「经济学人」A game of two graphs

The mathematics of network analysis helps them follow the action

To a human observer of football (the soccer sort), the on-pitch patterns—offence stretching and squeezing defence, counterattacks coalescing out of thin air—are as mesmerising as they are easy to follow. For an artificial-intelligence (AI) model, however, understanding what is going on is far from trivial. Raw video is stuffed with information, most of it irrelevant. The first thing an AI engineer, therefore, has to do is teach the model what matters and what doesn’t. For football tactics, player and ball positions are a good place to start. But a team isn’t just a collection of isolated players; it’s a network of relationships.

Such networks, known to mathematicians as graphs, are made up of nodes connected by edges. On a football pitch, each player is a distinctive node, with edges capturing interactions such as passes and tackles. A match can, thus, be represented as an evolving sequence of graphs, no two alike between kickoff and the final whistle.

AI models capable of parsing such information, known as graph neural networks (GNNs), can be used to identify which sorts of patterns spell danger for a team, and, consequently, what to avoid. Many scientific fields find them useful. At an upcoming conference hosted by the Massachusetts Institute of Technology, Joris Bekkers and Amod Sahasrabudhe, two sports analysts, will present a model they devised while at the United States Soccer Federation. This predicts counterattacks using a method originally devised to predict how atoms come together to form crystals.

All models are simplifications, and graph-based ones are no exception. For one thing, they are best at representing interactions between pairs of nodes. To capture a four-person defensive formation, other, yet more complex, structures may have to be called off the bench. (Feb 12th 2025)

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