# Centroid Conformity

The network centroid, which is a result of [f] from Equation (4.35), defines the most centrally located player, i.e., the most highly connected node in the network. The connectivity strength between other players and the centroid player can be calculated as follows:

This inter-player analysis is denoted as centroid conformity and corresponds to the adjacency between the centroid player ic [i] and the ith player, such that cc R1Xj4 is the vector of the centroid conformity of players. In

other words, ccil( [f ] presents the cooperation level of the ith player with the top-ranked player.

# Topological (Inter)dependency

The inter-player network analysis is based on the topological overlap presented in several works, such as Ravasz et al. (2003) and Horvath (2011), which represents the pair of players who cooperates with the same players. This measure also represents the overlap between two players even if they do not directly cooperate with one another. In other words, the topological overlap between the ith player and the /th player depends on the number of interactions they share with the same players, but it does not take into account the number of interactions between them.

The topological overlap is represented by a symmetric matrix, thus presenting the overlap between players, but neglecting the most independent player of the pair. Therefore, by using the concepts inherent to the clustering coefficient (Equation 4.34), one should consider not only the ‘shared’ interactions but also the influence of the conjoint interactions among players i and j. In other words, if two players interact with the same other players, then the cooperation between both of them allows building triangular relations between the other players. However, the ith player may be more dependable from the /th player if the former only interacts with the same players than player /th who, in turn, is able to interact with other players.

As a result, similarly to Ravasz et al. (2003) and Horvath (2011), one can define a topological dependency [;] =\_tdjj [f]Je R"*" as follows:

with i, j, I = 1,2,..., N. As a consequence, two players have a high topological dependency, i.e., tdj [f] = 1, if they interact with the same players and with one another. In other words, the more players they ‘share’, the stronger their cooperation and they are more likely to represent a small cluster.

Since Tj corresponds to a square matrix with the size equal to the number of players and since that it is not symmetric, i.e., tdj [f] tdj, [fj, it makes it difficult to compare the tdj [t] and tdj [f] pairs (Clemente et al., 2014). Therefore, to complement the previous analysis, the topological inter-dependency

ГиМ=[ад] e RVxV has been introduced as follows:

where Tj [f]r is the transpose of matrix Tj [f] and Ti(i [f] corresponds to an antisymmetric square matrix, i.e., tij [f] = —tijj [/]. In football and other team sports, one can easily observe dependencies between players, such that if tij [i] >0, then the ith player depends on the jth player to cooperate with the remaining teammates. Moreover, when associated with other network analysis (e.g., centroid player), the relative topological dependency allows identifying possible dependencies between players and even hierarchical relations (Clemente et al., 2014).

# Density

The player connectivity calculated in Equation (4.24) allows retrieving several other team network analyses, such as the network density, which can be defined as follows:

Within players’ networks, the density measures the overall cooperation among athletes. A density that tends to 1 indicates that all players strongly interact with each other.

# Heterogeneity

Another network analysis based on the connectivity of players is the network heterogeneity, which is closely related to the variation of connectivity across players (Albert, Jeong, & Barabasi, 2000; Watts, 2002), defined as the coefficient of variation of the connectivity distribution:

Since the heterogeneity is invariant with respect to multiplying the connectivity by a scalar, one could use the scaled connectivity instead. Many complex networks have been found to exhibit an approximate scale-free topology, which implies that these networks are very heterogeneous. In other words, a high heterogeneity of the football network means that the players exhibit a high level of performance and there is, collectively, a low level of cooperation between players (Clemente et al., 2014).

# Centralization

The network centrality, or degree centralization as Freeman (1978) addresses, can be defined as follows:

A centralization close to 1 means that one player strongly cooperates with all other players who, in turn, present a small (or inexistent) cooperation with each other. In contrast, a centralization of 0 indicates that all players cooperate equally among each other.

## Conclusion

This chapter summarized and mathematically defined a series of computational metrics proposed over the past years to model the athletic performance of athletes and teams. These metrics are described on a feature-engineering perspective, to be seen as measurable properties of sport performance and to feed AI approaches for pattern recognition (Nargesian et al., 2017). The presented feature selection has been achieved through the close collaboration between sport scientists, engineers, and mathematicians, many of whom share the authorship of this book.

It is expected that, in a near future, feature engineering shall be accomplished in a more automated manner, leading to major breakthroughs in artificial intelligence for performance analysis in sports. This would allow practitioners to automatically extract the most relevant and representative features without direct human input, additionally tackling the ‘curse of dimensionality’ by reducing the number of features, as already achieved in other domains (Tang, Kay, & He, 2016). Deep learning algorithms are paving the way into the automated extraction of complex data representations at high levels of abstraction. They inherently provide a layered, hierarchical architecture of learning and data representation, where higher-level (more abstract) features are defined in terms of lower-level (less abstract) features.

Notes

• 1 https://www.mathworks.com/matlabcentral/fileexchange/69381-sample-entropy.
• 2 https://www.mathworks.com/matlabcentral/fileexchange/24035-wgplot-weighted- graph-plot-a-better-version-of-gplot.

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ARTIFICIAL INTELLIGENCE FOR PATTERN RECOGNITION IN SPORTS