Power Law Behavior in Elementary Cellular Automata

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This Demonstration shows the formation of percolation clusters in elementary cellular automata with different colors. Also shown is a log-log plot of the size distribution. The first row in the graphic shows the initial spatial conditions, which can be random with a given occupation probability ("random"), a single occupied cell ("single 1") or a single empty cell ("single 0"). Try rules 30, 45, 60, 86, 99, 105, 129, 150, 151, 153, 169, 182, 183, 184, 195 and 225 under different occupation probabilities to see a strong linear relationship, whose slope, known as the power law exponent, has important implications in a wide variety of applications. Most of the slopes are less than 1, implying diverging means and variances of cluster sizes, which strongly suggests critical behavior.

Contributed by: Jorge Laval (August 2022)
Open content licensed under CC BY-NC-SA



More details are at [1]. The algorithm used to find the cluster sizes is at [2].


[1] J. Laval, "Power Law Behavior of Elementary Cellular Automata," Preprints, 2021, 2021060649. doi:10.20944/preprints202106.0649.v1.

[2] "Mimic a Procedural, Recursive Clustering Algorithm for Site Percolation Using Functional Programming." (Jul 9, 2021) mathematica.stackexchange.com/a/85191.

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