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# Football and graph theory

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### Football and graph theory

1. 1. Graph Theory in Football
2. 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. 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. 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. 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. 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. 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. 8. The networks for the Netherlands and Spain drawn before the final game, using the data and tactical formations of the semi-finals.
9. 9. I am out of my JOB!!! 
10. 10. The Factors used by us for our Prediction: Degree Centrality Betweeness Centrality Centre of a Graph
11. 11. 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.
12. 12. 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
13. 13. . 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.
14. 14. Player’s betweenness scores for Spain. Player’s betweenness scores for the Netherlands 0.12 6.17
15. 15. 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.
16. 16. 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.
17. 17. 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.
18. 18. Xavi  Iniesta Fabrega s Mapping Of The Spanish Attack
19. 19. 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.
20. 20. 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.
21. 21. 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
22. 22. Prachi Singhal and Umang Aggarwal Lady Shri Ram College