{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T23:11:03Z","timestamp":1778195463062,"version":"3.51.4"},"reference-count":56,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,16]],"date-time":"2019-02-16T00:00:00Z","timestamp":1550275200000},"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>This paper adopts the information and fractional calculus tools for studying the dynamics of a national soccer league. A soccer league season is treated as a complex system (CS) with a state observable at discrete time instants, that is, at the time of rounds. The CS state, consisting of the goals scored by the teams, is processed by means of different tools, namely entropy, mutual information and Jensen\u2013Shannon divergence. The CS behavior is visualized in 3-D maps generated by multidimensional scaling. The points on the maps represent rounds and their relative positioning allows for a direct interpretation of the results.<\/jats:p>","DOI":"10.3390\/e21020187","type":"journal-article","created":{"date-parts":[[2019,2,25]],"date-time":"2019-02-25T03:30:08Z","timestamp":1551065408000},"page":"187","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Entropy Analysis of Soccer Dynamics"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio M.","family":"Lopes","sequence":"first","affiliation":[{"name":"UISPA\u2013LAETA\/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4274-4879","authenticated-orcid":false,"given":"J. A.","family":"Tenreiro Machado","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4249-015 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.jsams.2013.03.016","article-title":"Evolution of World Cup soccer final games 1966\u20132010: Game structure, speed and play patterns","volume":"17","author":"Wallace","year":"2014","journal-title":"J. Sci. Med. Sport"},{"key":"ref_2","unstructured":"Dunning, E. (1999). 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