{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T08:46:07Z","timestamp":1770713167071,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2022,6,24]],"date-time":"2022-06-24T00:00:00Z","timestamp":1656028800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\/MCTES","award":["UIDB\/50008\/2020"],"award-info":[{"award-number":["UIDB\/50008\/2020"]}]},{"name":"EU funds","award":["UIDB\/50008\/2020"],"award-info":[{"award-number":["UIDB\/50008\/2020"]}]},{"name":"IPC-I2A","award":["UIDB\/50008\/2020"],"award-info":[{"award-number":["UIDB\/50008\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>Considering the Social Network Analysis approach and based on the creation of mathematical models, the aim of this study is to analyze the players\u2019 interactions of professional football teams in critical moments of the game. The sample consists in the analysis of a 2019\/2020 season UEFA Champions League match. The mathematical models adopted in the analysis of the players (micro analysis) and the game (macro analysis) were obtained through the uPATO software. The results of the networks indicated a performance pattern trend more robust in terms of the mathematical model: Network Density. As far as it concerned, we found that the Centroid Players had a decisive role in the level of connectivity and interaction of the team. Regarding the main critical moments of the game, the results showed that these were preceded by periods of great instability, obtaining a differentiated performance in the following mathematical models: Centrality, Degree Centrality, Closeness Centrality, and Degree Prestige. We concluded that the networks approach, in concomitance with the dynamic properties of mathematical models, and the critical moments of the game, can help coaches to better evaluate the level of interaction and connectivity of their players toward the actions imposed by opponents.<\/jats:p>","DOI":"10.3390\/app12136433","type":"journal-article","created":{"date-parts":[[2022,6,25]],"date-time":"2022-06-25T10:39:13Z","timestamp":1656153553000},"page":"6433","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Social Network Analysis: Mathematical Models for Understanding Professional Football in Game Critical Moments\u2014An Exploratory Study"],"prefix":"10.3390","volume":"12","author":[{"given":"Diana","family":"Assun\u00e7\u00e3o","sequence":"first","affiliation":[{"name":"Coimbra Business School, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4713-2759","authenticated-orcid":false,"given":"Isabel","family":"Pedrosa","sequence":"additional","affiliation":[{"name":"Coimbra Business School, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"ISTAR-IUL, 1649-026 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2433-5193","authenticated-orcid":false,"given":"Rui","family":"Mendes","sequence":"additional","affiliation":[{"name":"Coimbra School of Education, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"Applied Research Unit (IIA), Robocorp, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"Research Unit for Sport and Physical Activity\u2014CIDAF, University of Coimbra, 3040-248 Coimbra, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1812-2300","authenticated-orcid":false,"given":"Fernando","family":"Martins","sequence":"additional","affiliation":[{"name":"Coimbra School of Education, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"Applied Research Unit (IIA), Robocorp, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"Instituto de Telecomunica\u00e7\u00f5es, 6201-001 Covilh\u00e3, Portugal"}]},{"given":"Jo\u00e3o","family":"Francisco","sequence":"additional","affiliation":[{"name":"Coimbra School of Education, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2310-1560","authenticated-orcid":false,"given":"Ricardo","family":"Gomes","sequence":"additional","affiliation":[{"name":"Coimbra School of Education, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"Applied Research Unit (IIA), Robocorp, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"Research Unit for Sport and Physical Activity\u2014CIDAF, University of Coimbra, 3040-248 Coimbra, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3774-6661","authenticated-orcid":false,"given":"Gon\u00e7alo","family":"Dias","sequence":"additional","affiliation":[{"name":"Coimbra School of Education, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"Applied Research Unit (IIA), Robocorp, Polytechnic Institute of Coimbra, 3030-329 Coimbra, Portugal"},{"name":"Research Unit for Sport and Physical Activity\u2014CIDAF, University of Coimbra, 3040-248 Coimbra, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,24]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","unstructured":"Clemente, F., Martins, F.L., 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_3","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1080\/24748668.2014.11868752","article-title":"Network Analysis and Intra-Team Activity in Attacking Phases of Professional Football","volume":"14","author":"Gama","year":"2014","journal-title":"Int. J. Perform. Anal. Sport"},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"110369","DOI":"10.1016\/j.chaos.2020.110369","article-title":"Is a Social Network Approach Relevant to Football Results?","volume":"142","author":"Medina","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1630","DOI":"10.1038\/s41598-020-80091-w","article-title":"Social Network Structure and Composition in Former NFL Football Players","volume":"11","author":"Dhand","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1002\/cplx.21813","article-title":"Networks Metrics and Ball Possession in Professional Football","volume":"21","author":"Gama","year":"2016","journal-title":"Complexity"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Silva, F.G.M., Correia, A.F.P.P., Clemente, F., Martins, F.L., and Nguyen, Q.T. (2019). Ultimate Performance Analysis Tool (UPATO) Implementation of Network Measures Based on Adjacency Matrices for Team Sports, Springer.","DOI":"10.1007\/978-3-319-99753-7"},{"key":"ref_9","first-page":"353","article-title":"Homogeneous Distribution of Passing between Players of a Team Predicts Attempts to Shoot at Goal in Association Football: A Case Study with 10 Matches","volume":"24","author":"Gama","year":"2020","journal-title":"Nonlinear Dyn. Psychol. Life Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Martins, F., Gomes, R., Lopes, V., Silva, F., and Mendes, R. (2020). Node and Network Entropy-A Novel Mathematical Model for Pattern Analysis of Team Sports Behavior. Mathematics, 8.","DOI":"10.3390\/math8091543"},{"key":"ref_11","first-page":"85","article-title":"Small-world networks in professional football: Conceptual model and data","volume":"35","author":"Gama","year":"2015","journal-title":"Eur. J. Hum. Mov."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"70","DOI":"10.15446\/ing.investig.v34n3.43602","article-title":"Sistemas Inteligentes Para El An\u00e1lisis de F\u00fatbol: Centroide Ponderado","volume":"34","author":"Clemente","year":"2014","journal-title":"Ing. E Investig."},{"key":"ref_13","first-page":"75","article-title":"Networks and Centroid Metrics for Understanding Football","volume":"38","author":"Gama","year":"2016","journal-title":"S. Afr. J. Res. Sport Phys. Educ. Recreat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1650102","DOI":"10.1142\/S0129183116501023","article-title":"An ecological dynamics rationale to explain home advantage in professional football","volume":"27","author":"Gama","year":"2016","journal-title":"Int. J. Mod. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1080\/02640414.2012.703783","article-title":"Variability of Inter-Team Distances Associated with Match Events in Elite-Standard Soccer","volume":"30","author":"Frencken","year":"2012","journal-title":"J. Sports Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1080\/24748668.2016.11868891","article-title":"The Effect of Performance Indicators on the Time the First Goal Is Scored in Football Matches","volume":"16","author":"Pratas","year":"2016","journal-title":"Int. J. Perform. Anal. Sport"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1007\/s40279-017-0836-6","article-title":"What Performance Analysts Need to Know About Research Trends in Association Football (2012\u20132016): A Systematic Review","volume":"48","author":"Sarmento","year":"2017","journal-title":"Sports Med."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40279-016-0562-5","article-title":"Current Approaches to Tactical Performance Analyses in Soccer Using Position Data","volume":"47","author":"Memmert","year":"2017","journal-title":"Sports Med."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Janetzko, H., Sacha, D., Stein, M., Schreck, T., Keim, D.A., and Deussen, O. (2014, January 25\u201331). Feature-Driven Visual Analytics of Soccer Data. Proceedings of the 2014 IEEE Conference on Visual Analytics Science and Technology (VAST), Paris, France.","DOI":"10.1109\/VAST.2014.7042477"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1080\/16843703.2015.11673436","article-title":"Discovering the Drivers of Football Match Outcomes with Data Mining","volume":"12","author":"Carpita","year":"2016","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1055\/s-2006-924294","article-title":"Performance Characteristics According to Playing Position in Elite Soccer","volume":"28","author":"Baron","year":"2007","journal-title":"Int. J. Sports Med."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"81","DOI":"10.5628\/rpcd.09.01.81","article-title":"Trends of Tactical Performance Analysis in Team Sports: Bridging the Gap between Research, Training and Competition","volume":"9","author":"Garganta","year":"2009","journal-title":"Rev. Port. Ci\u00eancias Desporto"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"158","DOI":"10.52082\/jssm.2021.158","article-title":"Attacking Key Performance Indicators in Soccer: Current Practice and Perceptions from the Elite to Youth Academy Level","volume":"20","author":"Herold","year":"2021","journal-title":"J. Sports Sci. Med."},{"key":"ref_24","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. (2022, June 12). Ultimate Performance Analysis Tool (uPATO). Available online: http:\/\/uPATO.it.ubi.pt."},{"key":"ref_25","first-page":"57","article-title":"Observa\u00e7\u00e3o Como Instrumento No Processo de Avalia\u00e7\u00e3o Em Educa\u00e7\u00e3o F\u00edsica","volume":"6","author":"Mendes","year":"2012","journal-title":"Rev. Cient\u00edfica Exedra"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Martins, F., Gomes, R., Lopes, V., Silva, F., and Mendes, R. (2021). Mathematical Models to Measure the Variability of Nodes and Networks in Team Sports. Entropy, 23.","DOI":"10.3390\/e23081072"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1900","DOI":"10.3389\/fpsyg.2018.01900","article-title":"Using Network Science to Analyse Football Passing Networks: Dynamics, Space, Time, and the Multilayer Nature of the Game","volume":"9","author":"Busquets","year":"2018","journal-title":"Front. Psychol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1080\/17461391.2015.1042527","article-title":"Modelling Relationships between Match Events and Match Outcome in Elite Football","volume":"16","author":"Liu","year":"2016","journal-title":"Eur. J. Sport Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1016\/j.procs.2020.01.017","article-title":"Automated Goal Score Detection in Football Match Using Key Moments","volume":"165","author":"Penumala","year":"2019","journal-title":"Procedia Comput. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2566","DOI":"10.3389\/fpsyg.2019.02566","article-title":"An Extensive Comparative Analysis of Successful and Unsuccessful Football Teams in LaLiga","volume":"10","author":"Resta","year":"2019","journal-title":"Front. Psychol."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/12\/13\/6433\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:39:55Z","timestamp":1760139595000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/12\/13\/6433"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,24]]},"references-count":30,"journal-issue":{"issue":"13","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["app12136433"],"URL":"https:\/\/doi.org\/10.3390\/app12136433","relation":{},"ISSN":["2076-3417"],"issn-type":[{"value":"2076-3417","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,24]]}}}