Graph Theory in Football

Concepts of Graph theory Facts of FootballGraphical representationConstructing graphs using data from semi-finals Predicting Spain’s win-Spain did win!!Reasons for the predictionDegree CentralityBetweenness CentralityCentreConclusion

INDEX

Some basic concepts :- Graph: A simple graph G = (V, E) consists of V, a nonempty set

of vertices, and E, a set of unordered pairs of distinct elements of V called edges.

Here 1,2,3 and 4 are the vertices

• a, b, c and d are the edges

Arcs: An edge with a specified direction is called an arc

Path: It is a sequence of edges which connect a sequence of vertices

Directed Graph: A graph having arcs is called a directed graph

Directed Network: A directed graph with integer weight attached to each arc is called a directed network.

FACTS OF FOOTBALL• A team consists of exactly 1 goalkeeper, 3-5 defenders , 3-5 midfielders and 1-3 strikers

• A goalkeeper is a designated player charged with directly preventing the opposing team from scoring by intercepting shots at goal.

• A defender  is an outfield player whose primary role is to prevent the opposition from attacking.

• A mid-fielder plays in the middle of the field and does the job of both defenders and strikers as per the need.

• A striker is a player who plays nearest to the opposing team's goal, and is therefore principally responsible for scoring goals

Graphical representation of the game. We consider a football match to be analogous to a directed network.• The players of the team are represented as the vertices.• The passes exchanged between players are the arcs, the direction of the arc

will be in accordance with the direction of the pass.• We assume that the weight of each arc is one .• We assume that the diagraph will be a connected graph, i.e. every vertex is

adjacent to at least one other vertex in the graph. • The network is a tool for visualizing a team’s

strategy by fixing its vertices in positions roughly correspondingto the players’ formation on the pitch

0,1,2,3,4,5 and 6 are the players.

Arcs represent the passes between players

Constructing the NetworkThe data for the 2010 FIFA World Cup

games was downloaded from the official FIFA website

The networks were then constructed and analyzed using Wolfram Mathematica

As FIFA only provides the aggregate data over all the games, these networks were computed by dividing the number of passes by the total number of games played by each team.

The networks for the Netherlands and Spain drawn before the final game, using the data and tactical formations of the semi-finals.

Prediction: Spain will Win!

I am out of my JOB!!!

Result: Spain Won!!!!

The Factors used by us for our Prediction:

Degree Centrality

Betweeness Centrality

Centre of a Graph

Degree Centrality .

Degree Centrality is the number of arcs incident with a vertex.

This concept can be extended to a player by counting the number of passes he is involved in.

Jersey no. Player Name Position Centrality1 Stekelenburg Goalkeeper 482 Van Der Wiel Defender 313 Heitinga Defender 274 Mathijsen Defender 335 Van bronckhorst Defender 296 Van Bommel Mid-Fielder 297 Dirk Kuyt Forward 118 Nigel De Jong Mid-Fielder 329 Robin Van Persie Forward 19

10 Wesley Sneijder Mid-Fielder 3511 Arjen Robben Forward 1415 Braafheid (sub) Defender 4

Jersey No. Player Name Position Centrality1 Iker Casillas Goal-Keeper 313 Gerard Pique Defender 515 Carlos Puyol Defender 546 Andreas Iniestaa Mid-Fielder 437 David Villa Forward 138 Xavi Mid-Fielder 9511 Cap Devila Defender 5314 Xabi Alonso Mid-Fielder 4715 Sergio Ramos Defender 5416 Sergio Busquets Mid-Fielder 7518 Pedro Forward 139 Fernando Torres (sub) Forward 2

NETHERLAND

SPAIN

OBSERVATIONS

The Spanish Players have comparably higher degrees.

So, there is a lot of passing between all the different players, hence, they have a balanced network.

The Spanish team on an average has centrality 41 which is much higher than the Dutch team’s 23.

This means that the Dutch attack is Straight-forward while the Spanish play s more intricate.

The highest degree centrality for the Dutch side is that of the goal-keeper suggesting a lot of attacks on the Dutch goalpost.

BETWEENNESS CENTRALITY

Betweenness centrality quantifies the number of times a vertex acts as a bridge along the shortest path between two other vertices.

Betweenness does not measure how well-connected a player is, but rather how the ball-flow between other players depends on that particular player.

It thus provides a measure of the impact of removing that player from the game.

. .

.

Player’s betweenness scores for Spain.

Player’s betweenness scores for the Netherlands

0.12

6.17

On the Spanish ,The betweenness scores are low and uniformly distributed – a sign of a well-balanced passing strategy

Concentrated betweenness scores that are on the high side indicate a high dependence on few, too important players, whereas well distributed, low betweenness scores are an indication of a well-balanced passing strategy.The table gives us a good measure of the game-play robustness. By Blocking players with high betweenness centrality the opposing team can interrupt a teams natural flow.

OBSERVATIONS

Eccentricity of a vertex: It is the maximum distance of that vertex from any other vertex.

Graph Radius: It is the minimum distance between any two vertices of the graph.

Centre of a graph: Those vertices whose eccentricity equals the graph radius.For a game of football, the centre will be the

players whose distance from other players is minimum( here assumed as one)

The centre shall appear more in goal scoring attempts.

Centre

Graphical Analysis for a goal scoring Opportunity:

(We define a successful pass as one that was involved in an attack, i.e. the passes that resulted in shots on targets.)

Mapping Of the Dutch Attack.

Robben

Sneijder

Mapping Of The Spanish Attack

Xavi

Iniesta

Fabregas

•The Spanish players make extremely large no. of passes during the game as seen by the high network density of their graph, much more than their Dutch counterparts.

•The Spanish attack is often unpredictable because of the huge number of passing outlets.

•The Dutch attack seems much more traditional with most attacks being carried out by the strikers.

•The Spanish attack relies on swift passes between the players, which are evenly distributed among the midfield.

•The low no. of arcs in the Dutch team suggests a preference for quick attacks and counter attacks.

OBSERVATIONS

Conclusion.Using Graph theory to map the football world cup final, we

were able to determine the primary reason for Spanish triumph.

1. Spain had a well oiled network of connected players which made for a sound defense, a cohesive midfield and an effective attack.

2. The Dutch lost because their strategy was predictable and easily countered by Spanish who played a clever and an intricate game.

3. They should have considered passing more and involving players other than their attacking mid fielders and strikers in their goal-striking strategy.

4. Similarly, the outcome of the game could have been different had they succeed in blocking the players who belonged to the Spanish center, namely, Xavi, Iniesta and Fabregas.

5. Finally, our observations showed an unbalanced use of the pitch giving a clear preference for the left side by the Dutch.

Bibliography Eclat-volume –III Mathematics Journal

Edgar. G. Goodaire and Micheal M. Parmenter, Discrete Mathematics with graph theory 3rd Edition

http://www.maths.qmul.ac.uk/~ht/footballgraphs/hollandVspain.html

http://plus.maths.org/content/os/latestnews/may-aug10/football/index

www.fifa.com/live/competitions/worldcup/matchday=25/day=1/math=300061509/index.html

Compiled by:

Prachi Singhal and Umang Aggarwal

Lady Shri Ram College