Invariant Subspace Approach to Boolean (Control) Networks

TL;DR

This paper introduces an invariant subspace framework for Boolean networks and control networks, enabling analysis of their properties and a minimal realization approach to address computational complexity.

Contribution

It proposes the concept of invariant subspaces for Boolean and control networks, offering new insights into their dynamics and a method for minimal realization to reduce complexity.

Findings

Invariant subspace containing logical functions characterizes BN properties.

Invariant subspace of BCN remains invariant under dynamics.

Minimum realization of BCN reduces computational complexity.

Abstract

A logical function can be used to characterizing a property of a state of Boolean network (BN), which is considered as an aggregation of states. To illustrate the dynamics of a set of logical functions, which characterize 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. The dynamics of invariant subspace of BCN is also invariant. Finally, using outputs as the set of logical functions, the minimum realization of BCN is proposed, which provides a possible solution to overcome the computational complexity of large scale BNs/BCNs.