Multiplicative loops

Exact dynamics of the critical Kauffman model with connectivity one

The single-input Kauffman network is the simplest class of Boolean networks. But it exhibits intricate behaviour at the edge of order and chaos that is poorly understood. We show that the behaviour of the Kauffman network can be expressed as a product of the behaviours of its disconnected loops, revealing an underlying algebraic structure. Using this, we prove the number of attractors is $2^m$, where $m$ is the number of loop nodes.