{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:51:46Z","timestamp":1760147506011,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,7]],"date-time":"2023-02-07T00:00:00Z","timestamp":1675728000000},"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>Preferential attachment (PA) is a widely observed behavior in many living systems and has been used in modeling many networks. The aim of this work is to show that the mechanism of PA is a consequence of the fundamental principle of least effort. We derive PA directly from this principle in maximizing an efficiency function. This approach not only allows a better understanding of the different PA mechanisms already reported but also naturally extends these mechanisms with a non-power law probability of attachment. The possibility of using the efficiency function as a general measure of attachment efficiency is also investigated.<\/jats:p>","DOI":"10.3390\/e25020305","type":"journal-article","created":{"date-parts":[[2023,2,7]],"date-time":"2023-02-07T03:24:13Z","timestamp":1675740253000},"page":"305","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A System-Independent Derivation of Preferential Attachment from the Principle of Least Effort"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8801-6693","authenticated-orcid":false,"given":"Fran\u00e7ois Xavier","family":"Machu","sequence":"first","affiliation":[{"name":"Laboratoire SCIQ, ESIEA Group, 9 Rue V\u00e9sale, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ru Julie","family":"Wang","sequence":"additional","affiliation":[{"name":"Laboratoire SCIQ, ESIEA Group, 9 Rue V\u00e9sale, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean Louis","family":"Cheng","sequence":"additional","affiliation":[{"name":"Laboratoire SCIQ, ESIEA Group, 9 Rue V\u00e9sale, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jeremy","family":"Cocks","sequence":"additional","affiliation":[{"name":"Laboratoire SCIQ, ESIEA Group, 9 Rue V\u00e9sale, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiuping Alexandre","family":"Wang","sequence":"additional","affiliation":[{"name":"Laboratoire SCIQ, ESIEA Group, 9 Rue V\u00e9sale, 75005 Paris, France"},{"name":"IMMM, CNRS UMR 6283, Le Mans Universit\u00e9, Ave. O. 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Sci."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"4629","DOI":"10.1103\/PhysRevLett.85.4629","article-title":"Connectivity of Growing Random Networks","volume":"85","author":"Krapivsky","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"026114","DOI":"10.1103\/PhysRevE.78.026114","article-title":"Scale-free networks as preasymptotic regimes of superlinear preferential attachment","volume":"78","author":"Krapivsky","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_7","unstructured":"Zipf, G. (1949). Human Behavior and the Principle of Least Effort, Addison-Wesley."},{"key":"ref_8","unstructured":"Pareto, V. (1898). Cours D\u2019economie Politique, Librairie Droz."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","unstructured":"Pham, T., Sheridan, P., and Shimodaira, H. (2015). PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0137796"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Small, M., Li, Y., Stemler, T., and Judd, K. (2014). Super-star networks: Growing optimal scale-free networks via likelihood. arXiv.","DOI":"10.1103\/PhysRevE.91.042801"},{"key":"ref_12","unstructured":"Gao, F., and van der Vaart, A. (2022). Statistical Inference in Parametric Preferential Attachment Trees. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3738","DOI":"10.1214\/17-EJS1327","article-title":"Fitting the linear preferential attachment model","volume":"11","author":"Wan","year":"2017","journal-title":"Electron. J. Stat."},{"key":"ref_14","first-page":"362","article-title":"L\u2019inertie Mentale et la Loi du Moindre Effort","volume":"3","author":"Ferrero","year":"1894","journal-title":"Philos. Rev."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"111489","DOI":"10.1016\/j.chaos.2021.111489","article-title":"Principle of least effort vs. Maximum efficiency: Deriving Zipf-Pareto\u2019s laws, Chaos","volume":"153","author":"Wang","year":"2021","journal-title":"Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2150013","DOI":"10.1142\/S0219525921500132","article-title":"Study of a measure of efficiency as a tool for applying the principle of least effort to the derivation of the Zipf and the Pareto laws","volume":"24","author":"Machu","year":"2021","journal-title":"Adv. Complex Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6112","DOI":"10.1073\/pnas.0606779104","article-title":"Emergence of tempered preferential attachment from optimization","volume":"104","author":"Borgs","year":"2007","journal-title":"PNAS"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1749","DOI":"10.1242\/jeb.01588","article-title":"Scaling and power-laws in ecological systems","volume":"208","author":"Marquet","year":"2005","journal-title":"J. Exp. Biol."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Dorogovtsev, S., and Mendes, J. (2003). Evolution of Networks: From Biological Nets to the Internet and WWW, Oxford U. Press.","DOI":"10.1093\/acprof:oso\/9780198515906.001.0001"},{"key":"ref_20","unstructured":"Kaabouchi, A.E.L., Machu, F., Cocks, J., and Wang, Q. (2022, November 01). Uniqueness of the Efficiency Functional for Deriving the Zipf and the Pareto Laws from the Principle of Least effort. 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