networks

Complexity physics finds crucial tipping points in chess games

chess, mathematics, networks, phase transitions, Physics, Science, tipping points / Shannon Garcia / January 24, 2025

For his analysis, Barthelemy chose to represent chess as a decision tree in which each “branch” leads to a win, loss, or draw. Players face the challenge of finding the best move amid all this complexity, particularly midgame, in order to steer gameplay into favorable branches. That’s where those crucial tipping points come into play. Such positions are inherently unstable, which is why even a small mistake can have a dramatic influence on a match’s trajectory.

A case of combinatorial complexity

Barthelemy has re-imagined a chess match as a network of forces in which pieces act as the network’s nodes, and the ways they interact represent the edges, using an interaction graph to capture how different pieces attack and defend one another. The most important chess pieces are those that interact with many other pieces in a given match, which he calculated by measuring how frequently a node lies on the shortest path between all the node pairs in the network (its “betweenness centrality”).

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 matches played by the world’s top players over the last 200 years.

Barthelemy found that his metric could indeed identify tipping points in specific matches. Furthermore, when he averaged his analysis over a large number of games, an unexpected universal pattern emerged. “We observe a surprising universality: the average fragility score is the same for all players and for all openings,” Barthelemy writes. And in famous chess matches, “the maximum fragility often coincides with pivotal moments, characterized by brilliant moves that decisively shift the balance of the game.”

Specifically, fragility scores start to increase about eight moves before the critical tipping point position occurs and stay high for some 15 moves after that. “These results suggest that positional fragility follows a common trajectory, with tension peaking in the middle game and dissipating toward the endgame,” he writes. “This analysis highlights the complex dynamics of chess, where the interaction between attack and defense shapes the game’s overall structure.”

Physical Review E, 2025. DOI: 10.1103/PhysRevE.00.004300 (About DOIs).

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Why solving crosswords is like a phase transition

mathematics, networks, percolation, phase transitions, Physics, Science, statistical physics / Beth Washington / January 9, 2025

There’s also the more recent concept of “explosive percolation,” whereby connectivity emerges not in a slow, continuous process but quite suddenly, simply by replacing the random node connections with predetermined criteria—say, choosing to connect whichever pair of nodes has the fewest pre-existing connections to other nodes. This introduces bias into the system and suppresses the growth of large dominant clusters. Instead, many large unconnected clusters grow until the critical threshold is reached. At that point, even adding just one or two more connections will trigger one global violent merger (instant uber-connectivity).

Puzzling over percolation

One might not immediately think of crossword puzzles as a network, although there have been a couple of relevant prior mathematical studies. For instance, John McSweeney of the Rose-Hulman Institute of Technology in Indiana employed a random graph network model for crossword puzzles in 2016. He factored in how a puzzle’s solvability is affected by the interactions between the structure of the puzzle’s cells (squares) and word difficulty, i.e., the fraction of letters you need to know in a given word in order to figure out what it is.

Answers represented nodes while answer crossings represented edges, and McSweeney assigned a random distribution of word difficulty levels to the clues. “This randomness in the clue difficulties is ultimately responsible for the wide variability in the solvability of a puzzle, which many solvers know well—a solver, presented with two puzzles of ostensibly equal difficulty, may solve one readily and be stumped by the other,” he wrote at the time. At some point, there has to be a phase transition, in which solving the easiest words enables the puzzler to solve the more difficult words until the critical threshold is reached and the puzzler can fill in many solutions in rapid succession—a dynamic process that resembles, say, the spread of diseases in social groups.

In this sample realization, sites with black sites are shown in black; empty sites are white; and occupied sites contain symbols and letters. — In this sample realization, black sites are shown in black; empty sites are white; and occupied sites contain symbols and letters. Credit: Alexander K. Hartmann, 2024

Hartmann’s new model incorporates elements of several nonstandard percolation models, including how much the solver benefits from partial knowledge of the answers. Letters correspond to sites (white squares) while words are segments of those sites, bordered by black squares. There is an a priori probability of being able to solve a given word if no letters are known. If some words are solved, the puzzler gains partial knowledge of neighboring unsolved words, which increases the probability of those words being solved as well.

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Fungi may not think, but they can communicate

Biology, food, fungi, networks, nutrients, Science / Mike M. / November 2, 2024

Because the soil layer was so thin, most hyphae, which usually grow and spread underground by releasing spores, were easily seen, giving the researchers an opportunity to observe where connections were being made in the mycelium. Early hyphal coverage was not too different between the X and circle formations. Later, each showed a strong hyphal network, which makes up the mycelium, but there were differences between them.

While the hyphal network was pretty evenly distributed around the circle, there were differences between the inner and outer blocks in the X arrangement. Levels of decay activity were determined by weighing the blocks before and after the incubation period, and decay was pretty even throughout the circle, but especially evident on the four outermost blocks of the X. The researchers suggest that there were more hyphal connections on those blocks for a reason.

“The outermost four blocks, which had a greater degree of connection, may have served as “outposts” for foraging and absorbing water and nutrients from the soil, facilitated by their greater hyphal connections,” they said in the same study.

Talk to me

Fungal mycelium experiences what’s called acropetal growth, meaning it grows outward in all directions from the center. Consistent with this, the hyphae started out growing outward from each block. But over time, the hyphae shifted to growing in the direction that would get them the most nutrients.

Why did it change? Here is where the team thinks communication comes in. Previous studies found electrical signals are transmitted through hyphae. These signals sync up after the hyphae connect into one huge mycelium, much like the signals transmitted among neurons in organisms with brains. Materials such as nutrients are also transferred throughout the network.

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