For his analysis, Barthelemy chose to represent chess as a decision tree, with each “branch” leading to a win, loss, or draw. Players face the challenge of finding the best move amidst such complexity, especially in the mid-game, in order to steer the gameplay to a favorable branch. There is an important turning point involved. Such positions are inherently unstable, so even small mistakes can have a dramatic effect on the flow of the match.
Examples of combinatorial complexity
An example of Mehedrishvili-van Forest’s position.
Marc Barthelemy, 2025
Interaction graph showing White Knight as a key piece in the previous position.
Marc Barthelemy, 2025
Interaction graph showing White Knight as a key piece in the previous position.
Marc Barthelemy, 2025
An example of Mehedrishvili-van Forest’s position.
Marc Barthelemy, 2025
Interaction graph showing White Knight as a key piece in the previous position.
Marc Barthelemy, 2025
Barthelemy reimagined the game of chess as a network of forces where the pieces act as nodes in the network and the way the pieces interact represents the edges. We used an interaction graph to capture how different pieces attack and defend each other. The most important chess pieces are the ones that interact with many other pieces in a particular match, and he calculates the frequency with which a node lies on the shortest path between all pairs of nodes in the network (the It was calculated by measuring “centrality”).
He also calculated a so-called “vulnerability score” that shows how easy it is to remove important chess pieces from the board. He was then able to apply this analysis to over 20,000 actual chess matches played by the world’s top players over the past 200 years.
Barthelemy has found that his metrics can reliably identify turning points in a given match. Furthermore, when we averaged the analysis across a large number of games, an unexpected and universal pattern emerged. “We observed surprising universality: the average vulnerability score is the same for all players and all openings,” Barthelemy wrote. And in famous chess matches, “the greatest vulnerability often coincides with a key moment marked by a brilliant move that decisively changes the balance of the game.”
Specifically, the vulnerability score begins to increase at about 8 moves, when the critical tipping point position occurs, and then remains high for about 15 moves. “These results suggest that position vulnerability follows a common trajectory, with tension peaking in the middle and dissipating towards the end,” he wrote. “This analysis highlights the complex dynamics of chess, where the interplay between attack and defense forms the overall structure of the game.”
Physical Review E, 2025. DOI: 10.1103/PhysRevE.00.004300 (About DOI).
