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What’s Next in Complex Networks? Capturing the Concept of Attacking Play in Invasive Team Sports

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Abstract

The evolution of performance analysis within sports sciences is tied to technology development and practitioner demands. However, how individual and collective patterns self-organize and interact in invasive team sports remains elusive. Social network analysis has been recently proposed to resolve some aspects of this problem, and has proven successful in capturing collective features resulting from the interactions between team members as well as a powerful communication tool. Despite these advances, some fundamental team sports concepts such as an attacking play have not been properly captured by the more common applications of social network analysis to team sports performance. In this article, we propose a novel approach to team sports performance centered on sport concepts, namely that of an attacking play. Network theory and tools including temporal and bipartite or multilayered networks were used to capture this concept. We put forward eight questions directly related to team performance to discuss how common pitfalls in the use of network tools for capturing sports concepts can be avoided. Some answers are advanced in an attempt to be more precise in the description of team dynamics and to uncover other metrics directly applied to sport concepts, such as the structure and dynamics of attacking plays. Finally, we propose that, at this stage of knowledge, it may be advantageous to build up from fundamental sport concepts toward complex network theory and tools, and not the other way around.

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Notes

  1. Analyzing only at ball passing restricts the analysis of team performance to the attacking phase. In the current article, we do not attempt to directly resolve this limitation.

  2. An adjacency matrix, A, is a square matrix, with rows and columns representing nodes (e.g., players) with entry \(a_{ij}\) of A taking value 1 if there is a link between node i and node j; and 0 otherwise. Different types of networks lead to different matrix structures: undirected graphs are represented in symmetric adjacency matrices, the fact that the link between nodes i and j has no directionality is expressed in equality \(a_{ij} = a_{ji} ;\) in directed graphs (or digraphs), the links between nodes have a directionality; a link from node i to node j is expressed by entry \(a_{ij}\) taking value 1 independently of the value of \(a_{ji}\). In this article, the links represent actions by the players (e.g., making a pass) and are thus directed leading to digraphs. In what are called weighted graphs, the entries of the matrix can take other values \(w_{ij}\), called weights, that are not restricted to 0 or 1. The value taken by entry \(w_{ij}\) reflects the intensity or strength of that link. In an incidence matrix, E, rows represent nodes and columns represent links. The entry \(e_{ij}\) takes value 1 if the link j is incident on nodes i and j; 0 otherwise. In directed networks, values −1 and 1 are used to distinguish link origin and destination.

  3. Degree of a vertex \(i, {\text{hence }}v_{i}\) is given by the number of nodes that are directly connected with the focal node: \({\text{Centrality}}_{\text{degree}} \left( {v_{i} } \right) = {\text{degree}}\left( {v_{i} } \right) = \mathop \sum \nolimits_{j}^{N} a_{ij}\) where \(i\) is the focal node, \(j\) represents all other nodes, \(N\) is the total number of nodes, and \(a\) is the adjacency matrix, in which cell \(a_{ij}\) is defined as 1 if node \(i\) is connected to node \(j\); and 0 otherwise.

  4. Betweenness centrality expresses the degree in which one node lies on the shortest path between two other nodes: \({\text{Centrality}}_{\text{betweenness}} \left( {v_{i} } \right) = {\text{betweenness}}_{i} = \frac{{g_{st} (i)}}{{g_{st} }}\) where \(g_{st}\) is the number of shortest paths between vertices s and t, and \(g_{st} (i)\) is the number of those paths that pass through vertex i.

  5. Closeness centrality for each node, \(v_{i}\) is the inverse sum of the shortest distance, \({\text{distance}}\;(i,j)\) to all other nodes, j, from the focal node, i, or how long the information takes to spread from a given node to others \({\text{Closeness}}_{\text{centrality}} (v_{i} ) = {\text{closenness}}_{c} (i) = [\varSigma_{j = 1}^{N} {\text{distance}}\; ( {\text{i,j)}}]^{ - 1}\).

  6. Eigenvector centrality takes into consideration not only how many connections a vertex has (i.e., its degree), but also the degree of the vertices that it is connecting to. Each vertex \(i\) is assigned a weight \(x_{i} > 0\), which is defined to be proportional to the sum of the weights of all vertices that point to \(i: x_{i} = \lambda^{ - 1} \mathop \sum \nolimits_{j} A_{ij} x_{j}\) for some \(\lambda > 0\), or in matrix form: \(Ax = \lambda x\), where \(A\) is the (asymmetric) adjacency matrix of the graph, whose elements are \(A_{ij}\), and \(x\) is the vector whose elements are the \(x_{i}\), and \(\lambda\) is a constant (the eigenvalue).

  7. Preferential attachments, also known as cumulative advantage or ‘rich-get-richer paradigm’. This property means that every new vertex probability \((p_{i} )\) to connect the existing vertices is higher for those who have already a large number of connections (connectivity \(k_{i}\)). For example, in a given team sports with a ball, when a player attracts more interactions from the game’s beginning, his/her connectivity will increase at a higher rate when compared with his/her team mates as the game is played (network grows). Therefore, starting with a small number \((m_{0} )\) of players interacting at the beginning of the game, at every time step that a new player \(m( \le m_{0} )\) interacts with \(m\) different team mates already active in the game, for preferential attachment, there is a probability \(p_{i} (k_{i} ) = \frac{{k_{i} }}{{\mathop \sum \nolimits_{j} k_{j} }}\) that the new player \(i\) will interact with a certain team mate, depending on the connectivity \(k_{i}\) of the latter.

  8. The local clustering coefficient \(({\text{cc}}_{i} )\) for player \(i\) is defined by the proportion of actual edges/interactions \((e_{i} )\) between the \(n_{i} \ge 2\) common neighbors of a vertex/player i and the number of possible edges between them. \({\text{cc}}_{i} = \frac{{2e_{i} }}{{n_{i} \;(n_{i} - 1)}}\). The local clustering coefficient over the aggregate of all plays (Fig. 4) takes the following values: \({\text{cc}}_{\text{GK}} = 0,{\text{cc}}_{\text{LD}} = 1,{\text{cc}}_{\text{RD}} = 1, {\text{cc}}_{\text{MF}} = \frac{2}{3}, {\text{cc}}_{\text{CF}} = 1,\) where GK is the goalkeeper, LD/RD is the left/right defender, MF is the midfielder, and CF is the center forward.

  9. The \(j{\text{th}}\) play local clustering coefficient \(( {\text{cc}}_{i} )\) for player \(i\) in the \(j{\text{th}}\) attacking is defined in a similar manner to the local clustering coefficient but takes into account only the players’ projection network formed in the \(j{\text{th}}\) attacking play. The \(2{\text{nd}}\)lay local clustering coefficient, \({\text{cc}}_{i,2} ,\) (Fig. 3) takes the following values:

    \({\text{cc}}_{{{\text{GK}},2}} = 0,{\text{cc}}_{{{\text{LD}},2}} = 1,{\text{cc}}_{{{\text{RD}},2}} = \frac{1}{2},{\text{cc}}_{{{\text{MF}},2}} = 1,{\text{cc}}_{{{\text{CF}},2}} = 0\) where GK is the goalkeeper, LD/RD is the left/right defender, MF is the midfielder, and CF is the center forward.

  10. The \(k\) aggregation local clustering coefficient \(( {\text{cc}}_{i,k}^{*} )\) for player \(i\) is defined by the average of the local cluster coefficients for player \(i\) over the \(M_{k} (k_{1} \;{\text{to}}\;k_{M} )\) attacking plays that compose the \(k\) aggregation. \(cc_{i,k}^{*} = \frac{1}{{M_{k} }}.\) The aggregate play local clustering coefficient, for the k aggregate composed of attacking plays 1 and 2, has the following values for each of the players: \({\text{cc}}_{\text{GKk}}^{*} = 0,\;{\text{cc}}_{\text{LD,k}}^{*} = \frac{1}{2}\left( {0 + 1} \right) = \frac{1}{2},\;{\text{cc}}_{\text{RD,k}}^{*} = \frac{1}{2}\left( {0 + \frac{1}{2}} \right) = \frac{1}{4},\;{\text{cc}}_{\text{MF,k}}^{*} = \frac{1}{2},\;cc_{CF,k}^{*} = 0,\) where GK is the goalkeeper, LD/RD is the left/right defender, MF is the midfielder, and CF is the center forward.

  11. We define as the static network the static structure resulting from the aggregation over a time interval (e.g., the entire match) of all the observable edges (e.g., passes) within that interval.

  12. Voronoi diagrams are geometric constructions representing the nearest geographical region of a player, a subset of a team, or even a team.

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Authors and Affiliations

  1. ISCTE-Instituto Universitário de Lisboa, Lisbon, Portugal

    João Ramos & Rui J. Lopes

  2. Universidade Europeia, Laureate International Universities, Lisbon, Portugal

    João Ramos

  3. Instituto de Telecomunicações, Lisbon, Portugal

    Rui J. Lopes

  4. CIPER, Faculdade de Motricidade Humana, Universidade de Lisboa, Cruz Quebrada Dafundo, Lisbon, Portugal

    Duarte Araújo

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  1. João Ramos
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Correspondence to João Ramos.

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Funding

This work was partly supported by the Fundação para a Ciência e Tecnologia, under Grant UID/DTP/UI447/2013 to CIPER—Centro Interdisciplinar para o Estudo da Performance Humana (unit 447).

Conflict of interest

João Ramos, Rui J. Lopes, and Duarte Araújo have no conflicts of interest directly relevant to the content of this review.

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Ramos, J., Lopes, R.J. & Araújo, D. What’s Next in Complex Networks? Capturing the Concept of Attacking Play in Invasive Team Sports. Sports Med 48, 17–28 (2018). https://doi.org/10.1007/s40279-017-0786-z

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