Common and unique network dynamics in football games

Abstract

The sport of football is played between two teams of eleven players each using a spherical ball. Each team strives to score by driving the ball into the opposing goal as the result of skillful interactions among players. Football can be regarded from the network perspective as a competitive relationship between two cooperative networks with a dynamic network topology and dynamic network node. Many complex large-scale networks have been shown to have topological properties in common, based on a small-world network and scale-free network models. However, the human dynamic movement pattern of this network has never been investigated in a real-world setting. Here, we show that the power law in degree distribution emerged in the passing behavior in the 2006 FIFA World Cup Final and an international "A" match in Japan, by describing players as vertices connected by links representing passes. The exponent values γ ~ 3.1 are similar to the typical values that occur in many real-world networks, which are in the range of 2<γ<3, and are larger than that of a gene transcription network, γ ~ 1. Furthermore, we reveal the stochastically switched dynamics of the hub player throughout the game as a unique feature in football games. It suggests that this feature could result not only in securing vulnerability against intentional attack, but also in a power law for self-organization. Our results suggest common and unique network dynamics of two competitive networks, compared with the large-scale networks that have previously been investigated in numerous works. Our findings may lead to improved resilience and survivability not only in biological networks, but also in communication networks.

Figure 1. The distribution function of consecutiveness for each outgoing and incoming passing network.

a and b, Outgoing and incoming links for Italy in the first and second halves, respectively. The red and blue lines in panel a have slopes and , respectively, and those in panel b have slopes , and , respectively. c and d, Outgoing and incoming links for France in the first and second halves, respectively. Red and blue lines have slopes , , , and , respectively. e and f, Outgoing and incoming links for Japan in the first and second halves, respectively. Red and blue lines have slopes , , , and . g and h, Outgoing and incoming links for Ghana in the first and second halves, respectively. Red and blue lines have slopes , , , and . Solid lines show power law distributions, and dashed lines show power law with cut-off distributions (Text S1).

Figure 2. Relative ball touch frequencies for each player in each 5-min interval.

The goalkeeper, defender, midfielder, and forward are denoted as G, D, M, and F, respectively. The color gradation from red to white corresponds to an increase in the relative frequencies of ball touch from 0% to 50%. a–d. Italy, France, Japan, and Ghana, respectively.

Figure 3. Game momentum, represented by the number of triangles in each network.

a, Upper panel shows the number of triangles for each team during each 5-min interval of the 2006 World Cup Final. Differences between the numbers of triangles for both teams in each 5-min interval are shown in the lower panel as a bar chart. A green bar shows that the number of triangles was greater for Italy than for France, and a blue bar shows that France generated more triangles than Italy. The attacking phase also includes the shots added to the lower panel as green and blue arrows for Italy and France, respectively. The thick and thin arrows show the shots and the successful attacks without the shots, respectively. b, The case of the 2006 Kirin Challenge Cup shown in a manner similar to that for the 2006 World Cup in panel a. Fisher's exact tests were applied to determine whether more triangles generated by a team correlated with the more attacks during that time period. The -values for this correlation were 0.087 for the World Cup and 0.024 for the Kirin Cup.

Figure 4. Wiring diagrams for a football game.

All diagrams represent each player as a vertex and each pass as a link in each 5-min interval in the first 20 min of the first half. The size of each vertex shows its degree. France received a penalty kick and scored in the seventh minute. Italy scored in the 19th minute after a corner kick. The black line shows the tracking of the ball handling during each 5-min interval for the same team. a. Intra-group network for Italy. b. Inter-group network for both teams. The gray line also shows the tracking of the ball between teams. c. Intra-group network for France. G, D, M, and F denote goalkeeper, defender, midfielder, and forward, respectively, according to the 4-4-2 system of play, regardless of the system actually chosen by each team.