Structures of M-Invariant Dual Subspaces with Respect to a Boolean Network

TL;DR

This paper characterizes M-invariant dual subspaces of Boolean networks using equitable partitions, linking them to observability and providing methods to construct output functions for enhanced network analysis.

Contribution

It establishes a novel bijection between dual subspaces and state partitions, and offers a complete structural characterization of M-invariant dual subspaces in Boolean networks.

Findings

Dual subspaces correspond to equitable partitions of the state set.

A BN is observable if the smallest M-invariant dual subspace is trivial.

Method for constructing output functions to achieve observability.

Abstract

This paper presents the following research findings on Boolean networks (BNs) and their dual subspaces.First, we establish a bijection between the dual subspaces of a BN and the partitions of its state set. Furthermore, we demonstrate that a dual subspace is $M$-invariant if and only if the associated partition is equitable (i.e., for every two cells of the partition, every two states in the former have the same number of out-neighbors in the latter) for the BN's state-transition graph (STG). Here $M$ represents the structure matrix of the BN.Based on the equitable graphic representation, we provide, for the first time, a complete structural characterization of the smallest $M$-invariant dual subspaces generated by a set of Boolean functions. Given a set of output functions, we prove that a BN is observable if and only if the partition corresponding to the smallest $M$-invariant dual subspace generated by this set of functions is trivial (i.e., all partition cells are singletons). Building upon our structural characterization, we also present a method for constructing output functions that render the BN observable.