Invariant Subspace Approach to Boolean (Control) Networks

Abstract

A logical function can be used to characterize a property of states of a Boolean network (BN), which is considered as an aggregation of states. The dynamics of a set of logical functions are called the dual dynamics of the set. To illustrate the dual dynamics of a given set, which characterizes our concerned properties of a BN, the invariant subspace containing the set of logical functions is proposed, and its properties are investigated. Then, the invariant subspace of Boolean control network (BCN) is also proposed, and its dynamics are obtained. Finally, using outputs as the set of logical functions, the minimum output based dual dynamics is considered and proposed as the minimum realization of BCNs. The minimum realization might have much smaller size, which provides a possible solution to overcome the computational complexity of large scale BNs/BCNs. As an example, the proposed approaches for both BN and BCN are applied to an opinion dynamic network to demonstrate the efficiency of the technique proposed in this article.