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

TL;DR

The paper tackles the problem of understanding how dual subspace structures in Boolean networks relate to observable behavior. By establishing a bijection between dual subspaces and state partitions and proving that an $M$-invariant dual subspace corresponds exactly to an equitable partition, it provides a complete graph-theoretic characterization of the smallest $M$-invariant dual subspaces generated by given outputs. It then shows that BN observability is equivalent to the triviality of the partition associated with the unobservable subspace, and it offers a method to construct output functions that render the BN observable. The results yield a practical framework for analyzing dual dynamics via quotient digraphs and for designing observable outputs in large-scale BN/BCN models, enabling compact representations and systematic observability design.

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.

Figures (6)

• Figure 1: A digraph ${\mathcal{G}}$ and its three quotient digraphs.
• Figure 2: An illustration of $M$-invariant dual subspace using partitions.
• Figure 3: Given a connected STG and a dual subspace ${\mathcal{Z}}^{*}$, the process of finding ${\mathcal{P}}(\overline{{\mathcal{Z}}^{*}})$, the coarsest equitable partition finer than ${\mathcal{P}}({\mathcal{Z}}^{*})$ is illustrated, where ${\mathcal{P}}({\mathcal{Z}}^{*})=\{\{v_{1},v_{4},v_5\},\{v_{2},v_{3}\},\{v_{6,7}\},\{v_8\}\}$. We finally get ${\mathcal{P}}(\overline{{\mathcal{Z}}^{*}})=\{\{v_1,v_7\},\{v_2,v_5\},\{v_6\},\{v_3,v_4\},\{v_8\}\}$, whose quotient digraph is illustrated in (b).
• Figure 4: Proof of Theorem \ref{['the-5']}
• Figure 5: Given a connected STG and a dual subspace ${\mathcal{Z}}^{*}$, the process of finding ${\mathcal{P}}(\overline{{\mathcal{Z}}^{*}})$, the coarsest equitable partition finer than ${\mathcal{P}}({\mathcal{Z}}^{*})$ is illustrated, where ${\mathcal{P}}({\mathcal{Z}}^{*})=\{\{v_1, v_4, v_8\},\{v_2,v_5,v_7\},\{v_3,v_6,v_9,v_{10}\},\{v_{11},\ldots ,v_{16}\}\}$. The dotted edge between $v_{16}$ and $v_{12}$ means the path from $v_{16}$ to $v_{12}$. We finally get ${\mathcal{P}}(\overline{{\mathcal{Z}}^{*}})=\{\{v_{1},v_{4},v_{8}\},\{v_3,v_6\},\{v_{2},v_{5},v_{7}\},\{v_9,v_{10}\},\{v_{11}\},$$\ldots,\{v_{16}\}\}$, whose quotient digraph is illustrated in (c).

Theorems & Definitions (51)

• Definition 1
• Remark 1
• Lemma 1
• Definition 2
• Definition 3
• Example 1
• Definition 4
• Definition 5
• Definition 6
• Remark 2