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

$$\overline{ND} ( n_{i} ) = \frac{{\mathop \sum \nolimits_{j = 1}^{n} a_{ij} k_{j} }}{{k_{i} }}$$
(5.1)

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

$$\overline{ND}^{w} ( n_{i} ) = \frac{{\mathop \sum \nolimits_{j = 1}^{n} a_{ij} k_{j}^{w} }}{{k_{i}^{w} }}$$
(5.2)

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

$$\overline{ND}_{D} ( n_{i} ) = \frac{{\mathop \sum \nolimits_{j = 1}^{n} \left( {a_{ij} + a_{ji} } \right)\left( {k_{j}^{out} + k_{j}^{in} } \right)}}{{2 \times \left( {k_{i}^{out} + k_{i}^{in} } \right)}}$$
(5.3)

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

$$\overline{ND}_{D}^{w} ( n_{i} ) = \frac{{\mathop \sum \nolimits_{j = 1}^{n} \left( {a_{ij} + a_{ji} } \right)\left( {k_{j}^{w - out} + k_{j}^{w - in} } \right)}}{{2 \times \left( {k_{i}^{w - out} + k_{i}^{w - in} } \right)}}$$
(5.4)

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

$$r = \frac{{\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} k_{i} k_{j} }}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j > i} \\ \end{array} }}^{n} {a_{{ij}} } } }} - \left[ {\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}(k_{i} + k_{j} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j > i} \\ \end{array} }}^{n} {a_{{ij}} } } }}} \right]^{2} }}{{\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}(k_{i}^{2} + ~k_{j}^{2} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j > i} \\ \end{array} }}^{n} {a_{{ij}} } } }} - \left[ {\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}(k_{i} + k_{j} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j > i} \\ \end{array} }}^{n} {a_{{ij}} } } }}} \right]^{2} }}$$
(5.5)

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

$$r^{w} = \frac{{\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} a_{{ij}} k_{i}^{w} k_{j}^{w} }}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j > i} \\ \end{array} }}^{n} {a_{{ij}} } } }} - \left[ {\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} (k_{i}^{w} + k_{j}^{w} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j > i} \\ \end{array} }}^{n} {a_{{ij}} } } }}} \right]^{2} }}{{\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} ((k_{i}^{w} )^{2} + {\text{~(}}k_{j}^{w} {\text{)}}^{2} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j > i} \\ \end{array} }}^{n} {a_{{ij}} } } }} - \left[ {\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} (k_{i}^{w} + k_{j}^{w} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j > i} \\ \end{array} }}^{n} {a_{{ij}} } } }}} \right]^{2} }}$$
(5.6)

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

$$r_{D} = \frac{{\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} a_{{ij}} k_{i}^{{out}} k_{j}^{{in}} }}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{n} {a_{{ij}} } } }} - \left[ {\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} (k_{i}^{{out}} + k_{j}^{{in}} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{n} {a_{{ij}} } } }}} \right]^{2} }}{{\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} ((k_{i}^{{out}} )^{2} + {\text{~(}}k_{j}^{{in}} {\text{)}}^{2} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{n} {a_{{ij}} } } }} - \left[ {\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} (k_{i}^{{out}} + k_{j}^{{in}} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{n} {a_{{ij}} } } }}} \right]^{2} }}$$
(5.7)

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

$$r_{D}^{w} = = \frac{{\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} a_{{ij}} k_{i}^{{w - out}} k_{j}^{{w - in}} }}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{n} {a_{{ij}} } } }} - \left[ {\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} (k_{i}^{{w - out}} + k_{j}^{{w - in}} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{n} {a_{{ij}} } } }}} \right]^{2} }}{{\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} ((k_{i}^{{w - out}} )^{2} + {\text{~(}}k_{j}^{{w - in}} {\text{)}}^{2} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{n} {a_{{ij}} } } }} - \left[ {\frac{{\mathop \sum \nolimits_{{(n_{i} ,n_{j} ) \in E}} \frac{1}{2}a_{{ij}} (k_{i}^{{w - out}} + k_{j}^{{w - in}} )}}{{\sum\nolimits_{{i = 1}}^{n} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{n} {a_{{ij}} } } }}} \right]^{2} }}$$
(5.8)

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

$${\text{TOM}}_{ij} = \left\{ {\begin{array}{*{20}c} {\frac{{\mathop \sum \nolimits_{l \ne i,j} \,a_{il} a_{jl} + a_{ij} }}{{{ \hbox{min} }\left\{ {\mathop \sum \nolimits_{l \ne i} \,a_{il} - a_{ij} , \mathop \sum \nolimits_{l \ne j} \,a_{jl} - a_{ij} } \right\} + 1}}} & {i \ne j} \\ 1 & {i = j} \\ \end{array} } \right.$$
(5.9)

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

$$OS_{D}^{out} ( n_{i} ,n_{j} ) = \frac{{\left| {N^{out} (n_{i} )\mathop {\bigcap }\nolimits N^{out} (n_{j} )} \right|}}{{{ \hbox{min} }\left\{ {\left| {N^{out} (n_{i} )} \right|,\left| {N^{out} (n_{j} )} \right|} \right\}}}$$
(5.10)
$$OS_{D}^{in} ( n_{i} ,n_{j} ) = \frac{{\left| {N^{in} (n_{i} )\mathop {\bigcap }\nolimits N^{in} (n_{j} )} \right|}}{{{ \hbox{min} }\left\{ {\left| {N^{in} (n_{i} )} \right|,\left| {N^{in} (n_{j} )} \right|} \right\}}}$$
(5.11)

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

$$OS^{w} ( n_{i} ,n_{j} ) = \frac{{\mathop \sum \nolimits_{{n_{k} \in N(n_{i} )\mathop {\bigcap }\nolimits N(n_{j} )}} { \hbox{min} }\left( {w_{ik} ,w_{jk} } \right)}}{{{ \hbox{min} }\left\{ {\mathop \sum \nolimits_{{n_{k} \in N(n_{i} )}} w_{ik} ,\mathop \sum \nolimits_{{n_{k} \in N(n_{j} )}} w_{jk} } \right\}}}$$
(5.12)

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

$$OS_{D}^{w - out} ( n_{i} ,n_{j} ) = \frac{{\mathop \sum \nolimits_{{n_{k} \in N^{out} (n_{i} )\mathop {\bigcap }\nolimits N^{out} (n_{j} )}} { \hbox{min} }\left( {w_{ik} ,w_{jk} } \right)}}{{{ \hbox{min} }\left\{ {\mathop \sum \nolimits_{{n_{k} \in N^{out} (n_{i} )}} w_{ik} ,\mathop \sum \nolimits_{{n_{k} \in N^{out} (n_{j} )}} w_{jk} } \right\}}}$$
(5.13)
$$OS_{D}^{w - in} ( n_{i} ,n_{j} ) = \frac{{\mathop \sum \nolimits_{{n_{k} \in N^{in} (n_{i} )\mathop {\bigcap }\nolimits N^{in} (n_{j} )}} { \hbox{min} }\left( {w_{ki} ,w_{kj} } \right)}}{{{ \hbox{min} }\left\{ {\mathop \sum \nolimits_{{n_{k} \in N^{in} (n_{i} )}} w_{ki} ,\mathop \sum \nolimits_{{n_{k} \in N^{in} (n_{j} )}} w_{kj} } \right\}}}$$
(5.14)

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.