Physics and the World Cup: Untangling the networks behind “the beautiful game”

It’s a sport that goes by many names: soccer, football, “the beautiful game.” No matter what you call it, in a few short weeks billions of fans around the world will be watching as 48 national teams compete for the men’s championship title during the 2026 World Cup.

While successful tackles and goals are likely to dominate the highlight reels, much of a player’s time during a match is spent on passing — an essential aspect of the game that allows teams to move the ball upfield, maintain possession, and score.

To learn more, APS News spoke with Andres Chacoma, a researcher at Argentina’s National Scientific and Technical Research Council, and Ken Yamamoto, a researcher at the University of the Ryukyus in Japan. Using tools from network science and mathematical physics, these researchers have analyzed data from soccer matches and uncovered insights about passing’s importance.

Interviews have been lightly edited for brevity and clarity.

Ken, what are the benefits of using tools from fields like statistical physics to analyze data from soccer matches?

In complex systems, it is difficult to accurately predict the long-term behavior of an individual element, but some statistical properties can be described reliably.

Sports are complex systems — a player’s decision depends on their physical and mental condition and their interactions with others. When a player receives the ball, it’s difficult to estimate how long the player will keep it, but the statistical distribution of the length of time for ball possession can often be modeled by a gamma distribution, for example.

Andres, why do you use tools from network science to analyze matches?

Our research team is interested in "measuring" the interactions between players, such as passing patterns or structures related to marking, when a player stands close to an opponent so they can prevent their opponent from taking the ball. In this framework, network science tools are exceptionally useful.

Generally, we define the links of the network based on specific interactions. A link is established if, for example, two players exchange a pass. If they exchange multiple passes, we can add weight to that link. This results in a “passing network,” a mathematical object that allows us to see which players interact most frequently and in what areas.

Andres, tell us about the dataset with millions of in-game events from professional soccer games that you’ve analyzed.

That dataset is the result of an incredible effort by a group of Italian physicists led by Professor Luca Pappalard. While other datasets exist, many of them are proprietary — clubs tend to be protective of their data, perhaps out of concern that rivals might identify tactical vulnerabilities. Nevertheless, our group maintains an ongoing dialogue with clubs, urging them to at least release data from previous seasons. The dialogue can be challenging, but the culture is shifting. 

Ken, in a recent paper, you used a specific model known as a Pólya urn. Could you tell us more about this model?

The Pólya urn is a mathematical model where we repeatedly draw a colored ball from an urn and return it with additional balls of the same color. Because each draw increases the number of the color that’s just been drawn, colors that are selected frequently become more likely to be selected again in subsequent draws — a property known as preferential selection.

A similar mechanism exists in soccer: To avoid losing possession, players prefer to make safe passes, and successful passes are likely to be repeated. Using the Pólya urn framework, we found that a team’s passing behavior can be characterized by a single parameter — the number of balls added after each draw. This parameter can be estimated directly from pass sequence data, and its strength shows similar distributions across different leagues.

This parameter is closely related to playing style. In soccer matches, at least at the professional level, stronger preferential selection is associated with higher pass accuracy and lower pass difficulty.

Andres, tell us more about your 2020 paper on passing dynamics of teams in possession of the ball and your 2022 paper on defensive networks. Every ball possession is unique. They vary in duration, number of passes, player movement, and so on, and this randomness can be characterized statistically through distributions. In our 2020 paper, we proposed a simple model which suggested that possession intervals can be approximated by a particular type of drill called a "small-sided" training drill.

These results helped us realize that even a system as unconventional and complex as soccer can be modeled using physics, and that gave us the confidence to continue this line of research.

In our 2022 paper, we used network dynamics to understand marking. We analyzed individual player trajectories throughout several matches and set a proximity threshold: If two opponents were within a certain distance, a link was established. We found that the network would periodically "cluster," indicating moments of high-intensity marking, then "disintegrate" as players moved apart.

These networks could be measured to see if a team is marking incorrectly. If the distribution of high-intensity marking intervals changes, that could serve as a red flag for coaches to intervene.

Andres, in a recent paper, you identified differences in how high-performing and low-performing teams pass the ball. Tell us more about what you found.

In this work, we constructed networks based on, for instance, the last 50 passes, as we wanted to quantify differences in a network when a team was playing well versus poorly. If a team was positioned high up the pitch and registered a shot on goal, we categorized that as high performance. If they were pinned back and conceded a shot, it was labeled as low performance.

By analyzing match data from FC Barcelona, we found structural differences between these two states. And by looking at specific network metrics, we could see which player's involvement was associated with success. These metrics provide actionable information for a coach: If they see a pattern associated with low performance, they can make tactical adjustments to shift the network toward a more successful configuration.

In that paper, Andres, you also studied the “blocked player” problem. Can you explain?The problem is fascinating. In the paper, I use the example of Ousmane Dembélé from France and Ángel Di María from Argentina in the 2022 World Cup final. Knowing that Dembélé was a fast, attacking threat, the Argentine coach positioned Di María deep in Dembélé's zone. This forced Dembélé to stay back, effectively neutralizing his offensive impact. Then, when Di María was substituted for a more defensive player, France managed a tremendous comeback. Our analysis suggests that a team can modify its interaction patterns to compensate for a key player’s absence.

Ken, what has been the most surprising result from your research?

Soccer is simultaneously complex and simple. The match is built on sophisticated strategies, high-level decision-making, and the exceptional skills of players. Yet some aspects of soccer exhibit surprisingly simple statistical structures that can be uncovered through theoretical and data analyses.

Who will you be cheering for during this year’s World Cup, Ken?

I will be cheering for Japan as they face the Netherlands, Sweden, and Tunisia in the group stage. I am looking forward to brilliant plays, unexpected upsets, and dramatic moments, which go far beyond stochastic or other mathematical models.

How about you, Andres?

Of course, I will be cheering for the Argentine national team. I have every hope that our players can repeat the incredible campaign of 2022. As always, I will also be supporting our Latin American brothers; we all share the same deep passion for football.