{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T23:17:31Z","timestamp":1768346251019,"version":"3.49.0"},"reference-count":35,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T00:00:00Z","timestamp":1619049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Academia Sinica (Taiwan)","award":["AS-TP-109-M07"],"award-info":[{"award-number":["AS-TP-109-M07"]}]},{"name":"Ministry of Science and Technology (Taiwan)","award":["107-2118-M-001-011-MY3"],"award-info":[{"award-number":["107-2118-M-001-011-MY3"]}]},{"name":"Ministry of Science and Technology (Taiwan)","award":["109-2321-B-001-013"],"award-info":[{"award-number":["109-2321-B-001-013"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The degree distribution has attracted considerable attention from network scientists in the last few decades to have knowledge of the topological structure of networks. It is widely acknowledged that many real networks have power-law degree distributions. However, the deviation from such a behavior often appears when the range of degrees is small. Even worse, the conventional employment of the continuous power-law distribution usually causes an inaccurate inference as the degree should be discrete-valued. To remedy these obstacles, we propose a finite mixture model of truncated zeta distributions for a broad range of degrees that disobeys a power-law behavior in the range of small degrees while maintaining the scale-free behavior. The maximum likelihood algorithm alongside the model selection method is presented to estimate model parameters and the number of mixture components. The validity of the suggested algorithm is evidenced by Monte Carlo simulations. We apply our method to five disciplines of scientific collaboration networks with remarkable interpretations. The proposed model outperforms the other alternatives in terms of the goodness-of-fit.<\/jats:p>","DOI":"10.3390\/e23050502","type":"journal-article","created":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T13:59:14Z","timestamp":1619099954000},"page":"502","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Mixture Model of Truncated Zeta Distributions with Applications to Scientific Collaboration Networks"],"prefix":"10.3390","volume":"23","author":[{"given":"Hohyun","family":"Jung","sequence":"first","affiliation":[{"name":"Institute of Statistical Science, Academia Sinica, Taipei City 11529, Taiwan"},{"name":"Department of Statistics, Sungshin Women\u2019s University, Seoul 02844, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7417-8982","authenticated-orcid":false,"given":"Frederick Kin Hing","family":"Phoa","sequence":"additional","affiliation":[{"name":"Institute of Statistical Science, Academia Sinica, Taipei City 11529, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"123403","DOI":"10.1088\/1742-5468\/aaeb40","article-title":"Comparison of fitness and popularity: Fitness-popularity dynamic network model","volume":"2018","author":"Jung","year":"2018","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.ejor.2019.07.024","article-title":"A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market","volume":"281","author":"Mazzarisi","year":"2020","journal-title":"Eur. J. Oper. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1109\/JSTSP.2014.2310294","article-title":"Dynamic stochastic blockmodels for time-evolving social networks","volume":"8","author":"Xu","year":"2014","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1038\/s41467-017-00148-9","article-title":"Modelling sequences and temporal networks with dynamic community structures","volume":"8","author":"Peixoto","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1209\/epl\/i2001-00260-6","article-title":"Competition and multiscaling in evolving networks","volume":"54","author":"Bianconi","year":"2001","journal-title":"Europhys. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1137\/070710111","article-title":"Power-law distributions in empirical data","volume":"51","author":"Clauset","year":"2009","journal-title":"SIAM Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1137\/S003614450342480","article-title":"The structure and function of complex networks","volume":"45","author":"Newman","year":"2003","journal-title":"SIAM Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1080\/00107510500052444","article-title":"Power laws, Pareto distributions and Zipf\u2019s law","volume":"46","author":"Newman","year":"2005","journal-title":"Contemp. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1038\/43601","article-title":"Diameter of the world-wide web","volume":"401","author":"Albert","year":"1999","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1304","DOI":"10.1214\/20-AOAS1346","article-title":"PTEM: A popularity-based topical expertise model for community question answering","volume":"14","author":"Jung","year":"2020","journal-title":"Ann. Appl. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1002\/rsa.1009","article-title":"The degree sequence of a scale-free random graph process","volume":"18","author":"Riordan","year":"2001","journal-title":"Random Struct. Algorithms"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4633","DOI":"10.1103\/PhysRevLett.85.4633","article-title":"Structure of growing networks with preferential linking","volume":"85","author":"Dorogovtsev","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"066123","DOI":"10.1103\/PhysRevE.63.066123","article-title":"Organization of growing random networks","volume":"63","author":"Krapivsky","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"025101","DOI":"10.1103\/PhysRevE.63.025101","article-title":"Effect of the accelerating growth of communications networks on their structure","volume":"63","author":"Dorogovtsev","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1140\/epjb\/e2013-40920-6","article-title":"Degree correlation in scale-free graphs","volume":"86","author":"Fotouhi","year":"2013","journal-title":"Eur. Phys. J. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"14343","DOI":"10.1088\/0305-4470\/39\/46\/007","article-title":"The topology of an accelerated growth network","volume":"39","author":"Yu","year":"2006","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"056101","DOI":"10.1103\/PhysRevE.65.056101","article-title":"Geometric fractal growth model for scale-free networks","volume":"65","author":"Jung","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Dorogovtsev, S.N., and Mendes, J.F. (2002). Accelerated growth of networks. arXiv.","DOI":"10.1093\/acprof:oso\/9780198515906.001.0001"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"32558","DOI":"10.1038\/srep32558","article-title":"Joint estimation of preferential attachment and node fitness in growing complex networks","volume":"6","author":"Pham","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"043407","DOI":"10.1088\/1742-5468\/ab7754","article-title":"On the analysis of fitness change: Fitness-popularity dynamic network model with varying fitness","volume":"2020","author":"Jung","year":"2020","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_22","unstructured":"Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., and Upfal, E. (2000, January 12\u201314). Stochastic models for the web graph. Proceedings of the 41st Annual Symposium on Foundations of Computer Science, Redondo Beach, CA, USA."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.cnsns.2016.06.004","article-title":"A generalization of the power law distribution with nonlinear exponent","volume":"42","author":"Prieto","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Arnold, B.C. (2015). Pareto Distributions, Chapman and Hall\/CRC.","DOI":"10.1201\/b18141"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2492","DOI":"10.1103\/PhysRevLett.76.2492","article-title":"Composite power laws in shock fragmentation","volume":"76","author":"Meibom","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.1016\/S0301-9322(03)00139-3","article-title":"Power law and composite power law friction factor correlations for laminar and turbulent gas\u2013liquid flow in horizontal pipelines","volume":"29","author":"Padrino","year":"2003","journal-title":"Int. J. Multiph. Flow"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.socnet.2005.12.003","article-title":"A model for collaboration networks giving rise to a power-law distribution with an exponential cutoff","volume":"29","author":"Fenner","year":"2007","journal-title":"Soc. Netw."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"138701","DOI":"10.1103\/PhysRevLett.88.138701","article-title":"Truncation of power law behavior in \u201cscale-free\u201d network models due to information filtering","volume":"88","author":"Mossa","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1002\/rsa.20101","article-title":"The degree sequences and spectra of scale-free random graphs","volume":"29","author":"Jordan","year":"2006","journal-title":"Random Struct. Algorithms"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/S0378-4371(99)00291-5","article-title":"Mean-field theory for scale-free random networks","volume":"272","author":"Albert","year":"1999","journal-title":"Phys. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical mechanics of complex networks","volume":"74","author":"Albert","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_32","unstructured":"Jung, H., and Phoa, F.K.H. (2020). Analysis of a Finite Mixture of Truncated Zeta Distributions for Degree Distribution. Studies in Computational Intelligence, Proceedings of the International Conference on Complex Networks and Their Applications, Madrid, Spain, 1\u20133 December 2020, Springer."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v064.i02","article-title":"Fitting Heavy Tailed Distributions: The poweRlaw Package","volume":"64","author":"Gillespie","year":"2015","journal-title":"J. Stat. Softw."},{"key":"ref_34","unstructured":"Brent, R.P. (2013). Algorithms for Minimization without Derivatives, Dover."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.aap.2014.05.005","article-title":"Modelling road accident blackspots data with the discrete generalized Pareto distribution","volume":"71","author":"Prieto","year":"2014","journal-title":"Accid. Anal. Prev."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/5\/502\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:51:30Z","timestamp":1760161890000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/5\/502"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,22]]},"references-count":35,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["e23050502"],"URL":"https:\/\/doi.org\/10.3390\/e23050502","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,22]]}}}