{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T08:10:23Z","timestamp":1770711023493,"version":"3.49.0"},"reference-count":28,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T00:00:00Z","timestamp":1629331200000},"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>Pattern analysis is a widely researched topic in team sports performance analysis, using information theory as a conceptual framework. Bayesian methods are also used in this research field, but the association between these two is being developed. The aim of this paper is to present new mathematical concepts that are based on information and probability theory and can be applied to network analysis in Team Sports. These results are based on the transition matrices of the Markov chain, associated with the adjacency matrices of a network with n nodes and allowing for a more robust analysis of the variability of interactions in team sports. The proposed models refer to individual and collective rates and indexes of total variability between players and teams as well as the overall passing capacity of a network, all of which are demonstrated in the UEFA 2020\/2021 Champions League Final.<\/jats:p>","DOI":"10.3390\/e23081072","type":"journal-article","created":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T09:58:06Z","timestamp":1629367086000},"page":"1072","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Mathematical Models to Measure the Variability of Nodes and Networks in Team Sports"],"prefix":"10.3390","volume":"23","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"},{"name":"Instituto Polit\u00e9cnico de Coimbra, IIA, ROBOCORP, 3030-329 Coimbra, 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"},{"name":"CIDAF, FCDEF, Universidade de Coimbra, 3040-248 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":[[2021,8,19]]},"reference":[{"key":"ref_1","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. 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Exerc."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","unstructured":"Gon\u00e7alves, B., Coutinho, D., Santos, S., Lago-Penas, C., Jim\u00e9nez, S., and Sampaio, J. (2017). Exploring team passing networks and player movement dynamics in youth association football. PLoS ONE, 12.","DOI":"10.1371\/journal.pone.0171156"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Behera, R.K., Rath, S.K., Misra, S., Dama\u0161evi\u010dius, R., and Maskeli\u016bnas, R. (2019). Distributed centrality analysis of social network data using MapReduce. Algorithms, 12.","DOI":"10.3390\/a12080161"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Behera, R.K., Rath, S.K., Misra, S., Dama\u0161evi\u010dius, R., and Maskeli\u016bnas, R. (2017). Large scale community detection using a small world model. Appl. Sci., 7.","DOI":"10.3390\/app7111173"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"161","DOI":"10.2478\/hukin-2021-0048","article-title":"Coexistence of distinct performance models in high-level women\u2019s volleyball","volume":"78","author":"Laporta","year":"2021","journal-title":"J. Hum. Kinet."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Laporta, L., Afonso, J., and Mesquita, I. (2018). Interaction network analysis of the six game complexes in high-level volleyball through the use of Eigenvector Centrality. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0203348"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"209","DOI":"10.2478\/hukin-2019-0044","article-title":"Passing network analysis of positional attack formations in handball","volume":"70","author":"Korte","year":"2019","journal-title":"J. Hum. Kinet."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Novak, A.R., Palmer, S., Impellizzeri, F., Garvey, C., and Fransen, J. (2021). Description of collective team behaviours and team performance analysis of elite rugby competition via cooperative network analysis. Int. J. Perform. Anal. Sport.","DOI":"10.1080\/24748668.2021.1945882"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"35","DOI":"10.5114\/hm.2021.100322","article-title":"Effects of match location, quality of opposition, match outcome, and playing position on load parameters and players\u2019 prominence during official matches in proffesional soccer players","volume":"22","author":"Clemente","year":"2021","journal-title":"Hum. Mov."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1177\/1747954120905726","article-title":"Using passing network measures to determine the performance difference between foreign and domestic outfielder players in Chinese Football Super League","volume":"15","author":"Yu","year":"2020","journal-title":"Int. J. Sports Sci. Coach."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"109625","DOI":"10.1016\/j.chaos.2020.109625","article-title":"Player position relationships with centrality in the passing network of world cup soccer teams: Win\/loss match comparisons","volume":"133","author":"Clemente","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1565","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_17","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_18","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_19","doi-asserted-by":"crossref","first-page":"052314","DOI":"10.1103\/PhysRevE.98.052314","article-title":"Examination of Markov-chain approximation in football games based on time evolution of ball-passing networks","volume":"98","author":"Yamamoto","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_20","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_21","doi-asserted-by":"crossref","unstructured":"Martins, F., Gomes, R., Lopes, V., Silva, F., and Mendes, R. (2020). Node and network entropy\u2014A novel mathematical model for pattern analysis of team sports behavior. Mathematics, 8.","DOI":"10.3390\/math8091543"},{"key":"ref_22","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_23","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_24","unstructured":"Bishop, C.M. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_25","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_26","unstructured":"Han, T.S., and Kobayashi, K. (2002). Mathematics of Information and Coding, American Mathematical Society."},{"key":"ref_27","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_28","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. (2021, June 12). Ultimate Performance Analysis Tool (uPATO). Available online: http:\/\/uPATO.it.ubi.pt."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1072\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:47:10Z","timestamp":1760165230000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1072"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,19]]},"references-count":28,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23081072"],"URL":"https:\/\/doi.org\/10.3390\/e23081072","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,19]]}}}