Game of Football as a Complex Network

Network Analysis has long been used as a tool to analyze various processes and events in physical world ranging from social interactions , biological phenomenons , financial dynamics of corporations etc. But recently, the ideas from network analysis are used to explain the complexity of the game of football and to analyze the player performance in a team.

Since , football tends to be a low scoring game, saying that players with more goals and assists only have significant contribution to the performance of the team is simply undermining the

contribution of the other members of the team. So, people are trying to use network analysis to quantify the contribution of each player to the team performance  based on involvement of the player in the game and not only on the basis of limited information like goal scored and assists made.

Passing Network:

A directed network is built according to the following rules:
Each player represents a node and an arc between two players is weighted according to number of successful passes between those two players in the entire match.
To incorporate the additional information about the shots on target, two additional nodes are added which are named as "Shots on Target" and "Shots off Target".
This network captures the dynamics of the entire game to some extent.

Variants of Passing Network 

Network Analysis:

Each player in the team tries to move the ball towards the opposition goal through a series of passes among players of his team or  try to score a goal by shooting at opponents goal.

This idea can be incorporated in the network by finding the passing and shooting accuracy.

Passing Accuracy defines the fraction of passes initiated by a player that reach a teammate and 

Shooting Accuracy defines the fraction of shots that a player make and that do not miss the goal(i.e. directed  towards the "Shots on Target" node)

These two matrices define the ability of a payer to move the ball towards the opponent'g goal.

With this information about the passing network and the shooting and passing accuracy of each player in hand , we can find the probability that a particular path in the network leads to a shot on

target.

So, the most natural definition of the performance of a player taking into account the contribution to the team's play during the entire match, will be the "Flow Betweeness" of a particular player 

in the network.

The Flow Betweeness captures fraction of times a player lies on a path that leads to a shot on target. This seems to be a fair judgement of the contribution of a player in a move that can lead

to a goal. This method can further be improved by giving weight to performance of each player on the path that lead to "shots on target node" according to distance from the this node.

The match performance of player in a team is defined as the normalized value of the logarithm of the player's flow centrality in the match.

To substantiate the method defined above to measure the performance of a player , it was applied to the data collected from the Euro 2008 tournament.

Thus, this network centric approach towards football is able to quantify the player performance to great extent and can be extended to other areas which deal with evaluation of individual performance in a team environment.

Further Readings:

1: Duch J, Waitzman JS, Amaral LAN (2010) Quantifying the Performance of Individual Players in a Team Activity. PLoS ONE 5(6): e10937. doi:10.1371/journal.pone.0010937

2: Robert Kooij,Almerima Jamakovic,Frank van Kesteren,Tim de Koning,Ildiko Theisler,Pim Veldhoven The Dutch Soccer Team as a Social Network