Hidden Order of Boolean Networks

It is a common belief that the order of a Boolean network is mainly determined by its attractors, including fixed points and cycles. Using semi-tensor product (STP) of matrices and the algebraic state-space representation (ASSR) of Boolean networks, this paper reveals that in addition to this explicit order, there is certain implicit or hidden order, which is determined by the fixed points and limit cycles of their dual networks. The structure and certain properties of dual networks are investigated. Instead of a trajectory, which describes the evolution of a state, hidden order provides a global picture to describe the evolution of the overall network. It is our conjecture that the order of networks is mainly determined by the dual attractors via their corresponding hidden orders. The previously obtained results about Boolean networks are further extended to the k-valued case.