Viraj Patel

With its origins dating back as far as the second century [1], it has taken many changes for football to become the beautiful game that it is today and, just as nature has become more complex with evolution, so has football. In this article, I will be discussing the thermodynamic properties of a football match and how this relates to some of the cutting edge research being conducted here at Imperial College London [2].

A network (or graph) is a data structure commonly used to represent systems that require connections between different points in space. These points are called nodes and the connections are called edges. A network can be weighted (the edges have a strength) and/or directed (one can only move along an edge in a particular direction). In football, managers make their team remain in a formation throughout the game to ensure that the entire pitch is covered and many different plays can be made. The formation can be thought of as an unweighted, undirected network with the nodes representing players and the edges representing passing options [3], as shown in Figure 1 [4].

**The ****network theory**** of football**

The physics of football has been addressed by many scientists in the past, most of them examining the mechanics of kicking a ball [5]. In this article I will taking a different approach and examining the game as a whole. For this analysis, we need to be able to see how the formation progresses with time, so we use a passing network. This is a weighted, directed graph to show the number of passes between players and the directions of the passes, as shown in Figure 2 [6].

In a passing network, the size of the nodes represents the relative traffic (number of passes) of the node. We can now define some characteristics of a passing network. We define centrality (or closeness centrality) [7] for a given player as the inverse of the average number of passes that the player is directly involved in. Another important property is clustering, a measure of how tightly players interact during the game [8]. This is obtained by counting the total number of passing triangles that can be made around a given player at a given time [9]. As we can see, the centrality and clustering of a given player changes with time so, we can generate distributions for each player and find the mean clustering and mean centrality.

But, why is this important? Well, mathematicians and physicists have found that these properties play the largest role in describing the quality of a team and a player. It has been found that the higher the centrality of a player, the more important they are to the team (they tend to make more goal contributions) [10]. This makes perfect sense, a well-contributing player would need to have a well-connected passing network. In [10], a match between FC Barcelona and AC Milan showed that Barcelona had a larger mean centrality and won the game 4-0. Moreover, Lionel Messi had the highest individual centrality and was awarded Man of the Match [11].

Moreover, in football, the concept of a “front-three” has been used increasingly to describe the attackers of a team. A study based on the Japanese Football Division showed a positive correlation between the number of successful passes leading to a goal and the clustering coefficient of players in the front-three [12]. This also makes sense, as you’d expect the team to be well-connected in the attacking half when approaching the goal.

**The physics of football**

Now that some of the most important properties of a network have been established and their links to football have been described, we can move on to the more tackling (pun intended) part of this article. We know how maths is involved, but what about physics?

A football game begins at kick-off (which we can label as time t = 0) and each half lasts 45 minutes excluding injury time. At the beginning of each half and after each goal, there is a kick-off. During the kick-off the players remain stationary in formation and are in their most ordered state, so the entropy of the game is at a minimum. During the course of the game the entropy varies randomly, as we cannot predict when a pause of play will occur. We can, however, theorise that the entropy increases during the course of the game [13]. The players will break formation and move around more, hence moving into a less ordered state. In the same way that we used the mean centrality because it varied so much, we can examine the mean entropy. In this case, we find that the entropy decreases! This appears to break the second law of thermodynamics. It is due to the fact that the players become tired as the game progresses and move slower [14]. They also tend not to move around as much and generally remain in the initial formation.

There are many factors that affect the entropy of the game, including the player’s emotions, their energy, and how the game is going for them (if they’re having a particularly good game then they are more motivated to play well). Modelling these factors can help predict the progression of the game despite its randomness. To incorporate this, we can convert the passing network into a passing hypernetwork [15], a network of passing networks. This problem has now been extended to an n-dimensional problem (previously it had three spatial dimensions and one time dimension), so a hypernetwork is necessary due to its multi-level organisation [16].

**From football to physics**

Networks are becoming an increasingly popular way to represent complex systems in physics, because most complex systems can be broken down into smaller sub-systems (the nodes of the network). In a football match, the players are the sub-systems as they have their own state variables, but their behaviour also impacts the whole game (the overall system). Using our understanding of the thermodynamics of a football match, we can analyse the thermodynamics of complex physical systems that can best be represented by networks [17].

Networks can also be used to predict the motion of a randomly moving particle (Brownian motion) using an algorithm called a Random Walk. Random walk closeness centrality describes the average speed with which a randomly moving particle reaches a node from other nodes, and it is commonly used to rank nodes when producing a shortest path algorithm across a network [18]. This approach is called PageRank and only applies to static networks. By applying the same logic as in the analysis of a football match, physicists have found a similar approach for dynamic networks, called TempoRank [19]. Additionally, by incorporating hypernetworks for a multi-layered organisation, we are now able to define clustering for multi-level systems [20]. The Department of Physics at Imperial College have been analysing random walks [21] and their applications, particularly in optics [2].

##### References

[1] FootballHistory.org, “Football history,” [Online]. Available: https://www.footballhistory.org/. [Accessed 6 February 2021].

[2] J. Boutari, A. Feizpour, S. Barz, C. Di Franco, M. S. Kim, W. S. Kolthammer and I. A. Walmsley, “Large scale quantum walks by means of optical fiber cavities,” *Journal of Optics, *vol. 18, no. 9, p. 094007, 2016.

[3] C. Braham and M. Small, “Complex networks untangle competitive advantage in Australian football,” *Chaos: An Interdisciplinary Journal of Nonlinear Science, *vol. 28, no. 5, p. 053105, 2018.

[4] Cornell University, “FC Barcelona and the triadic closure property,” 18 September 2014. [Online]. Available: https://blogs.cornell.edu/info2040/2014/09/18/fc-barcelona-and-the-triadic-closure-property/. [Accessed 6 February 2021].

[5] T. Asai, T. Akatsuka and S. Haake, “The physics of football,” *Physics World, *vol. 11, no. 6, p. 25, 1998.

[6] A. Agusti, A. Lopez-Giraldo, I. Lopez-Giraldo, T. Cruz Santa Cruz and R. Faner, “Relevance of Systems Biology to Respiratory Medicine,” *Barcelona Respiratory Network, *vol. 2, 2016.

[7] G. Sabidussi, “The centrality index of a graph,” *Psychometrika, *vol. 31, no. 4, 1966.

[8] J. L. Pena and H. Touchette, “A network theory analysis of football strategies,” *arXiv preprint arXiv:1206.6904, *2012.

[9] J. Buldu, J. Busquets and I. Echegoyen, “Defining a historic football team: Using Network Science to analyze Guardiola’s F.C. Barcelona,” *Scientific reports, *vol. 9, no. 1, pp. 1-14, 2019.

[10] H. Song, “Passing Network of Football,” 22 January 2020. [Online]. Available: https://rpubs.com/ihatestudying/passing-network. [Accessed 7 February 2021].

[11] BBC, “Barca v AC Milan,” BBC, 12 March 2013. [Online]. Available: https://www.bbc.co.uk/sport/football/21747026. [Accessed 7 February 2021].

[12] T. Kawasaki, K. Sakaue, R. Matsubara and S. Ishizaki, “Football pass network based on the measurement of player position by using network theory and clustering,” *International Journal of Performance Analysis in Sport, *vol. 19, no. 3, pp. 381-392, 2019.

[13] J. H. Martinez, D. Garrido, J. L. Herrera-Diestra, J. Busquets, R. Sevilla-Escoboza and J. M. Buldu, “Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective,” *Entropy, *vol. 22, no. 2, p. 172, 2020.

[14] J. M. Buldu, J. Busquets, J. H. Martinez, J. L. Herrera-Diestra, I. Echegoyen and J. Luque, “Using network science to analyse football passing networks: Dynamics, space, time, and the multilayer nature of the game,” *Frontiers in psychology, *vol. 9, p. 1900, 2018.

[15] J. Ramos, R. J. Lopes, P. Marques and D. Araujo, “Hypernetworks Reveal Compound Variables That Capture Cooperative and Competitive Interactions in a Soccer Match,” *Frontiers in psychology, *vol. 8, p. 1379, 2017.

[16] J. H. Johnson, “Hypernetworks: Multidimensional relationships in multilevel systems,” *The European Physical Journal Special Topics, *vol. 225, no. 6, pp. 1037-1052, 2016.

[17] G. Oster, A. Perelson and A. Katchalsky, “Network Thermodynamics,” *Nature, *vol. 234, no. 5329, pp. 393-399, 1971.

[18] S. White and P. Smyth, “Algorithms for Estimating Relative Importance in Networks,” in *Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining*, 2003.

[19] L. E. C. Rocha and N. Masuda, “Random walk centrality for temporal networks,” *New Journal of Physics, *vol. 16, no. 6, p. 063023, 2014.

[20] L. Bottcher and M. A. Porter, “Classical and quantum random-walk centrality measures in multilayer networks,” *arXiv preprint arXiv:2012.07157, *2020.

[21] T. S. Evans and J. P. Saramaki, “Scale-free networks from self-organisation,” *Physical Review E, *vol. 72, no. 2, p. 026138, 2005.

[22] J. Boutari, A. Feizpour, S. Barz, C. Di Franco, M. S. Kim, W. S. Kolthammer and I. A. Walmsley, “Large scale quantum walks by means of optical fiber cavities,” *Journal of Optics, *vol. 18, no. 9, p. 094007, 2016.