Single-input Boolean networks

Exact dynamics 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 surprisingly intricate behaviour. We introduce a formalism for expressing the dynamics of multiple loops as a product of the dynamics of individual loops. Using this, we prove that the number of attractors scales as $2^m$, where m is the number of nodes in loops—which is as fast as possible.