Exact behavior of the critical Kauffman model with connectivity one

The critical Kauffman model with connectivity one is the simplest class of critical Boolean networks. Nevertheless, it exhibits intricate behavior at the boundary of order and chaos. We show that the model is equivalent to a deceptively simple algebraic system of polynomials which count the number and length of cycles. The polynomial for multiple loops is the product of the polynomials for individual loops. Using this perspective, we prove that the number of cycles scales as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msup><a:mn>2</a:mn><a:mi>m</a:mi></a:msup></a:math>, where <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>m</b:mi></b:math> is the number of nodes in loops—as fast as possible and faster than previously believed. Published by the American Physical Society 2024