For his analysis, Barthelemy chose to represent chess as a decision tree in which each “branch” leads to a victory, a loss or a draw. Players are challenged to find the best move amidst all this complexity, particularly in the mid-game, to take the game towards favorable branches. That’s where those crucial turning points come into play. These positions are inherently unstable, which is why even a small error can have a dramatic influence on the trajectory of a match.
A case of combinatorial complexity
Example of a position taken from Mehedlishvili-Van Forrest.
Marc Barthélemy, 2025
Example of a position taken from Mehedlishvili-Van Forrest.
Marc Barthélemy, 2025
Interaction graph showing the white knight as the key piece in the previous position.
Marc Barthélemy, 2025
Interaction graph showing the white knight as the key piece in the previous position.
Marc Barthélemy, 2025
Example of a position taken from Mehedlishvili-Van Forrest.
Marc Barthélemy, 2025
Interaction graph showing the white knight as the key piece in the previous position.
Marc Barthélemy, 2025
Barthelemy has reimagined a chess game as a network of forces in which the pieces act as nodes in the network and the ways they interact represent the edges, using an interaction graph to capture how different pieces attack and defend each other. . The most important chess pieces are those that interact with many other pieces in a given game, which he calculated by measuring the frequency with which a node is on the shortest path between all pairs of nodes in the network (its “centrality”). of intermediation”).
It also calculated so-called “fragility scores,” which indicate how easy it is to remove those critical chess pieces from the board. And he was able to apply this analysis to more than 20,000 real chess games played by the best players in the world over the last 200 years.
Barthelemy discovered that his metric could identify turning points in specific matches. Furthermore, when he averaged his analysis over a large number of games, an unexpected universal pattern emerged. “We observed a surprising universality: the average fragility score is the same for all players and for all openings,” Barthelemy writes. And in famous chess games, “maximum fragility often coincides with crucial moments, characterized by brilliant moves that decisively change the balance of the game.”
Specifically, frailty scores begin to increase about eight moves before the critical inflection point position occurs and remain high for about 15 moves after that. “These results suggest that positional fragility follows a common trajectory, with tension peaking in the middlegame and dissipating toward the end,” he writes. “This analysis highlights the complex dynamics of chess, where the interaction between attack and defense shapes the overall structure of the game.”
Physical Review E, 2025. DOI: 10.1103/PhysRevE.00.004300 (About DOIs).