{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T07:57:52Z","timestamp":1770710272398,"version":"3.49.0"},"reference-count":29,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,9]],"date-time":"2020-09-09T00:00:00Z","timestamp":1599609600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Pattern analysis is a well-established topic in team sports performance analysis, and is usually centered on the analysis of passing sequences. Taking a Bayesian approach to the study of these interactions, this work presents novel entropy mathematical models for Markov chain-based pattern analysis in team sports networks, with Relative Transition Entropy and Network Transition Entropy applied to both passing and reception patterns. To demonstrate their applicability, these mathematical models were used in a case study in football\u2014the 2016\/2017 Champions League Final, where both teams were analyzed. The results show that the winning team, Real Madrid, presented greater values for both individual and team transition entropies, which indicate that greater levels of unpredictability may bring teams closer to victory. In conclusion, these metrics may provide information to game analysts, allowing them to provide coaches with accurate and timely information about the key players of the game.<\/jats:p>","DOI":"10.3390\/math8091543","type":"journal-article","created":{"date-parts":[[2020,9,9]],"date-time":"2020-09-09T09:01:09Z","timestamp":1599642069000},"page":"1543","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Node and Network Entropy\u2014A Novel Mathematical Model for Pattern Analysis of Team Sports Behavior"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1812-2300","authenticated-orcid":false,"given":"Fernando","family":"Martins","sequence":"first","affiliation":[{"name":"Instituto Polit\u00e9cnico de Coimbra, ESEC, UNICID-ASSERT, 3030-329 Coimbra, Portugal"},{"name":"Instituto de Telecomunica\u00e7\u00f5es, Delega\u00e7\u00e3o da Covilh\u00e3, 6201-001 Covilh\u00e3, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2310-1560","authenticated-orcid":false,"given":"Ricardo","family":"Gomes","sequence":"additional","affiliation":[{"name":"Instituto Polit\u00e9cnico de Coimbra, ESEC, UNICID-ASSERT, 3030-329 Coimbra, Portugal"},{"name":"Instituto Polit\u00e9cnico de Coimbra, IIA, ROBOCORP, 3030-329 Coimbra, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5577-1094","authenticated-orcid":false,"given":"Vasco","family":"Lopes","sequence":"additional","affiliation":[{"name":"Department of Informatics, Universidade da Beira Interior, 6201-001 Covilh\u00e3, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0611-6157","authenticated-orcid":false,"given":"Frutuoso","family":"Silva","sequence":"additional","affiliation":[{"name":"Instituto de Telecomunica\u00e7\u00f5es, Delega\u00e7\u00e3o da Covilh\u00e3, 6201-001 Covilh\u00e3, Portugal"},{"name":"Department of Informatics, Universidade da Beira Interior, 6201-001 Covilh\u00e3, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2433-5193","authenticated-orcid":false,"given":"Rui","family":"Mendes","sequence":"additional","affiliation":[{"name":"Instituto Polit\u00e9cnico de Coimbra, ESEC, UNICID-ASSERT, 3030-329 Coimbra, Portugal"},{"name":"Instituto Polit\u00e9cnico de Coimbra, IIA, ROBOCORP, 3030-329 Coimbra, Portugal"},{"name":"CIDAF, FCDEF, Universidade de Coimbra, 3040-248 Coimbra, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,9]]},"reference":[{"key":"ref_1","first-page":"80","article-title":"General network analysis of national soccer teams in FIFA World Cup 2014","volume":"15","author":"Clemente","year":"2015","journal-title":"Int. J. Perf. Anal. Sport"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1515\/hukin-2015-0013","article-title":"Using Network Metrics in Soccer: A Macro-Analysis","volume":"45","author":"Clemente","year":"2015","journal-title":"J. Hum. Kinet."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"McLean, S., Salmon, P.M., Gorman, A.D., Read, G.J.M., and Solomon, C. (2017). What\u2019s in a game? A systems approach to enhancing performance analysis in football. PLoS ONE, 12.","DOI":"10.1371\/journal.pone.0172565"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.jsams.2010.10.459","article-title":"Network as a novel tool for studying team ball sports as complex social system","volume":"14","author":"Passos","year":"2011","journal-title":"J. Sci. Med. Sport"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.humov.2017.10.001","article-title":"A social network analysis of the goal scoring passing networks of the 2016 European Football Championships","volume":"57","author":"Mclean","year":"2018","journal-title":"Hum. Mov. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.1007\/s40279-017-0695-1","article-title":"Team Sports Performance Analysed Through the Lens of Social Network Theory: Implications for Research and Practice","volume":"47","author":"Ribeiro","year":"2017","journal-title":"Sports Med."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1080\/1091367X.2010.495559","article-title":"The Application of Social Network Analysis to Team Sports","volume":"14","author":"Lusher","year":"2010","journal-title":"Meas. Phys. Educ. Exerc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1519\/JSC.0000000000002725","article-title":"Influence of Situational Variables, Team Formation, and Playing Position on Match Running Performance and Social Network Analysis in Brazilian Professional Soccer Players","volume":"34","author":"Aquino","year":"2020","journal-title":"J. Strength Cond. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2631","DOI":"10.1080\/02640414.2019.1589919","article-title":"Network-based centrality measures and physical demands in football regarding player position: Is there a connection? A preliminary study","volume":"37","author":"Castellano","year":"2019","journal-title":"J. Sports Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.humov.2017.08.022","article-title":"Applying graphs and complex networks to football metric interpretation","volume":"57","author":"Zuniga","year":"2018","journal-title":"Hum. Mov. Sci."},{"key":"ref_11","unstructured":"Martins, F.M.L., Silva, F., Clemente, F., Gomes, A.J.P., Correia, A., Nguyen, Q., Sequeiros, J.B., Ribeiro, J.S., and Lopes, V.F. (2020, March 12). Ultimate Performance Analysis Tool (uPATO). Available online: http:\/\/uPATO.it.ubi.pt."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Silva, F., Nguyen, Q., Correia, A., Clemente, F., and Martins, F.M.L. (2019). Ultimate Performance Analysis Tool (uPATO): Implementation of Network Measures Based on Adjacency Matrices for Team Sports, Springer International Publishing.","DOI":"10.1007\/978-3-319-99753-7"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Yamamoto, K., and Narizuka, T. (2018). Examination of Markov-chain approximation in football games based on time evolution of ball-passing networks. Phys. Rev. E, 98.","DOI":"10.1103\/PhysRevE.98.052314"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.physa.2014.06.037","article-title":"Statistical Properties of Position-Dependent Ball-Passing Networks in Football Games","volume":"412","author":"Narizuka","year":"2014","journal-title":"Phys. A"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wasserman, S., and Faust, K. (1994). Social Network Analysis: Methods and Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511815478"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Clemente, F., Martins, F., and Mendes, R. (2016). Social Network Analysis Applied to Team Sports Analysis, Springer International Publishing.","DOI":"10.1007\/978-3-319-25855-3"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Manning, C.D., Raghavan, P., and Sch\u00fctze, M. (2008). Introduction to Information Retrieval, Cambridge University Press.","DOI":"10.1017\/CBO9780511809071"},{"key":"ref_18","unstructured":"Bishop, C.M. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Tuckwell, H. (1988). Elementary Applications of Probability Theory, Chapman and Hall Ltd.","DOI":"10.1007\/978-94-009-1221-2"},{"key":"ref_21","unstructured":"Han, T.S., and Kobayashi, K. (2002). Mathematics of Information and Coding, American Mathematical Society."},{"key":"ref_22","unstructured":"Rao, C.R., and Gudivada, V.N. (2018). Computational Analysis and Understanding of Natural Languages: Principles, Methods and Applications, Elsevier."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Marinescu, D.C., and Marinescu, G.M. (2011). Classical and Quantum Information, Academic Press.","DOI":"10.1016\/B978-0-12-383874-2.00003-5"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Pina, T.J., Paulo, A., and Ara\u00fajo, D. (2017). Network Characteristics of Successful Performance in Association Football. A Study on the UEFA Champions League. Front. Psychol., 8.","DOI":"10.3389\/fpsyg.2017.01173"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Neuman, Y., Israeli, N., Vilenchik, D., and Cohen, Y. (2018). The Adaptive Behavior of a Soccer Team: An Entropy-Based Analysis. Entropy, 20.","DOI":"10.3390\/e20100758"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-020-58549-8","article-title":"From physical to social interactions: The relative entropy model","volume":"10","author":"Neuman","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"43591","DOI":"10.1109\/ACCESS.2019.2907067","article-title":"Modeling Small Systems through the Relative Entropy Lattice","volume":"7","author":"Neuman","year":"2019","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Mart\u00ednez, J.H., Garrido, D., Herrera-Diestra, J.L., Busquets, J., Sevilla-Escoboza, R., and Buld\u00fa, J.M. (2020). Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective. Entropy, 22.","DOI":"10.3390\/e22020172"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"51","DOI":"10.18280\/ijht.34S107","article-title":"Design and flow in basketball","volume":"34","author":"Arriaza","year":"2016","journal-title":"Int. J. Heat Technol."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/8\/9\/1543\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:08:21Z","timestamp":1760177301000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/8\/9\/1543"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,9]]},"references-count":29,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["math8091543"],"URL":"https:\/\/doi.org\/10.3390\/math8091543","relation":{},"ISSN":["2227-7390"],"issn-type":[{"value":"2227-7390","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,9]]}}}