Complexity physics finds crucial turning points in chess games

For his analysis, Barthelemy chose to represent chess as a decision tree in which each “branch” leads to a victory, loss or drawing. Players face the challenge of finding the best movement in the midst of all this complexity, especially midgame, to send gameplay to favorable branches. That is where those crucial turning points come into play. Such positions are inherently unstable, which is why even a small error can have a dramatic influence on the process of a competition.

A case of combination complexity

Barthelemy has re -devised a chess match as a network of forces in which documents act as the nodes of the network, and the ways in which they act on each other represent the edges, using an interaction craft to determine how different pieces attack and defend each other. The most important chess pieces are those who interact with many other pieces in a certain match, which he calculated by measuring how often a junction on the shortest path lies between all button pairs in the network (the “intermediate centralality”)).

He 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 actual chess competitions that have been played by the world's top players in the last 200 years.

Barthelemy discovered that his metric could indeed identify tip points in specific competitions. Moreover, when he on average his analysis over a large number of games, an unexpected universal pattern arose. “We observe a surprising universality: the average fragility score is the same for all players and for all openings,” writes Barthelemy. And in famous chess competitions, “the maximum vulnerability often coincides with crucial moments, characterized by brilliant movements that shift the balance of the game decisively.”

In particular, fragility scores begin to increase about eight movements before the critical turning point position takes place and then about 15 movements remain high. “These results suggest that positional fragility follows a common process, with tension that peaks in the middle game and disappears to the endgame,” he writes. “This analysis emphasizes the complex dynamics of chess, whereby the interaction between attack and defense forms the general structure of the game.”

Physical assessment E, 2025. DOI: 10.1103/Physreve.00.004300 (Over Dois).