Hidden Order of Boolean Networks

TL;DR

This paper uncovers a hidden order in Boolean networks using algebraic methods, revealing that dual network attractors significantly influence the network's overall order beyond observable trajectories.

Contribution

It introduces the concept of hidden order via dual networks and extends Boolean network analysis to the k-valued case, providing a new global perspective.

Findings

Hidden order is determined by dual network attractors.

Dual networks' structure influences the overall network order.

Results extended to k-valued Boolean networks.

Abstract

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.