{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T23:59:04Z","timestamp":1786665544775,"version":"build-2736575974"},"reference-count":45,"publisher":"Public Library of Science (PLoS)","issue":"2","license":[{"start":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T00:00:00Z","timestamp":1770249600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>Network centralization, driven by hub nodes, impacts communication efficiency, structural integration, and dynamic processes such as diffusion and synchronization. Although numerous centralization measures exist, a major challenge lies in determining measures that are both theoretically sound and empirically reliable across different network contexts. To resolve this challenge, we normalize 11 measures of network centralization and assess them systematically using an axiomatic framework and numerical simulations. Our axiomatic assessment tests each measure against the six postulates of centralization, ensuring consistency with minimal theoretical requirements. In addition, our numerical assessment examines the behavior of normalized centralization measures over different random graphs. Our results indicate major differences among the measures, despite their common aim of quantifying centralization. Together, our assessments point to the relative suitability of three measures: normalized betweenness centralization, normalized closeness centralization, and normalized degree centralization. Applying these three measures to real-world networks from diverse domains reveals meaningful variation in the organization of the networks with respect to hubs. Normalized betweenness centralization highlights path-based dominance; normalized closeness centralization reflects accessibility and efficiency of reach; and normalized degree centralization captures degree-based hub concentration. When used jointly, the three measures demonstrate the required sensitivity to varying levels of centralization and provide complementary aspects of network centralization that no single measure can offer alone. Our dual evaluation framework clarifies conceptual differences among existing measures and offers practical guidance for selecting reliable centralization metrics.<\/jats:p>","DOI":"10.1371\/journal.pone.0341630","type":"journal-article","created":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T18:37:47Z","timestamp":1770316667000},"page":"e0341630","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":2,"title":["Evaluating global measures of network centralization: Axiomatic and numerical assessments"],"prefix":"10.1371","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4364-751X","authenticated-orcid":true,"given":"Majid","family":"Saberi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5870-9253","authenticated-orcid":true,"given":"Samin","family":"Aref","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,2,5]]},"reference":[{"issue":"3","key":"pone.0341630.ref001","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":"LC Freeman","year":"1978","journal-title":"Social Networks."},{"issue":"1","key":"pone.0341630.ref002","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.socnet.2004.11.008","article-title":"Centrality and network flow","volume":"27","author":"SP Borgatti","year":"2005","journal-title":"Social Networks."},{"key":"pone.0341630.ref003","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511815478","volume-title":"Social network analysis: methods and applications","author":"S Wasserman","year":"1994"},{"key":"pone.0341630.ref004","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199206650.001.0001","volume-title":"Networks: an introduction","author":"MEJ Newman","year":"2010"},{"issue":"2","key":"pone.0341630.ref005","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1137\/S003614450342480","article-title":"The structure and function of complex networks","volume":"45","author":"MEJ Newman","year":"2003","journal-title":"SIAM Rev."},{"key":"pone.0341630.ref006","volume-title":"Network science","author":"AL Barab\u00e1si","year":"2016"},{"key":"pone.0341630.ref007","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":"S Boccaletti","year":"2006","journal-title":"Physics Reports."},{"issue":"6","key":"pone.0341630.ref008","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1007\/s11747-019-00678-2","article-title":"What makes online content viral? The contingent effects of hub users versus non\u2013hub users on social media platforms","volume":"47","author":"Q Wang","year":"2019","journal-title":"J Acad Mark Sci."},{"key":"pone.0341630.ref009","doi-asserted-by":"crossref","first-page":"101498","DOI":"10.1016\/j.ijdrr.2020.101498","article-title":"Crowd or hubs: information diffusion patterns in online social networks in disasters","volume":"46","author":"C Fan","year":"2020","journal-title":"International Journal of Disaster Risk Reduction."},{"issue":"12","key":"pone.0341630.ref010","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1016\/j.tibs.2009.07.007","article-title":"Protein-protein interaction networks: how can a hub protein bind so many different partners?","volume":"34","author":"C-J Tsai","year":"2009","journal-title":"Trends Biochem Sci."},{"issue":"4","key":"pone.0341630.ref011","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pone.0005344","article-title":"Identifying hubs in protein interaction networks","volume":"4","author":"RR Vallabhajosyula","year":"2009","journal-title":"PLoS One."},{"key":"pone.0341630.ref012","doi-asserted-by":"crossref","first-page":"101959","DOI":"10.1016\/j.tre.2020.101959","article-title":"Analysis of the air cargo transport network using a complex network theory perspective","volume":"138","author":"A Bombelli","year":"2020","journal-title":"Transportation Research Part E: Logistics and Transportation Review."},{"key":"pone.0341630.ref013","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.jtrangeo.2013.10.011","article-title":"Classifying airports according to their hub dimensions: an application to the US domestic network","volume":"33","author":"H Rodr\u00edguez-D\u00e9niz","year":"2013","journal-title":"Journal of Transport Geography."},{"issue":"12","key":"pone.0341630.ref014","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1016\/j.tics.2013.09.012","article-title":"Network hubs in the human brain","volume":"17","author":"MP van den Heuvel","year":"2013","journal-title":"Trends Cogn Sci."},{"issue":"1","key":"pone.0341630.ref015","doi-asserted-by":"crossref","first-page":"2176","DOI":"10.1038\/s41598-021-81767-7","article-title":"Topological impact of negative links on the stability of resting-state brain network","volume":"11","author":"M Saberi","year":"2021","journal-title":"Sci Rep."},{"issue":"5","key":"pone.0341630.ref016","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1080\/01969722.2020.1871228","article-title":"Centralization measures for social networks","volume":"52","author":"R Palak","year":"2021","journal-title":"Cybernetics and Systems."},{"issue":"10","key":"pone.0341630.ref017","doi-asserted-by":"crossref","first-page":"1810","DOI":"10.3390\/math8101810","article-title":"Group degree centrality and centralization in networks","volume":"8","author":"M Krnc","year":"2020","journal-title":"Mathematics."},{"issue":"1","key":"pone.0341630.ref018","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":"LC Freeman","year":"1977","journal-title":"Sociometry."},{"issue":"3","key":"pone.0341630.ref019","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/0378-8733(81)90014-9","article-title":"The degree variance: an index of graph heterogeneity","volume":"3","author":"TAB Snijders","year":"1981","journal-title":"Social Networks."},{"issue":"1","key":"pone.0341630.ref020","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/s41109-020-00273-3","article-title":"Normalised degree variance","volume":"5","author":"KM Smith","year":"2020","journal-title":"Appl Netw Sci."},{"key":"pone.0341630.ref021","doi-asserted-by":"crossref","first-page":"066102","DOI":"10.1103\/PhysRevE.82.066102","article-title":"Quantifying network heterogeneity","volume":"82","author":"E Estrada","year":"2010","journal-title":"Phys Rev E Stat Nonlin Soft Matter Phys."},{"issue":"8","key":"pone.0341630.ref022","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pone.0011976","article-title":"Characterizing the community structure of complex networks","volume":"5","author":"A Lancichinetti","year":"2010","journal-title":"PLoS One."},{"issue":"1","key":"pone.0341630.ref023","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/s13721-016-0140-7","article-title":"Employing the Gini coefficient to measure participation inequality in treatment-focused Digital Health Social Networks","volume":"5","author":"T van Mierlo","year":"2016","journal-title":"Netw Model Anal Health Inform Bioinform."},{"key":"pone.0341630.ref024","volume-title":"Handbook of income distribution","author":"AB Atkinson","year":"2014"},{"issue":"4","key":"pone.0341630.ref025","doi-asserted-by":"crossref","first-page":"696","DOI":"10.1137\/090761070","article-title":"Network properties revealed through matrix functions","volume":"52","author":"E Estrada","year":"2010","journal-title":"SIAM Rev."},{"issue":"52","key":"pone.0341630.ref026","article-title":"Eigenvector centrality for characterization of protein allosteric pathways","volume":"115","author":"CFA Negre","year":"2018","journal-title":"Proc Natl Acad Sci U S A."},{"issue":"3","key":"pone.0341630.ref027","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":"F Tutzauer","year":"2009","journal-title":"Communication Monographs."},{"issue":"4","key":"pone.0341630.ref028","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.socnet.2005.07.003","article-title":"Exact bounds for degree centralization","volume":"28","author":"CT Butts","year":"2006","journal-title":"Social Networks."},{"issue":"4","key":"pone.0341630.ref029","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1093\/comnet\/cnv005","article-title":"Assortativity in complex networks","volume":"3","author":"R Noldus","year":"2015","journal-title":"J Complex Netw."},{"issue":"17","key":"pone.0341630.ref030","doi-asserted-by":"crossref","first-page":"2079","DOI":"10.4236\/am.2016.717166","article-title":"Systemic risk of conventional and islamic banks: comparison with graphical network models","volume":"07","author":"SQ Hashem","year":"2016","journal-title":"AM."},{"issue":"3","key":"pone.0341630.ref031","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1080\/01969722.2022.2162737","article-title":"A centralization measure for social networks assessment","volume":"55","author":"R Palak","year":"2023","journal-title":"Cybernetics and Systems."},{"key":"pone.0341630.ref032","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.ins.2014.03.133","article-title":"Extremality of degree-based graph entropies","volume":"278","author":"S Cao","year":"2014","journal-title":"Information Sciences."},{"issue":"01","key":"pone.0341630.ref033","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1142\/S0219525905000385","article-title":"The GINI coefficient\u2019s application to general complex networks","volume":"08","author":"H-B Hu","year":"2005","journal-title":"Advs Complex Syst."},{"issue":"7","key":"pone.0341630.ref034","doi-asserted-by":"crossref","first-page":"078902","DOI":"10.1088\/0256-307X\/27\/7\/078902","article-title":"Natural connectivity of complex networks","volume":"27","author":"W Jun","year":"2010","journal-title":"Chinese Phys Lett."},{"key":"pone.0341630.ref035","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.neuroimage.2013.05.041","article-title":"The WU-Minn human connectome project: an overview","volume":"80","author":"DC Van Essen","year":"2013","journal-title":"Neuroimage."},{"issue":"1","key":"pone.0341630.ref036","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1145\/1217299.1217301","article-title":"Graph evolution: densification and shrinking diameters","volume":"1","author":"J Leskovec","year":"2007","journal-title":"ACM Transactions on Knowledge Discovery from Data."},{"issue":"1","key":"pone.0341630.ref037","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1080\/15427951.2009.10129177","article-title":"Community structure in large networks: natural cluster sizes and the absence of large well-defined clusters","volume":"6","author":"J Leskovec","year":"2009","journal-title":"Internet Mathematics."},{"key":"pone.0341630.ref038","unstructured":"Klimt B, Yang Y. Introducing the Enron corpus. In: CEAS; 2004. p. 92\u20136."},{"key":"pone.0341630.ref039","first-page":"1","article-title":"Learning to discover social circles in ego networks","volume":"25","author":"J Leskovec","year":"2012","journal-title":"Advances in Neural Information Processing Systems."},{"key":"pone.0341630.ref040","doi-asserted-by":"crossref","unstructured":"McAuley JJ, Leskovec J. From amateurs to connoisseurs: modeling the evolution of user expertise through online reviews. In: Proceedings of the 22nd International Conference on World Wide Web; 2013. p. 897\u2013908.","DOI":"10.1145\/2488388.2488466"},{"issue":"6887","key":"pone.0341630.ref041","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1038\/nature750","article-title":"Comparative assessment of large-scale data sets of protein-protein interactions","volume":"417","author":"C von Mering","year":"2002","journal-title":"Nature."},{"issue":"4","key":"pone.0341630.ref042","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1086\/jar.33.4.3629752","article-title":"An information flow model for conflict and fission in small groups","volume":"33","author":"WW Zachary","year":"1977","journal-title":"Journal of Anthropological Research."},{"issue":"1","key":"pone.0341630.ref043","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.socnet.2009.02.004","article-title":"Introduction to stochastic actor-based models for network dynamics","volume":"32","author":"TAB Snijders","year":"2010","journal-title":"Social Networks."},{"key":"pone.0341630.ref044","article-title":"The igraph software package for complex network research","volume":"1695","author":"G Csardi","year":"2006","journal-title":"InterJournal Complex Systems."},{"key":"pone.0341630.ref045","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-24277-4","volume-title":"ggplot2: elegant graphics for data analysis","author":"H Wickham","year":"2016"}],"container-title":["PLOS One"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0341630","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T18:37:54Z","timestamp":1770316674000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0341630"}},"subtitle":[],"editor":[{"given":"Amit","family":"Nanavati","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2026,2,5]]},"references-count":45,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2,5]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0341630","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,5]]}}}