Concepts of Graph theory
Facts of Football
Constructing graphs using data from
Predicting Spain’s win-Spain did
Reasons for the prediction
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
Here 1,2,3 and 4 are the vertices
b, c and d are the edges
Arcs: An edge with a specified direction is called an arc
It is a sequence of edges which connect a
sequence of vertices
Directed Graph: A graph having arcs is called a directed
Directed Network: A directed graph with integer
weight attached to each arc is called a
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
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
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
The networks for the Netherlands and Spain drawn before
the final game, using the data and tactical formations of
The Factors used by us
for our Prediction:
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
Van Der Wiel
Nigel De Jong
Robin Van Persie
Fernando Torres (sub)
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.
scores for the
On the Spanish ,The S
betweenness scores are low and
uniformly distributed – a sign of a well-balanced passing
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.
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
The centre shall appear more in goal scoring attempts.
Graphical Analysis for a goal scoring
(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.
Mapping Of The Spanish Attack
•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
•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
•The low no. of arcs in the Dutch team suggests a
preference for quick attacks and counter attacks.
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
2. The Dutch lost because their strategy was predictable and
easily countered by Spanish who played a clever and an
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.
Eclat-volume –III Mathematics Journal
Edgar. G. Goodaire and Micheal M. Parmenter, Discrete
Mathematics with graph theory 3rd Edition
Prachi Singhal and Umang Aggarwal
Lady Shri Ram College