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Edited by: Nancy J. Cooke, Arizona State University, United States

Reviewed by: Jamie Gorman, Georgia Institute of Technology, United States; Mustafa Demir, Arizona State University, United States; Aaron Likens, Arizona State University, United States

*Correspondence: João Ramos

This article was submitted to Performance Science, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The combination of sports sciences theorization and social networks analysis (SNA) has offered useful new insights for addressing team behavior. However, SNA typically represents the dynamics of team behavior during a match in dyadic interactions and in a single cumulative snapshot. This study aims to overcome these limitations by using hypernetworks to describe illustrative cases of team behavior dynamics at various other levels of analyses. Hypernetworks simultaneously access cooperative and competitive interactions between teammates and opponents across space and time during a match. Moreover, hypernetworks are not limited to dyadic relations, which are typically represented by edges in other types of networks. In a hypernetwork,

Coaches, players, and scientists have long tried to understand team behavior dynamics during a game, aiming to develop interventions and training plans that may increase team performance (Araújo and Davids,

These interactions, based on informational and physical constraints have been studied by network theorical approaches, like social network analysis (SNA). SNA is a powerful tool to capture and study interpersonal relations in team sports (Araújo and Davids, _{i}, …, _{j}〉 (Johnson,

In the present study, we have extended the approach by Johnson and Iravani (

By considering the domain specificity of soccer matches to tag the sets of players formed (e.g., 2 vs. 1 corresponds to a set with two attackers and one defender) as these tags describe local dominance (Duarte et al.,

By including the spatiotemporal occurrence of the different sets of players by counting their frequency and location;

By analyzing and relating the dynamics of the sets with players velocity in specific events (goal scoring opportunities);

By studying, for the same events of interest, the formation and dynamics of higher level simplices; notably, the relations between simplices of simplices.

The present approach is applied to a set of matches in order to investigate how the proposed compound variables can be useful on characterizing the behavior of players and teams at different levels and the relationships between these levels and match context, e.g., team local dominance and current match result.

As a first step in this approach, it is necessary, at each level of analysis, to identify the meaningful relations for the match dynamics, and represent them using different criteria for selecting the players in each set (i.e., connected by a hyperedge; Johnson, _{1}, _{2, …}, _{n}). The compound variables adopted in this study reflect and capture this space and temporal features, e.g., local dominance and the dynamics, i.e., changes on, players' sets.

In Figure _{1} and _{2},), a defender (_{1}), a goalkeeper (_{0}), and a goal (_{a}). These nodes are connected by two hyperedges at Level _{1}, _{2}, _{1}〉 and 〈_{0}, _{a}〉 in one time frame, and 〈_{1}, _{1}〉 and 〈_{2}, _{0}, _{a}〉 on the next.

Multilevel hypernetwork representation (from bottom to top). Each level corresponds to a different abstraction level (Level

For a more complete description of the system's dynamics, each tuple identified in the hypernetwork can be extended by an element, _{1}, _{2}, _{1}〉 and 〈_{0}, _{a}〉 on one frame lead to the sets and 〈_{1}, _{1}〉 and 〈_{2}, _{0}, _{a}〉 on the next. When a player observes the game searching for the best action possibilities offered by the other players' positioning, the entire configuration of team-mates and opponents has to be perceived. Such sets of players, either in 1vs.1, 2vs.1, or 2vs.2, or any other set, may be related to one another, regarding the players' general configuration. Thus, when one player decides to move, the entire configuration is affected. Johnson and Iravani (

In this study, we propose several compound variables to describe the players' cooperative and competitive behavior dynamics during a soccer match. The simplest of these variables depicts the dominant interactions in each set, and is expressed by two values representing the number of attacking and defending players, for example, 2 vs. 1 corresponds to a set with two attackers and one defender. In Figure _{1} = (2 _{2} = (0 _{1} = 〈_{1}, _{2}, _{1}; (2 _{2} = 〈_{0}, _{a}; (0

At higher complexity levels, the hypernetwork can represent the interactions between related simplices, or simplices of simplices (see Figure

In this study we put forward the hypothesis that hypernetworks and compound variables over these hypernetworks can capture relevant features of soccer team dynamics during a match. We validate qualitatively this hypothesis by applying the proposed method to a set of matches of a focal team within different contexts and by analysis the results thus obtained. The aim of this study was therefore to operationalize a method addressing different levels of hypernetworks on soccer matches and by providing a study case for tackling the following questions:

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Five matches were analyzed from a pool of 11 matches of the English Premier League season 2010–2011 provided by STATS (formally Prozone). This data set was selected because it contained no errors, such as, missing or duplicated positioning data, and because the

Matches and their score were: Team A vs. Team B (1–0); Team A vs. Team C (1–0); Team A vs. Team D (1–0); Team E vs. Team A (2–1) and Team F vs. Team A (0–0). The details for each match are presented in Table

Matches' details indicating the result and changes in the team structure due to sent-offs, substitutions, or injuries (without substitution).

Results | 1–0 | 1–0 | 1–0 | 2–1 | 0–0 |

Substitutions | 3–3 | 3–3 | 3–3 | 3–3 | 2–2 |

Sent-offs | 0–0 | 0–0 | 0–0 | 0–0 | 1–1 |

Injuries (without substitution) | 0–0 | 0–0 | 0–0 | 0–0 | 0–0 |

For each match, raw data consisted of two-dimensional player displacement coordinates provided by STATS. These data were obtained by a multiple-camera match analysis system whereby the movements of the 22 players during the match were recorded with eight cameras positioned at the top of the stadium. The frames were processed at 10 Hz through an automated system that synchronized the video files. The effective playing area was 80 m wide and 120 m long, including the out-of-bound locations such as, set-plays. A computer procedure for computing the simplices' hyperedges set with the proximity criterion was implemented using GNU Octave version 4.2.0 and applied to each frame. This criterion has the advantage of being non-parametric; the corresponding pseudo-code for this algorithm is provided in Figure

Each simplex was represented graphically by the convex hull computation (the minimum convex area containing all players in the simplex) and included the velocity of each player (vector velocity considering the instant

To represent the field positioning of the different types of simplices, we used heat maps for the frequency of simplices occurrence. This type of graphical representation allowed us to capture the most frequent type of simplices for each time period, as well as their geographical position in the field.

For analyzing specific time points, we represented simplices (

Our results revealed how the matches' hypernetworks are characterized from

We analyzed the structure at

We computed the relative frequencies of the simplices structures at Level

Histogram for the most frequently occurring simplices structures in the 5 matches: 1vs.1; 2vs.1; 1vs.2; 2vs.2; 3vs.1; 1vs.3. The matches (and score) were: Team A vs. team B (1–0); Team A vs. Team C (1–0); Team A vs. Team D (1–0); Team E vs. Team A (2–1); and Team F vs. Team A (0–0).

By computing the frequencies for the “local dominance tag” compound variable it is possible to investigate for each game the most frequent cooperation and competition interactions sets.

Figure

By identifying the relevant events in a match, such as, changes in the score, at

Simplices in a sequence of nine frames (58′23″ to 58′31″) leading to a goal by Team A. Visiting players are attacking from right to left (represented in green), while home players are attacking from left to right (represented in red, including the opponents' goal). A simplex is represented by the polygon (or a line when there are only two players) defining the convex hull (or envelope) that links the nodes (players or goal). A velocity vector for each player is also presented.

The example in Figure

Higher-order simplices (simplices of simplices) in a sequence of five frames before team A scores a goal. Higher-order simplices are represented by the polygon (and lines) forming the convex hull (−) that connects the geographical centers of the

The example of

The different levels of analysis of a hypernetwork can capture various degrees of team behavior dynamics, from player, to simplices, and to interactions between simplices across space and time.

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Important interpretations can be inferred from the simplices at

Concerning simplices with an unbalanced number of players, 2vs.1 occurred more often in the center of the pitch and in the opponent middle field (similarly to 2vs.2 in the match lost against team E). The 1vs.2 simplices were also detected more often in the middle fields. Simplices 3vs.1 were distributed in the center of Team A's middle field, however, in the match against team E, they were more distant from their own goal (in the middle field). In the opposite way, in the matches against teams B and F, there were some notable occurrences of 3vs.1 simplices near team's A goal. Moreover, in these matches, 1vs.3 occurred near the center but more toward team A's middle field, suggesting that team B and F “forced” team A players away from their goal.

The results obtained considered both geographical placement and context dependency, and showed that the use of simplices formation captured match properties, such as, local dominance. These properties emerge in each match event resulting from the local interaction between players of both teams. Multilevel hypernetworks proved to be a useful method in answering to chief problems such as, the relation among micro (e.g., players' positions), meso (e.g., local dominance), and macro levels (e.g., match result). Moreover, the use of hypernetworks allows that the analysis can consider more than the typical (in SNA)

The analysis of the dynamics of simplices interactions at

The results show that certain moves performed by the player who scored the goal (player _{7}) had significant impact on some simplices transformations, for example, at instants 58′27″, 58′28″, 58′29″, 58′30″, and goal scored. Player _{10} had an important role in promoting balance in the simplex that scored the goal (with player _{7}), by maintaining defender _{19} distant from his teammate _{16}. Moreover, player _{19} appeared to be facing the defender's dilemma, hesitating between defending his opponent (player _{10}) and supporting his teammate (player _{16}). Player _{24} was also essential in the attack play leading to the goal scored, as he lost the ball but kept pursuing it, almost reaching player _{7} and thereby including him into his simplex. Finally, player _{6} broke the central simplex (containing teammate _{7}) by attracting a defender toward him and hence reducing the number of players in the central middle field.

Results showed that by considering the temporal sequence of simplices transformations during critical events of the match (e.g., from ball recovery to scoring a goal) the dynamics of interaction among players is captured. Moreover, it is possible to analyze how interactions among players led to changes in simplices' structures and, consequently to such critical events (e.g., a goal scoring opportunity). Multilevel hypernetworks offer a fine temporal grain of analysis of how the micro-meso-macro level relationships emerge.

This study showed that the hypernetworks' analysis by considering simplices of simplices reveal the degree of connection between sub-sets of players.

We have applied multilevel hypernetworks analysis, and a set of associated compound variables, to selected soccer matches by using positional variables for all players involved.

The interactions between players, as well as the sets of these interactions (simplices), were assessed based on interpersonal distance, more specifically

Our results revealed a pattern in these interactions' dynamics that was independent of the type (home or away) and score of the match. Specifically, in every match analyzed the most frequently occurring simplices structures were, by decreasing order of frequency, 1vs.1, 2vs.1 and 1vs.2, 2vs.2, and finally, 3vs.1 and 1vs.3.

However, these simplices show differences in their distribution on the pitch, and this is particularly evident for unbalanced simplices such as, 2vs.1, 1vs.2, 3vs.1, and 1vs.3. These differential distributions are consistent with the match result (wins vs. losses) and the opponent team's strength.

We analyzed the changes in local dominance at

Finally, our last and global analysis level revealed how all the simplices were connected, but most importantly, it enabled to permanently connect all the simplices into larger hypersimplices, including the goal and goalkeeper simplex, and also the defenders and attackers who were distant from the goal.

These results may significantly contribute to improve training and playing strategies. We highlight the importance of mastering 1vs.1 situations (with and without the ball), as this structure occurs more frequently in all types of matches. For example, coaches could design exercises to train players to rapidly transform any structure into a 1vs.1 structure. Unbalanced situations such as, 2vs.1 and 3vs.1 typically reveal which team is dominating the match, particularly when those structures occur on the attacking side of that team's field. Thus, designing training exercises that create an overload for the attacking team may allow players to better adapt to such situations in a match. Finally, we found that as an attacking team moves closer to the goal, changes in player speed become more pronounced. It is therefore likely that encouraging such speed changes during training may facilitate the players' positioning inside finishing areas during a match.

Moreover, when players are connected with other players (in cooperation or competition) forming simplices, where the smaller simplices are also connected with other simplices, team coordination develops due to attunement to shared affordances and the creation of team synergies (Araújo and Davids,

In the context of this article the criterion, closest player, for the formation of hyperedges was the only one used. The results presented at different levels of analysis are therefore conditioned and limited by this criterion. At the same time all these results where possible with only this parsimonious criterion and without any other assumptions.

Other limitation of the study is that there is no data about ball positioning, nor about “ball flux” (e.g., passes between the players). This type of interactions between players could be included by extending the proposed method with additional layers. In such layers, ball flux could be represented either as a link between players' or simplices, or alternatively as an additional term in the relationship,

Multilevel hypernetworks is a promising framework for soccer performance analysis that reveals important features of cooperative and competitive interactions during attacking plays. By considering space and time in multilevel analyses involving interactions between two or more players, we can obtain a richer understanding of real-world complex systems.

JR, main contribution regarding theoretical approach, method and results production. RL, significant contribution regarding method, software computation, and results production. PM, significant contribution on results reading and discussion and the impact to practitioners. DA, significant contribution regarding performance analysis and the general impression.

PM is affiliated with the commercial company, City Football Services. The other authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewers AL and MD declared their shared affiliation with the handling editor, and the handling Editor states that the process nevertheless met the standards of a fair and objective review

Pseudocode for building the simplex hyperedge set.