Abstract
In team sports the teammates cooperates with interdependency between them. One of the examples it is the forward player that depend from the backward players to receive the ball. For that reason, it is important to understand how teammates cooperate and autonomous or dependent are a specific player. Moreover, using such idea will be also possible to identify some clusters and patterns of interactions inside the team. For that reason, the aim of this chapter is to present the network measurements that allows to perform a meso-analysis and identify the specific interactions between teammates.
5.1 Average Neighbor Degree
Definition 5.1
(Rubinov and Sporns 2010) Given one unweighted graph G with n vertices. The average neighbors degree index \(\overline{ND} ( n_{i} )\) of a vertex \(n_{i}\), of G is calculated as
where \(a_{ij}\) are elements of the adjacency matrix of a G, \(k_{i}\) degree centrality index of vertex \(n_{i}\).
Definition 5.2
(Sierra 2015) Given one weighted graph G with n vertices. The average neighbors degree index \(\overline{ND}^{w} ( n_{i} )\) of a vertex \(n_{i}\), of G is calculated as
where \(a_{ij}\) are elements of the weighted adjacency matrix of a G, \(k_{i}^{w}\) degree centrality index of vertex \(n_{i}\).
Definition 5.3
(Rubinov and Sporns 2010) Given one unweighted digraph G with n vertices. The average neighbors degree index \(\overline{ND}_{D} ( n_{i} )\) of a vertex \(n_{i}\), of G is calculated as
where \(a_{ij}\) are elements of the adjacency matrix of a G, \(k_{i}^{out}\) and \(k_{i}^{in}\) degree index of vertex \(n_{i}\).
Definition 5.4
(Rubinov and Sporns 2010) Given one weighted digraph G with n vertices. The average neighbors degree index \(\overline{ND}_{D}^{w} ( n_{i} )\) of a vertex \(n_{i}\), of G is calculated as
where \(a_{ij}\) are elements of the weighted adjacency matrix of a G, \(k_{i}^{w - out}\) and \(k_{i}^{w - in}\) degree index of vertex \(n_{i}\).
5.1.1 Team Sports Network Interpretation
The Average neighbor degree measures the correlation levels between pairs of players. This measure allows identifying the patterns of interactions and the strength of interaction between two players. Can be used to identify patterns of cooperation inside the team and to associate these interactions with macro levels of analysis.
5.2 Assortativity Coefficient
Definition 5.5
(Ciglan et al. 2013) Given one unweighted graph G with n vertices. The assortativity coefficient index, \(r\), of G is calculated as
where \(a_{ij}\) are elements of the adjacency matrix of a G, \(k_{i}\) degree index of vertex \(n_{i}\).
Definition 5.6
(Schott and Wunderlich 2014) Given one weighted graph G with n vertices. The weighted assortativity coefficient index, \(r^{w}\), of G is calculated as
where \(a_{ij}\) are elements of the weighted adjacency matrix of a G, \(k_{i}^{w}\) degree index of vertex \(n_{i}\).
Definition 5.7
(Rubinov and Sporns 2010) Given one unweighted digraph G with n vertices. The directed assortativity coefficient index, \(r_{D}\), of G is calculated as
where \(a_{ij}\) are elements of the adjacency matrix of a G, \(k_{i}^{out}\) and \(k_{i}^{in}\) degree index of vertex \(n_{i}\).
Definition 5.8
(Schott and Wunderlich 2014) Given one weighted digraph G with n vertices. The weighted directed assortativity coefficient index, \(r_{D}^{w}\), of G is calculated as
where \(a_{ij}\) are elements of the adjacency matrix of a G, \(k_{i}^{w - out}\) and \(k_{i}^{w - in}\) degree index of vertex \(n_{i}\).
Remark 5.1
We also can calculate the (weighted) assortativity coefficient for the indegree and outdegree of (weighted) digraphs (Schott and Wunderlich 2014; Piraveenan et al. 2012).
5.2.1 Team Sports Network Interpretation
Assortativity is often operationalized as a correlation between two players. A network is called assortative if the players with higher degree have the tendency to connect with other players that also have high degree of connectivity (Pavlopoulos et al. 2011). If the players with higher degree have the tendency to connect with other players with low degree then the network is called disassortative (Pavlopoulos et al. 2011).
5.3 Topological Overlap
Definition 5.9
(Horvath 2011) Given one unweighted graph G with n vertices. The topological overlap measure index, \({\text{TOM}}_{ij}\), between vertices \(n_{i}\) and \(n_{j}\) is calculated as
where \(a_{ij}\) are elements of the adjacency matrix of a G.
Remark 5.2
If \(0 \le a_{ij} \le 1\) then \(0 \le {\text{TOM}}_{ij} \le 1\) (Li and Horvath 2007).
Definition 5.10
(Schott and Wunderlich 2014) Given one unweighted digraph G with n vertices. The overlap similarity index between vertices \(n_{i}\) and \(n_{j}\) at incoming or outgoing is calculated as, respectively
where \(\left| {N^{out} (n_{i} )} \right|\) is the number of all outgoing neighbors of \(n_{i}\) and \(\left| {N^{in} (n_{i} )} \right|\) is the number of all incoming neighbors of \(n_{i}\).
Definition 5.11
(Schott and Wunderlich 2014) Given one weighted graph G with n vertices. The weighted overlap similarity index, \(OS^{w} ,\) between vertices \(n_{i}\) and \(n_{j}\) is calculated as, respectively
where \(w_{ik}\) is the weight of \((n_{i} ,n_{k} ) \in E\) and \(N\left( {n_{i} } \right)\) is a set of all neighbors of \(n_{i}\).
Definition 5.12
(Schott and Wunderlich 2014) Given one weighted digraph G with n vertices. The weighted overlap similarity index between vertices \(n_{i}\) and \(n_{j}\) at incoming or outgoing is calculated as, respectively
where \(w_{ki}\) is the weight of \((n_{k} ,n_{i} ) \in E\).
5.3.1 Team Sports Network Interpretation
The topological overlap measure represents the pair of players that cooperates with the same players (Clemente et al. 2014). Topological overlap measure may also represent the overlap between two players even if they do not participate in the same attacking plays with one another (Clemente et al. 2014). Thus, this measure allows to identify the patterns of interaction between triads without consider the specific pairs relations.
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Clemente, F.M., Martins, F.M.L., Mendes, R.S. (2016). Meso Level of Analysis: Subgroups in Teams. In: Social Network Analysis Applied to Team Sports Analysis. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25855-3_5
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