1. Graph Theory in
Football

2. INDEX
Concepts of Graph theory
Facts of Football
Graphical representation
Constructing graphs using data from
semi-finals
Predicting Spain’s win-Spain did
win!!
Reasons for the prediction
Degree Centrality
Betweenness Centrality
Centre
Conclusion

3. 
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

4. 

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.

5. 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

6. 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 corresponding
to the players’ formation on the pitch
 0,1,2,3,4,5 and 6 are the players.
 Arcs represent the passes between
players

7. Constructing the
Network World Cup
The data for the 2010 FIFA
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.

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

10. I am out of
my JOB!!! 

11. The Factors used by us
for our Prediction:
Degree Centrality
Betweeness Centrality
Centre of a Graph

12. 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.

13. N
E
T
H
E
R
L
A
N
D
Jersey no.
1
2
3
4
5
6
7
8
9
10
11
15
Jersey No.
1
3
5
S
6
7
P
8
A
11
I
14
N 15
16
18
9
Player Name
Stekelenburg
Van Der Wiel
Heitinga
Mathijsen
Van bronckhorst
Van Bommel
Dirk Kuyt
Nigel De Jong
Robin Van Persie
Wesley Sneijder
Arjen Robben
Braafheid (sub)
Player Name
Iker Casillas
Gerard Pique
Carlos Puyol
Andreas Iniestaa
David Villa
Xavi
Cap Devila
Xabi Alonso
Sergio Ramos
Sergio Busquets
Pedro
Fernando Torres (sub)
Position
Goalkeeper
Defender
Defender
Defender
Defender
Mid-Fielder
Forward
Mid-Fielder
Forward
Mid-Fielder
Forward
Defender
Position
Goal-Keeper
Defender
Defender
Mid-Fielder
Forward
Mid-Fielder
Defender
Mid-Fielder
Defender
Mid-Fielder
Forward
Forward
Centrality
48
31
27
33
29
29
11
32
19
35
14
4
Centrality
31
51
54
43
13
95
53
47
54
75
13
2

15. .
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.

16. Player’s
betweenness scores
for Spain.
Player’s betweenness
scores for the
Netherlands
0.12
6.17

17. OBSERVATION
On the Spanish ,The S
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.

18. Centre
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.

19. 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.)
Robben 
Sneijder 
Mapping Of the Dutch Attack.

20. Xavi 
Iniesta
Fabrega
s
Mapping Of The Spanish Attack

21. OBSERVATIONS
•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.

22. 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.

23. 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/holland
Vspain.html

http://plus.maths.org/content/os/latestnews/mayaug10/football/index

www.fifa.com/live/competitions/worldcup/matchday=25/
day=1/math=300061509/index.html

24. Prachi Singhal and Umang Aggarwal
Lady Shri Ram College