-title":"Sequential seeding strategy for social influence diffusion with improved entropy-based centrality","volume":"545","author":"Ni","year":"2020","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"103740","DOI":"10.1016\/j.compbiomed.2020.103740","article-title":"EMDIP: An Entropy Measure to Discover Important Proteins in PPI networks","volume":"120","author":"Bashiri","year":"2020","journal-title":"Comput. Biol. Med."},{"key":"ref_62","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. Stat. Mech. Its Appl."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"124630","DOI":"10.1016\/j.physa.2020.124630","article-title":"Entropy based flow transfer for influence dissemination in networks","volume":"555","author":"Saxena","year":"2020","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1959","DOI":"10.1016\/j.disc.2006.03.054","article-title":"Sums of powers of the degrees of a graph","volume":"306","year":"2006","journal-title":"Discret. Math."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1080\/0022250X.2001.9990249","article-title":"A faster algorithm for betweenness centrality","volume":"25","author":"Brandes","year":"2001","journal-title":"J. Math. Sociol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/12\/1417\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:45:22Z","timestamp":1760179522000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/12\/1417"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,15]]},"references-count":65,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["e22121417"],"URL":"https:\/\/doi.org\/10.3390\/e22121417","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,15]]}}}