March 3, 2023

F. C. Sheldon, T. M. A. Fink

The Kauffman model of genetic computation highlights the importance of criticality at the border of order and chaos. But our understanding of its behavior is incomplete, and much of what we do know relies on heuristic arguments. To better understand the model and obtain more rigorous insights, we show that there are fundamental links between the critical Kauffman model and aspects of number theory. Using these connections, we prove that the number of attractors and the mean attractor length grow faster than previously believed. Our work suggests that techniques from number theory, which are less familiar to the physics community, may be the right tools for fully cracking the Kauffman model.

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Joshua E. S. Duke University, Durham, NC Socolar, Stuart A. Bios Group, Santa Fe, NM Kauffman

Random Boolean networks, originally invented as models of genetic regulatory networks, are simple models for a broad class of complex systems that show rich dynamical structures. From a biological perspective, the most interesting networks lie at or near a critical point in parameter space that divides ``ordered'' from ``chaotic'' attractor dynamics. In the ordered regime, we show rigorously that the average number of relevant nodes (the ones that determine the attractor dyna...