{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T08:15:46Z","timestamp":1784621746568,"version":"3.55.0"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2015,3,9]],"date-time":"2015-03-09T00:00:00Z","timestamp":1425859200000},"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>Weakest-link scaling is used in the reliability analysis of complex systems. It is characterized by the extensivity of the hazard function instead of the entropy. The Weibull distribution is the archetypical example of weakest-link scaling, and it describes variables such as the fracture strength of brittle materials, maximal annual rainfall, wind speed and earthquake return times. We investigate two new distributions that exhibit weakest-link scaling, i.e., a Weibull generalization known as the \u03ba-Weibull and a modified gamma probability function that we propose herein. We show that in contrast with the Weibull and the modified gamma, the hazard function of the \u03ba -Weibull is non-extensive, which is a signature of inter-dependence between the links. We also investigate the impact of heterogeneous links, modeled by means of a stochastic Weibull scale parameter, on the observed probability distribution.<\/jats:p>","DOI":"10.3390\/e17031103","type":"journal-article","created":{"date-parts":[[2015,3,9]],"date-time":"2015-03-09T11:47:19Z","timestamp":1425901639000},"page":"1103-1122","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Weakest-Link Scaling and Extreme Events in Finite-Sized Systems"],"prefix":"10.3390","volume":"17","author":[{"given":"Dionissios","family":"Hristopulos","sequence":"first","affiliation":[{"name":"School of Mineral Resources Engineering, Technical University of Crete, Chania 73100, Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Manolis","family":"Petrakis","sequence":"additional","affiliation":[{"name":"School of Mineral Resources Engineering, Technical University of Crete, Chania 73100, Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giorgio","family":"Kaniadakis","sequence":"additional","affiliation":[{"name":"Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,3,9]]},"reference":[{"key":"ref_1","unstructured":"Sornette, D. (2004). Critical Phenomena in Natural Sciences, Springer."},{"key":"ref_2","first-page":"115","article-title":"Les valeurs extr\u00eames des distributions statistiques","volume":"5","author":"Gumbel","year":"1935","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1115\/1.4010337","article-title":"A statistical distribution function of wide applicability","volume":"18","author":"Weibull","year":"1951","journal-title":"J. Appl. Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"021122","DOI":"10.1103\/PhysRevE.82.021122","article-title":"Randomized central limit theorems: A unified theory","volume":"82","author":"Eliazar","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Barlow, R.E., and Proschan, F. (1996). Mathmatical Theory of Reliability, SIAM.","DOI":"10.1137\/1.9781611971194"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"064108","DOI":"10.1103\/PhysRevB.70.064108","article-title":"Structural disorder effects on the tensile strength distribution of heterogeneous brittle materials with emphasis on fiber networks","volume":"70","author":"Hristopulos","year":"2004","journal-title":"Phys. Rev. B"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s10704-009-9317-8","article-title":"Statistics of strength of ceramics: Finite weakest-link model and necessity of zero threshold","volume":"154","author":"Pang","year":"2008","journal-title":"Int. J. Fract."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1007\/s00603-007-0154-7","article-title":"Weibull statistical analysis of granite bending strength","volume":"41","author":"Amaral","year":"2008","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0040-1951(74)90030-4","article-title":"Probability of earthquake occurrence as obtained from a Weibull distribution analysis of crustal strain","volume":"23","author":"Hagiwara","year":"1974","journal-title":"Tectonophysics"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/0040-1951(76)90075-5","article-title":"Recurrence of great earthquakes at subduction zones","volume":"35","author":"Rikitake","year":"1976","journal-title":"Tectonophysics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0040-1951(91)90122-9","article-title":"Assessment of earthquake hazard in the Tokyo area, Japan","volume":"199","author":"Rikitake","year":"1991","journal-title":"Tectonophysics"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"238501","DOI":"10.1103\/PhysRevLett.97.238501","article-title":"Space-Time Clustering and Correlations of Major Earthquakes","volume":"97","author":"Holliday","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1111\/j.1365-246X.2007.03479.x","article-title":"Recurrence-time and frequency-slip statistics of slip events on the creeping section of the San Andreas fault in central California","volume":"170","author":"Abaimov","year":"2007","journal-title":"Geophys. J. Int."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1007\/s00024-008-0331-y","article-title":"Earthquakes: Recurrence and Interoccurrence Times","volume":"165","author":"Abaimov","year":"2008","journal-title":"Pure Appl. Geophys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1016\/j.physa.2012.09.011","article-title":"Strength statistics and the distribution of earthquake interevent times","volume":"392","author":"Hristopulos","year":"2013","journal-title":"Physica A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1175\/1520-0450(1984)023<1173:ROWSFE>2.0.CO;2","article-title":"Review of Weibull statistics for estimation of wind speed distributions","volume":"23","author":"Conradsen","year":"1984","journal-title":"J. Clim. Appl. Meteorol."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Van den Brink, H., and K\u00f6nnen, G. (2008). The statistical distribution of meteorological outliers. Geophys. Res. Lett., 35.","DOI":"10.1029\/2008GL035967"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0378-4371(03)00019-0","article-title":"Superstatistics","volume":"322","author":"Beck","year":"2003","journal-title":"Physica A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1785\/BSSA0870040789","article-title":"The paradox of the expected time until the next earthquake","volume":"87","author":"Sornette","year":"1997","journal-title":"Bull. Seismol. Soc. Am."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Chakrabarti, B.K., and Benguigui, L.G. (1997). Statistical Physics of Fracture and Breakdown in Disordered Systems, Clarendon Press.","DOI":"10.1093\/oso\/9780198520566.001.0001"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1103\/PhysRevLett.80.1445","article-title":"Size Scaling of Strength in Heterogeneous Materials","volume":"80","author":"Curtin","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1080\/00018730300741518","article-title":"Statistical models of fracture","volume":"55","author":"Alava","year":"2006","journal-title":"Adv. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"214012","DOI":"10.1088\/0022-3727\/42\/21\/214012","article-title":"Size effects in statistical fracture","volume":"42","author":"Alava","year":"2009","journal-title":"J. Phys. D"},{"key":"ref_24","unstructured":"Ditlevsen, O.D., and Madsen, H.O. (1996). Structural Reliability Methods, Wiley."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Mouslopoulou, V., and Hristopulos, D.T. (2011). Patterns of tectonic fault interactions captured through geostatistical analysis of microearthquakes. J. Geophys. Res. Solid Earth., 116.","DOI":"10.1029\/2010JB007804"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.physd.2004.01.007","article-title":"Superstatistics","volume":"193","author":"Cohen","year":"2004","journal-title":"Physica D"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"052142","DOI":"10.1103\/PhysRevE.89.052142","article-title":"Finite-size Effects on Return Interval Distributions for Weakest-link-scaling Systems","volume":"89","author":"Hristopulos","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_28","first-page":"11484","article-title":"Scaling of strength and lifetime probability distributions of quasi-brittle structures based on atomistic fracture mechanics","volume":"1061","author":"Le","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA."},{"key":"ref_29","first-page":"405","article-title":"The statistical theory of the strength of bundles of threads","volume":"183","author":"Daniels","year":"1945","journal-title":"Proc. R. Soc. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1115\/1.3157595","article-title":"Asymptotic distributions for the failure of fibrous materials under series-parallel structure and equal load sharing","volume":"48","author":"Smith","year":"1981","journal-title":"J. Appl. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"036108","DOI":"10.1103\/PhysRevE.72.036108","article-title":"Statistical mechanics in the context of special relativity II","volume":"72","author":"Kaniadakis","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3201","DOI":"10.1016\/j.physa.2008.01.109","article-title":"The \u03ba-generalized distribution: A new descriptive model for the size distribution of incomes","volume":"387","author":"Clementi","year":"2008","journal-title":"Physica"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1140\/epjb\/e2009-00161-0","article-title":"Maximum Entropy Principle and power-law tailed distributions","volume":"70","author":"Kaniadakis","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Clementi, F., Gallegati, M., and Kaniadakis, G. (A \u03ba-generalized statistical mechanics approach to income analysis, 2009). A \u03ba-generalized statistical mechanics approach to income analysis, arXiv:0902.0075.","DOI":"10.1088\/1742-5468\/2009\/02\/P02037"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3983","DOI":"10.3390\/e15103983","article-title":"Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions","volume":"15","author":"Kaniadakis","year":"2013","journal-title":"Entropy"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.jmps.2006.05.007","article-title":"Activation energy based extreme value statistics and size effect in brittle and quasi-brittle fracture","volume":"55","author":"Pang","year":"2007","journal-title":"J. Mech. Phys. Solids."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"e1002461","DOI":"10.1371\/journal.pcbi.1002461","article-title":"Power-Law Inter-Spike Interval Distributions Infer a Conditional Maximization of Entropy in Cortical Neurons","volume":"8","author":"Tsubo","year":"2012","journal-title":"PLoS Comput. Biol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"065504","DOI":"10.1103\/PhysRevLett.108.065504","article-title":"Fracture Strength of Disordered Media: Universality, Interactions, and Tail Asymptotics","volume":"108","author":"Manzato","year":"2012","journal-title":"Phys. Rev. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/3\/1103\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:43:18Z","timestamp":1760215398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/3\/1103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,9]]},"references-count":38,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2015,3]]}},"alternative-id":["e17031103"],"URL":"https:\/\/doi.org\/10.3390\/e17031103","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,3,9]]}}}