# Structural Controllability of Boolean Control Networks under Partial Information

Authors: Qinyao Pan; Jie Zhong; Shalin Tong; Bowen Li; Xiaoxu Liu
Journal: Complexity
 (2021)

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## Abstract

It is worth noting that both nodes’ coupling connections and logical updating functions play a vital role in state evolutions of Boolean networks (BNs). In this paper, a new concept named structural controllability (SC) about Boolean control networks (BCNs) with known partial information on nodes’ connections is studied. Then, by referring to semi-tensor product (STP) techniques, several types of SC are presented according to different issues of Boolean functions. Thereafter, several necessary and sufficient conditions are derived for SC of BCNs. Finally, a biological model of the lactose operon inEscherichia coli is simulated to show the effectiveness of the main theoretical results.


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## Body

## 1. Introduction

Boolean networks (BNs) are known as discrete-time logical dynamics, firstly proposed in [1], where nodes evolute the corresponding states according to several Boolean functions. In a BN, every gene can be chosen from a set with logical variables 1 and 0. Therefore, a BN is illustrated according to certain Boolean functions g1,…,gn:1,0n2⟶1,0n. A digraph (named as network structure digraph) with nodes a1,…,an can well show the connections among every node (named as network structure), where there exists a directed edge from node ai to node aj if gj depends on ai, i.e., if there is a sequence of variables a˜∈1,0n−1 satisfying gja˜,ai≠gja˜,¬ai, where a˜=a1,…,ai−1,ai+1,…,an. Recently, many important properties of BNs have been issued [2–18], and so on.Recently, great attention has been paid to BN owing to a new concept called semi-tensor product (STP) of matrices [19]. STP is firstly proposed by Cheng and his collaborators [19], which is a generalization of conventional matrix product. STP can be used as an efficient tool to convert BNs into corresponding algebraic representations. Then, many fundamental results about BNs have been issued [20–22]. Please refer to [23, 24] for more applications of STP.It is worth noting that stability and controllability are of importance in different research fields, such as impulsive differential systems [25, 26], functional differential equations [27], switched systems, and Boolean networks [28–30]. If we take external control inputs into consideration, BNs will be Boolean control networks (BCNs) established by Akutsu [28] for the first time. Note that it has been increasingly challenging for researches concerning BNs and BCNs [29, 30]. Recently, the problem of controllability has been studied rapidly; some fundamental results have been established. To list a few, one of the main results on controllability has been issued [31], based on an input-state incidence matrix, where controllability is illustrated by the positiveness of a controllability matrix. Moreover, by Perron–Frobenius theory, controllability has been studied in [20]. In the past few years, another important issue (called pinning control method) gets great attention with less control cost and lower computational complexity [32–34].It is noted that both nodes’ coupling connections and logical updating functions play a vital role in state evolutions of BNs. During the past few decades, a relationship between nodes’ connection and some concepts of BNs, such as stability and oscillatority, has been issued in [35–39]. Note that in real-world, not all the messages about nodes coupling and the updating Boolean functions are available, that is, only part of the nodes’ connections are available. For instance, a new research concept named structural controllability (SC) concerning BCNs is well addressed in [40], whose full messages of nodes’ connections are available while Boolean functions are unavailable. Therefore, it is important to analyze dynamical properties with only part of the messages on nodes’ connections, i.e., all the messages of BNs are unavailable.Therefore, based on the above analysis, some contributions of this paper are given: (i) SC of BCNs with partial information of nodes’ connections available has been firstly issued, where partial nodes’ connections are available; (ii) four cases of SC are defined owing to different types of network structure and logical functions. Then, several types of SC matrices are constructed using STP and Hadamard product, and some criteria are issued; (iii) SC of BCNs with partial information of nodes’ connections available is a simple extension of general controllability of BCNs with full information of both network structure and logical functions.The rest of the paper is organized as follows: in Section2, some preliminaries are given. In Section 3, the main results of SC are given, and some criteria are obtained. In Section 4, to explain the main results, one biological model is provided. In Section 5, a brief conclusion is established.Compared with the conference version of this paper [41], in this paper, much more detailed descriptions of the theoretical results about SC have been established. In [41], only two cases of SC are considered. However, in the revision, totally four cases of SC are discussed, and several necessary and sufficient conditions are derived for SC of BCNs, which is a generalization of the theoretical results in [41]. In addition, in this paper, some comments and detailed comparisons with the existing references have been presented, as well as a new figure has been added to better illustrate the implication relationships between four different types of SC.Here, some notations are given:ℕ+ denotes the set consisting of positive integers. s:t=s,s+1,…,t, where s<t and s,t∈ℕ+. δnm is the m-th column of the identity matrix In. Δn=δnm|1≤m≤n. ColkBColB denotes the m-th column (columns) of matrix B. ℜq×rLq×r denotes the set of q×r real matrices (logical matrices). Bu×v is the set consisting of u×v Boolean matrices. M=δnj1,…,δnjp:=δnj1,…,jp is a logical matrix. Boolean addition is defined as u+ℬv:=u∨v, where u,v∈D. A+ℬBij=aij+ℬbij, where A=aij∈Bn×s, B=bij∈Bn×s. Blki (A) represents the i-th block of matrix A. A Boolean matrix satisfies all the entries belonging to D=0,1. Wm,n=In⊗δm1,In⊗δm2,…,In⊗δmm. “⊗” denotes the known Kronecker product, “∗” denotes the known Khatri–Rao product, “∘” denotes the known Hadamard product, and “<⋉” denotes STP of two given matrices.


## 2. Preliminaries

Firstly, here, some basic illustrations on STP and BCNs are introduced.Definition 1.
The STP of two given matricesC∈ℜm×n and D∈ℜp×q is defined as C⋉D=C⊗Is/nD⊗Is/p, where s is the least common multiple of n and p.Define the bijective relationship “∼” between D:=1,0 and Δ2 as γ∼δ22−γ, which naturally extends to γ1,…,γn∼⋉i=1nδ22−γi between Dn and Δ2ν. Equivalently, Boolean functions can be transferred into an algebraic representation.Lemma 1.
Given a logical functiong:Dn⟶D, its multilinear form is given as gx1,…,xν∼Mg∼⋉i=1nxi, where Mg∈L2×2ν is the structure matrix, uniquely determined by g.Some well-known structure matrices of binary operators are presented based on Lemma1, such as negation “¬” Mn=δ221, conjunction “∧” Mc=δ21222, disjunction “∨” Md=δ21112, conditional “⟶” Mi=δ21211, and biconditional “↔” Me=δ21221.Normally, a BCN consisting ofν nodes and κ external inputs can be described:(1)xiτ+1=gixkτk∈Θi,x¯kτk∈Θ¯i,ϑkτk∈Ξi,ϑ¯kτk∈Ξ¯i,i∈1:ν,where the negation of x is denoted by x¯, i.e., x¯=¬x. Here, x1,…,xν∈D express the states, ϑi∈D expresses the control input, and Θi⊆1:ν and Ξi⊆1:κ are the index sets of the in-neighbors of i-th node. For each i∈1:ν, we call sets Θi,Ξi as the activating coupling node sets of the i-th node. In addition, for each i∈1:ν, Θ¯i⊆1:ν\Θi and Ξ¯i⊆1:κ\Ξi are the index sets of coupling nodes of i-th node, and we call sets Θ¯i,Ξ¯i as the inhibiting couplings node sets of the i-th node. gi:DΘi+Θ¯i+Ξi+Ξ¯i⟶D, i∈1:ν, are Boolean functions, which only consist of “∧” and “∨.” Therefore, the logical dynamics of (1) is only connected by logical operators “∧,” “∨,” and “¬.” Thus, given a BCN (1), one can always obtain its corresponding algebraic representation using Lemma 1.


### 2.1. Problem Formulation

It is noted that dynamical evolutions of BN (1) are determined by the Boolean functions g1,…,gν, as well as its coupling sets Θi, Θ¯i, Ξi, and Ξ¯i, i∈1:ν. In practical biological applications, due to experimental impacts and other reasons, we might only know the in-neighbors of some nodes, but cannot acquire the coupling relationship for each node. Therefore, assume that only partial information of (1) is known in this paper. Here, we assume that the sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈i1,…,iμ⊆1:ν, are given beforehand, while binary operators among different nodes in Θi, Θ¯i, Ξi, and Ξ¯i, i∈i1,…,iμ⊆1:ν, are unavailable, which implies that for nodes xi1,…,xiμ, the corresponding activating coupling nodes, and inhibiting coupling nodes are given, whereas binary operators among nodes are unavailable.For simple illustrations, suppose only activating coupling nodes and inhibiting coupling nodes for nodesx1,…,xμ, are given, which is as follows:(2)xiτ+1=gixkτk∈Θi,x¯kτk∈Θ¯i,ϑkτk∈Ξi,ϑ¯kτk∈Ξ¯i,i∈1:μ,xjτ+1=gjxkτk∈Θj,x¯kτk∈Θ¯j,ϑkτk∈Ξj,ϑ¯kτk∈Ξ¯j,j∈μ+1:ν,where Θi,Θ¯i,Ξi,andΞ¯i,i∈1:μ, are known and Θj,Θ¯j,Ξj, and Ξ¯j,j∈μ+1:ν, are unknown. The binary operators among all the nodes are unknown, which implies that the logical operators between nodes in Θi,Θ¯i,Ξi, and Ξ¯i,i∈1:ν, are unknown.Based on Lemma1, one has equivalent algebraic form of logical dynamics of (2). Given sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈1:μ, we make the following ordering assumptions to simplify calculation:(3)Θi=a1i,…,aΘii,a1i<⋯<aΘii∈1:ν,Θ¯i=b1i,…,bΘ¯ii,b1i<⋯<bΘ¯ii∈1:ν,Ξi=c1i,…,cΞii,c1i<⋯<cΞii∈1:κ,Ξ¯i=d1i,…,dΞ¯ii,d1i<⋯<dΞ¯ii∈1:κ.Then, under assumption (3), one has that(4)xiτ+1=xa1iτσ1ixa2iτσ2i,…,σΘi−1ixaΘiiτσΘiix¯b1iτσΘi+1ix¯b2iτσΘi+2i,…,σΘi+Θ¯i−1ix¯bΘ¯iiτσΘi+Θ¯iiϑc1iτσ^1iϑc2iτσ^2i,…<σ^Ξi−1iϑcΞiiτσ^Ξiiϑ¯d1iτσ^Ξi+1iϑ¯d2iτσ^Ξi+2i,…,σ^Ξi+Ξ¯i−1iϑ¯dΞ¯iiτ,i∈1:μ,where binary operators σji∈∨,∧, j∈1:Θi+Θ¯i, i∈1:ν, σ^ji∈∨,∧, j∈1:Ξi+Ξ¯i−1, and i∈1:ν. Here, operators σji, j∈1:Θi+Θ¯i, σ^ji, and j∈1:Ξi+Ξ¯i−1 are binary operators connecting the in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i,i∈1:ν.Given binary operatorsσj∈∨,∧, j∈1:Θi+Θ¯i, and σ^j∈∨,∧, j∈1:Ξi+Ξ¯i−1, one has that(5)xiτ+1=Oi⋉j=1Θixai,jτ〆j=1Θ¯iMn〆xbi,jτ⋉j=1Ξiϑci,jτ⋉j=1Ξ¯iMn⋉ϑdi,jτ,where matrix Oi is uniquely calculated according to binary operators σj∈∨,∧, j∈1:Θi+Θ¯i, and σ^j∈∨,∧, j∈1:Ξi+Ξ¯i−1. Thus, there totally exist 2Θi+Θ¯i+Ξi+Ξ¯i−1 different kinds of structure matrices for matrices Oi, i∈1:μ. Given each i∈1:μ, illustrate all the 2Θi+Θ¯i+Ξi+Ξ¯i−1 kinds of matrices Oi in (5) by set Γi.Denotingxτ=⋉i=1νxiτ and ϑτ=⋉i=1κϑiτ, one can further calculate a unique matrix Ti∈L2×2ν+κ, i∈1:μ, based on Lemma 1, satisfying(6)⋉j=1Θixai,jτ⋉j=1Θ¯iMn⋉xbi,jτ⋉j=1Ξiϑci,jτ⋉j=1Ξ¯iMn⋉ϑdi,jτ=Tiϑτxτ.Thus, forx1,…,xμ, the following equation is obtained:(7)xiτ+1=OiTiϑτxτ.Letx1τ=⋉i=1μxiτ∈Δ2μ; it leads to a simplification illustration of (4) for nodes x1,…,xμ, using Khatri–Rao product,(8)x1τ+1=F1ϑτxτ,where F1∈Λ1:≜∗i=1μOi⋉Ti:Oi∈Γi,i∈1:μ. Then, one has Λ1=2∑i=1μΘi+Θ¯i+Ξi+Ξ¯i−μ. Denoting r=Λ1, assume that Λ1=F11,F12,…,F1r, where matrices F11,F12,…,F1r∈L2μ×2ν+κ.Then, letx2τ=⋉i=μ+1νxiτ∈Δ2ν−μ; one can obtain algebraic form for the last ν−μ nodes:(9)x2τ+1=F2ϑτxτ,where F2∈L2ν−μ×2ν+κ expresses the structure matrix determined by in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈μ+1:ν, and its binary operators. Therefore, it leads to totally s=2ν−μ2ν+κ cases for matrix F2. Denote the set of all the 2ν−μ2ν+κ different kinds of F2 by Λ2, that is, F2∈Λ2≜F21,F22,…,F2s, where F21,F22,…,F2s∈L2ν−μ×2ν+κ. Therefore, considering (2), one has that(10)x1τ+1=F1ϑτxτ,x2τ+1=F2ϑτxτ,where F1∈Λ1=F11,F12,…,F1r and F2∈Λ2=F21,F22,…,F2s. Multiplying (10) yields an algebraic form of (2),(11)xτ+1=F1ϑτxτF2ϑτxτ≔Lϑτxτ,where matrix L is called the state transition matrix of system (2) choosing from a set Ω≜F1∗F2,F1∈Λ1,F2∈Λ2. In addition, one can calculate that Ω=rs and represent all of the state transition matrices by Ω≜L1,L2,…,Lrs, where L1,L2,…,Lrs∈L2ν×2ν+κ. This implies that each matrix Li, i∈1:rs, is one possible state transition matrix for system (2), where only the corresponding in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈1:μ, for those first μ nodes, are known.


## 2.1. Problem Formulation

It is noted that dynamical evolutions of BN (1) are determined by the Boolean functions g1,…,gν, as well as its coupling sets Θi, Θ¯i, Ξi, and Ξ¯i, i∈1:ν. In practical biological applications, due to experimental impacts and other reasons, we might only know the in-neighbors of some nodes, but cannot acquire the coupling relationship for each node. Therefore, assume that only partial information of (1) is known in this paper. Here, we assume that the sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈i1,…,iμ⊆1:ν, are given beforehand, while binary operators among different nodes in Θi, Θ¯i, Ξi, and Ξ¯i, i∈i1,…,iμ⊆1:ν, are unavailable, which implies that for nodes xi1,…,xiμ, the corresponding activating coupling nodes, and inhibiting coupling nodes are given, whereas binary operators among nodes are unavailable.For simple illustrations, suppose only activating coupling nodes and inhibiting coupling nodes for nodesx1,…,xμ, are given, which is as follows:(2)xiτ+1=gixkτk∈Θi,x¯kτk∈Θ¯i,ϑkτk∈Ξi,ϑ¯kτk∈Ξ¯i,i∈1:μ,xjτ+1=gjxkτk∈Θj,x¯kτk∈Θ¯j,ϑkτk∈Ξj,ϑ¯kτk∈Ξ¯j,j∈μ+1:ν,where Θi,Θ¯i,Ξi,andΞ¯i,i∈1:μ, are known and Θj,Θ¯j,Ξj, and Ξ¯j,j∈μ+1:ν, are unknown. The binary operators among all the nodes are unknown, which implies that the logical operators between nodes in Θi,Θ¯i,Ξi, and Ξ¯i,i∈1:ν, are unknown.Based on Lemma1, one has equivalent algebraic form of logical dynamics of (2). Given sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈1:μ, we make the following ordering assumptions to simplify calculation:(3)Θi=a1i,…,aΘii,a1i<⋯<aΘii∈1:ν,Θ¯i=b1i,…,bΘ¯ii,b1i<⋯<bΘ¯ii∈1:ν,Ξi=c1i,…,cΞii,c1i<⋯<cΞii∈1:κ,Ξ¯i=d1i,…,dΞ¯ii,d1i<⋯<dΞ¯ii∈1:κ.Then, under assumption (3), one has that(4)xiτ+1=xa1iτσ1ixa2iτσ2i,…,σΘi−1ixaΘiiτσΘiix¯b1iτσΘi+1ix¯b2iτσΘi+2i,…,σΘi+Θ¯i−1ix¯bΘ¯iiτσΘi+Θ¯iiϑc1iτσ^1iϑc2iτσ^2i,…<σ^Ξi−1iϑcΞiiτσ^Ξiiϑ¯d1iτσ^Ξi+1iϑ¯d2iτσ^Ξi+2i,…,σ^Ξi+Ξ¯i−1iϑ¯dΞ¯iiτ,i∈1:μ,where binary operators σji∈∨,∧, j∈1:Θi+Θ¯i, i∈1:ν, σ^ji∈∨,∧, j∈1:Ξi+Ξ¯i−1, and i∈1:ν. Here, operators σji, j∈1:Θi+Θ¯i, σ^ji, and j∈1:Ξi+Ξ¯i−1 are binary operators connecting the in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i,i∈1:ν.Given binary operatorsσj∈∨,∧, j∈1:Θi+Θ¯i, and σ^j∈∨,∧, j∈1:Ξi+Ξ¯i−1, one has that(5)xiτ+1=Oi⋉j=1Θixai,jτ〆j=1Θ¯iMn〆xbi,jτ⋉j=1Ξiϑci,jτ⋉j=1Ξ¯iMn⋉ϑdi,jτ,where matrix Oi is uniquely calculated according to binary operators σj∈∨,∧, j∈1:Θi+Θ¯i, and σ^j∈∨,∧, j∈1:Ξi+Ξ¯i−1. Thus, there totally exist 2Θi+Θ¯i+Ξi+Ξ¯i−1 different kinds of structure matrices for matrices Oi, i∈1:μ. Given each i∈1:μ, illustrate all the 2Θi+Θ¯i+Ξi+Ξ¯i−1 kinds of matrices Oi in (5) by set Γi.Denotingxτ=⋉i=1νxiτ and ϑτ=⋉i=1κϑiτ, one can further calculate a unique matrix Ti∈L2×2ν+κ, i∈1:μ, based on Lemma 1, satisfying(6)⋉j=1Θixai,jτ⋉j=1Θ¯iMn⋉xbi,jτ⋉j=1Ξiϑci,jτ⋉j=1Ξ¯iMn⋉ϑdi,jτ=Tiϑτxτ.Thus, forx1,…,xμ, the following equation is obtained:(7)xiτ+1=OiTiϑτxτ.Letx1τ=⋉i=1μxiτ∈Δ2μ; it leads to a simplification illustration of (4) for nodes x1,…,xμ, using Khatri–Rao product,(8)x1τ+1=F1ϑτxτ,where F1∈Λ1:≜∗i=1μOi⋉Ti:Oi∈Γi,i∈1:μ. Then, one has Λ1=2∑i=1μΘi+Θ¯i+Ξi+Ξ¯i−μ. Denoting r=Λ1, assume that Λ1=F11,F12,…,F1r, where matrices F11,F12,…,F1r∈L2μ×2ν+κ.Then, letx2τ=⋉i=μ+1νxiτ∈Δ2ν−μ; one can obtain algebraic form for the last ν−μ nodes:(9)x2τ+1=F2ϑτxτ,where F2∈L2ν−μ×2ν+κ expresses the structure matrix determined by in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈μ+1:ν, and its binary operators. Therefore, it leads to totally s=2ν−μ2ν+κ cases for matrix F2. Denote the set of all the 2ν−μ2ν+κ different kinds of F2 by Λ2, that is, F2∈Λ2≜F21,F22,…,F2s, where F21,F22,…,F2s∈L2ν−μ×2ν+κ. Therefore, considering (2), one has that(10)x1τ+1=F1ϑτxτ,x2τ+1=F2ϑτxτ,where F1∈Λ1=F11,F12,…,F1r and F2∈Λ2=F21,F22,…,F2s. Multiplying (10) yields an algebraic form of (2),(11)xτ+1=F1ϑτxτF2ϑτxτ≔Lϑτxτ,where matrix L is called the state transition matrix of system (2) choosing from a set Ω≜F1∗F2,F1∈Λ1,F2∈Λ2. In addition, one can calculate that Ω=rs and represent all of the state transition matrices by Ω≜L1,L2,…,Lrs, where L1,L2,…,Lrs∈L2ν×2ν+κ. This implies that each matrix Li, i∈1:rs, is one possible state transition matrix for system (2), where only the corresponding in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈1:μ, for those first μ nodes, are known.


## 3. Main Results

In this section, several types of controllability of (2) are presented, where only the corresponding in-neighbors of nodes x1,…,xμ, are known.


### 3.1. First Class of Structural Controllability

At the beginning, only the corresponding in-neighbor setsΘi,Θ¯i,Ξi,andΞ¯i for x1,…,xμ, are supposed to be known, the corresponding Boolean functions gi, i∈1:μ, and both the in-neighbor sets Θj,Θ¯j,Ξj,andΞ¯j and the Boolean functions gj, j∈μ+1:ν, are unknown. Under these assumptions, reachability and controllability for (2) are firstly defined as follows.Firstly, consider the first class structural reachability (FCSR) between two given statesx0 and xd under any Boolean functions gi, i∈1:ν.Definition 2.
Consider system (2) and the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for x1,…,xμ, are available. The destination state xd∈Δ2ν achieves FCSR from the initial state x0∈Δ2ν, if x0 can be steered to xd for arbitrary Boolean functions g1,…,gν. In addition, (2) achieves first class structural controllability (FCSC) if for any initial state x0 and destination state xd, xd achieves FCSR from x0.By consideringΞ=L1,L2,…,Lrs, divide L1,…,Lrs, as follows: Li=Blk1Li,Blk2Li,…,Blk2κLi, and denote(12)Ci=∑l=12νMiBl,∈L2ν×2ν,i∈1:rs,where Mi=Li⋉12κ∈L2ν×2ν, and matrix Ci, i∈1:rs, is the controllability matrix when state transition matrix of system (2) is Li. Then, introduce the following Boolean matrix in order to issue FCSC,(13)S1=C1∘C2∘⋯∘Crs∈L2ν×2ν,where “∘” represents the well-known Hadamard matrix product. Here, name S1 as FCSC matrix of (2) with partial in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi available, i∈1:μ.By referring to matrixS1 established in (13), the following criteria for FCSC are established.Theorem 1.
Consider system (2) with available sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ, then state xd=δ2νπ∈Δ2νπ∈1:2ν achieves FCSR from state x0=δ2νη∈Δ2νη∈1:2ν if and only if the following condition S1π,η=1 holds.Proof.
According to the definition of Hadamard product “∘,” one can easily obtain that S1π,η=1 if and only if for each i∈1:rs, Ciπ,η=1. For system xτ+1=Liϑτxτ, i∈1:rs, according to [31], one has that xd=δ2νπ is reachable from x0=δ2νη if and only if Ciπ,η=1. Thus, by Definition 2, xd=δ2νπ achieves FCSC from x0=δ2νη if and only if condition S1π,η=1 holds.
Then, FCSC for (2) with available partial in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈1:μ, is established.Theorem 2.
Given system (2) with available neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ, (2) achieves FCSC if and only if condition S1=12ν×2ν holds.Proof.
From Theorem1, for any initial state x0 and destination state xd∈Δ2ν, xd achieves FCSR from x0 if and only if S1=12ν×2ν. Thus, according to Definition 2, system (2) achieves FCSC.


### 3.2. Second Class of Structural Controllability

Based on Definition2, FCSR is introduced on reachability between the initial state x0 and destination state xd for any Boolean functions g1,…,gν. Hence, assume that state x0 can be steered to state xd under certain possible Boolean mappings gi,i∈1:ν, then the following second class of structural controllability is established.Definition 3.
Given system (2) and the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, for nodes x1,…,xμ, are available, the destination state xd∈Δ2ν achieves second class structural reachability (SCSR) from the initial state x0∈Δ2ν, if there is certain Boolean functions g1,…,gν satisfying that xd can be reachable from x0. In addition, (2) achieves SCSC if for any x0,xd∈Δ2ν, xd achieves second class structural controllability (SCSR) from x0.By referring to established matricesCi in (12), i∈1:rs, define(14)S2=∑i=1rsCBi,∈L2ν×2ν.We nameS2 as the SCSC matrix of (2) with available sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi, i∈1:μ. According to S2 defined in (14), the following necessary and sufficient criteria are derived.Theorem 3.
Consider system (2) with available neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ, then state xd=δ2νπ∈Δ2ν achieves SCSR from state x0=δ2νη∈Δ2ν if and only if condition S2π,η=1 holds.Proof.
S2π,η=1 if and only if there is a sequence of Boolean matrices Ck, k∈1:rs, such that Ckπ,η=1. For system xτ+1=Lkϑτxτ, according to [31], one has that xd=δ2νπ can be reachable from x0=δ2νη if and only if Ckπ,η=1. Thus, by Definition 3, xd=δ2νπ achieves SCSR from x0=δ2νη if and only if condition S2π,η=1 holds.Theorem 4.
Considering system (2), the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi, i∈1:μ, are available, then (2) achieves SCSC if and only if condition S2=12ν×2ν holds.Proof.
From Theorem3, for any two states x0,xd∈Δ2ν, state xd achieves SCSR from state x0 if and only if condition S2=12ν×2ν holds. Thus, according to Definition 3, system (2) achieves SCSC.
In the above, controllability of (2) with known in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ, under the situations of the arbitrariness and existence of logical functions g1,…,gν, has been established, which are shown in Theorems 1 and 3. In the following, controllability of (2) under the situation of arbitrary logical functions gi, i∈1:μ, and certain possible logical functions gj, j∈μ+1:ν, will be studied.


### 3.3. Third Class of Structural Controllability

In the following sequel, we consider the reachability between two given statesx0 and xd under any Boolean functions gi, i∈1:μ, and certain possible logical functions gi, i∈μ+1:ν.Definition 4.
Considering system (2) and the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, for nodes x1,…,xμ, are available, the destination state xd∈Δ2ν achieves third class structural reachability (TCSR) from the initial state x0∈Δ2ν, if for any Boolean functions gi,i∈1:μ, there are certain Boolean mappings gj,j∈μ+1:ν satisfying that xd can be reachable from x0. In addition, (2) achieves third class structural controllability (TCSC), if for any x0,xd∈Δ2ν, xd achieves TCSR from x0.For convenience, denote the set ofL in (11) by Ξ=F1∗F2,F1∈Λ1,F2∈Λ2≜L1,1,L1,2,…,L1,s,L2,1,L2,2,…,L2,s,…,Lr,1,Lr,2,…,Lr,s, where Li,j=F1i∗F2j, i∈1:r, j∈1:s. Divide Li,j into 2κ equal blocks as Li,j=Blk1Li,j,Blk2Li,j,…,Blk2κLi,j, and denote(15)Ci,j=∑l=12νMi,jBl,∈L2ν×2ν,i∈1:r,j∈1:s,where Mi,j=Li,j<imes12κ∈L2ν×2ν, and matrix Ci,j is the corresponding controllability matrix when state transition matrix of system (2) is Li,j. In addition, introduce the following Boolean matrix in order to issue TCSC,(16)S3=T1∘T2∘⋯∘Tr,∈L2ν×2ν,where Ti=∑j=1sCBi,j,∈L2ν×2ν, i∈1:r. We name S3 as TCSC matrix of (2) with available sets Θi,Θ¯i,Ξi, and Ξ¯i for x1,…,xμ.By referring to matrixS3 defined in (16), the following criteria for TCSC are established.Theorem 5.
Given system (2), for i∈1:μ, the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi are available, then state xd=δ2νπ∈Δ2ν achieves LCSR from state x0=δ2νη∈Δ2ν if and only if condition S3π,η=1 holds.Proof.
S3π,η=1 if and only if Tiπ,η=1,i∈1:r, and there is a sequence of Boolean matrices Ci,k,k∈1:s, such that Ci,kπ,η=1 for every Ti. For system xτ+1=Li,kϑτxτ, according to [31], one has that xd=δ2νπ is reachable from x0=δ2νη if and only if Ci,kπ,η=1. Thus, by Definition 4, state xd=δ2νπ achieves TCSR from state x0=δ2νη if and only if condition S3π,η=1 holds.Theorem 6.
Given system (2), the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi, i∈1:μ, are available, then (2) achieves TCSC if and only if condition S3=12ν×2ν holds.Proof.
From Theorem5, for any two states x0,xd∈Δ2ν, state xd achieves TCSR from state x0 if and only if condition S3=12ν×2ν holds. Thus, according to Definition 4, system (2) achieves TCSC.Remark 1.
More strictly, if forgj, j∈μ+1:ν, logical functions are the same, then for any functions gi,i∈1:μ, we define the TCSC matrix as S3=∑j=1sT˜Bj,∈L2ν×2ν, where T˜j=C1,j∘C2,j∘⋯∘Cr,j∈L2ν×2ν,j∈1:s.


### 3.4. Last Class of Structural Controllability

Based on Definition4, TCSR is introduced on reachability between the initial state x0 and destination state xd for any Boolean functions gi,i∈1:μ and certain Boolean functions gi, i∈μ+1:ν. Thus, if destination state xd is reachable from the initial state x0 under certain Boolean functions gi, i∈1:μ and any Boolean functions gi,i∈μ+1:ν, the following last class of structural controllability is established.Definition 5.
Considering system (2), for i∈1:μ, the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi are available. The destination state xd∈Δ2ν achieves last class structural reachability (LCSR) from initial state x0∈Δ2ν, if there is certain Boolean functions g1,…,gp satisfying that xd can be reachable from x0 for any Boolean functions gi,i∈μ+1:ν. Moreover, (2) achieves last class structural controllability (LCSC) if for any x0,xd∈Δ2ν, state xd achieves LCSR from state x0.By referring to matrixCi,j established in (15), i∈1:r,j∈1:s, define(17)S4=∑i=1rBT^i∈L2ν×2ν,where T^i=Ci,1∘Ci,2∘⋯∘Ci,s,i∈1:r. We call S4 as LCSC matrix of (2) with available sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ.By referring to matrixS4 defined in (17), the following criteria for LCSC are obtained.Theorem 7.
Given system (2), for i∈1:μ, the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi are available, then the destination state xd=δ2νπ∈Δ2ν achieves LCSR from the initial state x0=δ2νη∈Δ2ν if and only if condition S4π,η=1 holds.Proof.
S4π,η=1 if and only if there exists a sequence of matrices T^k,k∈1:r, such that T^kπ,η=1, and for every matrix Ck,j,j∈1:s, Ck,jπ,η=1. For system xτ+1=Lk,jϑτxτ, according to [31], one has that xd=δ2νπ is reachable from x0=δ2νη if and only if Ck,jπ,η=1. Thus, by Definition 5, xd=δ2νπ achieves LCSR from x0=δ2νη if and only if condition S4π,η=1 holds.Theorem 8.
Given system (2), for i∈1:μ, the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi are available, then (2) achieves LCSC if and only if condition S4=12ν×2ν holds.Proof.
From Theorem7, for any initial state x0 and destination state xd∈Δ2ν, state xd achieves LCSR from state x0 if and only if condition S4=12ν×2ν holds. Thus, according to Definition 5, system (2) achieves LCSC.Remark 2.
In this paper, four types of SC of BCNs with partial information available have been studied, where one only knows part of messages on nodes’ connections. Among the abovementioned four different kinds of structural controllability, the relationships are shown as follows: FCSC implies TCSC and LCSC; the third and the last class of SC both imply SCSC. In addition, FCSC is the most conservative definition and all other types of SC can be implied by FCSC, and the diagram describing the detailed relationships among different kinds of SC is shown as Figure1.Figure 1
The implication relationships among different types of SC (I, II, III, and IV represent abbreviations of FCSC, SCSC, TCSC, and LCSC).Remark 3.
By comparing with the recent theoretical results [39, 40], the established theoretical results in this paper are more general. If p=n, that is, one knows all the in-neighbors of (2), then the results in [40] can be directly obtained from FCSC in this paper. Moreover, if p=n, FCSC in this paper implies common controllability [39], which means that system (2) is always controllable no matter which functions are chosen. Thus, the study of these four cases of SC of BCNs with partial information available is a simple extension of common controllability, which provides a new insight for further studying and reducing the high computational complexity of BCNs.


## 3.1. First Class of Structural Controllability

At the beginning, only the corresponding in-neighbor setsΘi,Θ¯i,Ξi,andΞ¯i for x1,…,xμ, are supposed to be known, the corresponding Boolean functions gi, i∈1:μ, and both the in-neighbor sets Θj,Θ¯j,Ξj,andΞ¯j and the Boolean functions gj, j∈μ+1:ν, are unknown. Under these assumptions, reachability and controllability for (2) are firstly defined as follows.Firstly, consider the first class structural reachability (FCSR) between two given statesx0 and xd under any Boolean functions gi, i∈1:ν.Definition 2.
Consider system (2) and the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for x1,…,xμ, are available. The destination state xd∈Δ2ν achieves FCSR from the initial state x0∈Δ2ν, if x0 can be steered to xd for arbitrary Boolean functions g1,…,gν. In addition, (2) achieves first class structural controllability (FCSC) if for any initial state x0 and destination state xd, xd achieves FCSR from x0.By consideringΞ=L1,L2,…,Lrs, divide L1,…,Lrs, as follows: Li=Blk1Li,Blk2Li,…,Blk2κLi, and denote(12)Ci=∑l=12νMiBl,∈L2ν×2ν,i∈1:rs,where Mi=Li⋉12κ∈L2ν×2ν, and matrix Ci, i∈1:rs, is the controllability matrix when state transition matrix of system (2) is Li. Then, introduce the following Boolean matrix in order to issue FCSC,(13)S1=C1∘C2∘⋯∘Crs∈L2ν×2ν,where “∘” represents the well-known Hadamard matrix product. Here, name S1 as FCSC matrix of (2) with partial in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi available, i∈1:μ.By referring to matrixS1 established in (13), the following criteria for FCSC are established.Theorem 1.
Consider system (2) with available sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ, then state xd=δ2νπ∈Δ2νπ∈1:2ν achieves FCSR from state x0=δ2νη∈Δ2νη∈1:2ν if and only if the following condition S1π,η=1 holds.Proof.
According to the definition of Hadamard product “∘,” one can easily obtain that S1π,η=1 if and only if for each i∈1:rs, Ciπ,η=1. For system xτ+1=Liϑτxτ, i∈1:rs, according to [31], one has that xd=δ2νπ is reachable from x0=δ2νη if and only if Ciπ,η=1. Thus, by Definition 2, xd=δ2νπ achieves FCSC from x0=δ2νη if and only if condition S1π,η=1 holds.
Then, FCSC for (2) with available partial in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, i∈1:μ, is established.Theorem 2.
Given system (2) with available neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ, (2) achieves FCSC if and only if condition S1=12ν×2ν holds.Proof.
From Theorem1, for any initial state x0 and destination state xd∈Δ2ν, xd achieves FCSR from x0 if and only if S1=12ν×2ν. Thus, according to Definition 2, system (2) achieves FCSC.


## 3.2. Second Class of Structural Controllability

Based on Definition2, FCSR is introduced on reachability between the initial state x0 and destination state xd for any Boolean functions g1,…,gν. Hence, assume that state x0 can be steered to state xd under certain possible Boolean mappings gi,i∈1:ν, then the following second class of structural controllability is established.Definition 3.
Given system (2) and the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, for nodes x1,…,xμ, are available, the destination state xd∈Δ2ν achieves second class structural reachability (SCSR) from the initial state x0∈Δ2ν, if there is certain Boolean functions g1,…,gν satisfying that xd can be reachable from x0. In addition, (2) achieves SCSC if for any x0,xd∈Δ2ν, xd achieves second class structural controllability (SCSR) from x0.By referring to established matricesCi in (12), i∈1:rs, define(14)S2=∑i=1rsCBi,∈L2ν×2ν.We nameS2 as the SCSC matrix of (2) with available sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi, i∈1:μ. According to S2 defined in (14), the following necessary and sufficient criteria are derived.Theorem 3.
Consider system (2) with available neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ, then state xd=δ2νπ∈Δ2ν achieves SCSR from state x0=δ2νη∈Δ2ν if and only if condition S2π,η=1 holds.Proof.
S2π,η=1 if and only if there is a sequence of Boolean matrices Ck, k∈1:rs, such that Ckπ,η=1. For system xτ+1=Lkϑτxτ, according to [31], one has that xd=δ2νπ can be reachable from x0=δ2νη if and only if Ckπ,η=1. Thus, by Definition 3, xd=δ2νπ achieves SCSR from x0=δ2νη if and only if condition S2π,η=1 holds.Theorem 4.
Considering system (2), the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi, i∈1:μ, are available, then (2) achieves SCSC if and only if condition S2=12ν×2ν holds.Proof.
From Theorem3, for any two states x0,xd∈Δ2ν, state xd achieves SCSR from state x0 if and only if condition S2=12ν×2ν holds. Thus, according to Definition 3, system (2) achieves SCSC.
In the above, controllability of (2) with known in-neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ, under the situations of the arbitrariness and existence of logical functions g1,…,gν, has been established, which are shown in Theorems 1 and 3. In the following, controllability of (2) under the situation of arbitrary logical functions gi, i∈1:μ, and certain possible logical functions gj, j∈μ+1:ν, will be studied.


## 3.3. Third Class of Structural Controllability

In the following sequel, we consider the reachability between two given statesx0 and xd under any Boolean functions gi, i∈1:μ, and certain possible logical functions gi, i∈μ+1:ν.Definition 4.
Considering system (2) and the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i, for nodes x1,…,xμ, are available, the destination state xd∈Δ2ν achieves third class structural reachability (TCSR) from the initial state x0∈Δ2ν, if for any Boolean functions gi,i∈1:μ, there are certain Boolean mappings gj,j∈μ+1:ν satisfying that xd can be reachable from x0. In addition, (2) achieves third class structural controllability (TCSC), if for any x0,xd∈Δ2ν, xd achieves TCSR from x0.For convenience, denote the set ofL in (11) by Ξ=F1∗F2,F1∈Λ1,F2∈Λ2≜L1,1,L1,2,…,L1,s,L2,1,L2,2,…,L2,s,…,Lr,1,Lr,2,…,Lr,s, where Li,j=F1i∗F2j, i∈1:r, j∈1:s. Divide Li,j into 2κ equal blocks as Li,j=Blk1Li,j,Blk2Li,j,…,Blk2κLi,j, and denote(15)Ci,j=∑l=12νMi,jBl,∈L2ν×2ν,i∈1:r,j∈1:s,where Mi,j=Li,j<imes12κ∈L2ν×2ν, and matrix Ci,j is the corresponding controllability matrix when state transition matrix of system (2) is Li,j. In addition, introduce the following Boolean matrix in order to issue TCSC,(16)S3=T1∘T2∘⋯∘Tr,∈L2ν×2ν,where Ti=∑j=1sCBi,j,∈L2ν×2ν, i∈1:r. We name S3 as TCSC matrix of (2) with available sets Θi,Θ¯i,Ξi, and Ξ¯i for x1,…,xμ.By referring to matrixS3 defined in (16), the following criteria for TCSC are established.Theorem 5.
Given system (2), for i∈1:μ, the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi are available, then state xd=δ2νπ∈Δ2ν achieves LCSR from state x0=δ2νη∈Δ2ν if and only if condition S3π,η=1 holds.Proof.
S3π,η=1 if and only if Tiπ,η=1,i∈1:r, and there is a sequence of Boolean matrices Ci,k,k∈1:s, such that Ci,kπ,η=1 for every Ti. For system xτ+1=Li,kϑτxτ, according to [31], one has that xd=δ2νπ is reachable from x0=δ2νη if and only if Ci,kπ,η=1. Thus, by Definition 4, state xd=δ2νπ achieves TCSR from state x0=δ2νη if and only if condition S3π,η=1 holds.Theorem 6.
Given system (2), the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi, i∈1:μ, are available, then (2) achieves TCSC if and only if condition S3=12ν×2ν holds.Proof.
From Theorem5, for any two states x0,xd∈Δ2ν, state xd achieves TCSR from state x0 if and only if condition S3=12ν×2ν holds. Thus, according to Definition 4, system (2) achieves TCSC.Remark 1.
More strictly, if forgj, j∈μ+1:ν, logical functions are the same, then for any functions gi,i∈1:μ, we define the TCSC matrix as S3=∑j=1sT˜Bj,∈L2ν×2ν, where T˜j=C1,j∘C2,j∘⋯∘Cr,j∈L2ν×2ν,j∈1:s.


## 3.4. Last Class of Structural Controllability

Based on Definition4, TCSR is introduced on reachability between the initial state x0 and destination state xd for any Boolean functions gi,i∈1:μ and certain Boolean functions gi, i∈μ+1:ν. Thus, if destination state xd is reachable from the initial state x0 under certain Boolean functions gi, i∈1:μ and any Boolean functions gi,i∈μ+1:ν, the following last class of structural controllability is established.Definition 5.
Considering system (2), for i∈1:μ, the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi are available. The destination state xd∈Δ2ν achieves last class structural reachability (LCSR) from initial state x0∈Δ2ν, if there is certain Boolean functions g1,…,gp satisfying that xd can be reachable from x0 for any Boolean functions gi,i∈μ+1:ν. Moreover, (2) achieves last class structural controllability (LCSC) if for any x0,xd∈Δ2ν, state xd achieves LCSR from state x0.By referring to matrixCi,j established in (15), i∈1:r,j∈1:s, define(17)S4=∑i=1rBT^i∈L2ν×2ν,where T^i=Ci,1∘Ci,2∘⋯∘Ci,s,i∈1:r. We call S4 as LCSC matrix of (2) with available sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes x1,…,xμ.By referring to matrixS4 defined in (17), the following criteria for LCSC are obtained.Theorem 7.
Given system (2), for i∈1:μ, the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi are available, then the destination state xd=δ2νπ∈Δ2ν achieves LCSR from the initial state x0=δ2νη∈Δ2ν if and only if condition S4π,η=1 holds.Proof.
S4π,η=1 if and only if there exists a sequence of matrices T^k,k∈1:r, such that T^kπ,η=1, and for every matrix Ck,j,j∈1:s, Ck,jπ,η=1. For system xτ+1=Lk,jϑτxτ, according to [31], one has that xd=δ2νπ is reachable from x0=δ2νη if and only if Ck,jπ,η=1. Thus, by Definition 5, xd=δ2νπ achieves LCSR from x0=δ2νη if and only if condition S4π,η=1 holds.Theorem 8.
Given system (2), for i∈1:μ, the neighbor sets Θi,Θ¯i,Ξi, and Ξ¯i for nodes xi are available, then (2) achieves LCSC if and only if condition S4=12ν×2ν holds.Proof.
From Theorem7, for any initial state x0 and destination state xd∈Δ2ν, state xd achieves LCSR from state x0 if and only if condition S4=12ν×2ν holds. Thus, according to Definition 5, system (2) achieves LCSC.Remark 2.
In this paper, four types of SC of BCNs with partial information available have been studied, where one only knows part of messages on nodes’ connections. Among the abovementioned four different kinds of structural controllability, the relationships are shown as follows: FCSC implies TCSC and LCSC; the third and the last class of SC both imply SCSC. In addition, FCSC is the most conservative definition and all other types of SC can be implied by FCSC, and the diagram describing the detailed relationships among different kinds of SC is shown as Figure1.Figure 1
The implication relationships among different types of SC (I, II, III, and IV represent abbreviations of FCSC, SCSC, TCSC, and LCSC).Remark 3.
By comparing with the recent theoretical results [39, 40], the established theoretical results in this paper are more general. If p=n, that is, one knows all the in-neighbors of (2), then the results in [40] can be directly obtained from FCSC in this paper. Moreover, if p=n, FCSC in this paper implies common controllability [39], which means that system (2) is always controllable no matter which functions are chosen. Thus, the study of these four cases of SC of BCNs with partial information available is a simple extension of common controllability, which provides a new insight for further studying and reducing the high computational complexity of BCNs.


## 4. Simulations

Here, simulations on a Boolean model of the lactose operon inEscherichia coli are given to illustrate the obtained results.Example 1.
Firstly, a lactose operon model inEscherichia coli modeled by a BCN is considered [42]:(18)w1τ+1=ξ2τ∨w3τ∧ξ¯1τ,w2τ+1=w1τ,w3τ+1=ξ2τ∧w2τ∨w¯2τ∧w3τ∧ξ¯1τ,where w1∈D denotes the biological structure of mRNA, w2∈D denotes the biological structure of lacZ polypeptide, w3∈D denotes the intracellular lactose, ξ1∈D denotes the external glucose, and ξ2∈D denotes the external lactose.
Using STP tool, the corresponding algebraic form of system (18) is given as follows: wτ+1=Luτwτ, where L=δ86,6,6,6,8,8,8,8,6,6,6,6,8,8,8,8,1,1,1,2,3,3,3,4,2,6,1,6,4,8,3,8. Then, split matrix L into four blocks and obtain its controllability matrix C=∑l=18MBl∈B8×8, where M=L⋉14∈B8×8. After a straightforward calculation, one has that C=δ81+δ82+δ83+δ84+δ86+δ88,δ81+δ82+δ83+δ84+δ86+δ88,…,δ81+δ82+δ83+δ84+δ86+δ88∈B8×8. By referring to [31], one can conclude that system (18) is not controllable.
Suppose that only in-neighbors for nodesw1 and w2 are given, but the detailed logical functions are unknown. Thus, the following system is considered:(19)w1τ+1=f1w3τ,ξ2τ,ξ¯1τ,w2τ+1=f2w1τ,w3τ+1=f3wjτj∈Θ3,w¯jτj∈Θ¯3,ξjτj∈Ξ3,ξ¯τj∈Ξ¯3,where Θ3 and Ξ3Θ¯3andΞ¯3 express the unavailable sets of the activating indegrees (inhibiting indegrees) of node w3.
Based on equations (5)–(9), a simple equivalent algebraic illustration of (19) can be derived:(20)x1τ+1=F1ξτwτ,w2τ+1=F2ξτwτ,where w1=w1w2, w2=w3, ξ=ξ1ξ2, F1∈Λ1=F11,F12,F13,F14, matrix F2∈Λ2=F21,…,F2232, and the cardinality of Λ2 is 232. Here, F11=δ43,3,3,3,4,4,4,4,3,3,3,3,4,4,4,4,1,3,1,3,2,4,2,4,3,3,3,3,4,4,4,4, F12=δ43,3,3,3,4,4,4,4,3,3,3,3,4,4,4,4,1,1,1,1,2,2,2,2,1,3,1,3,2,4,2,4, F13=δ41,3,1,3,2,4,2,4,3,3,3,3,4,4,4,4,1,1,1,1,2,2,2,2,1,1,1,1,2,2,2,2, and F14=δ41,1,1,1,2,2,2,2,1,3,1,3,2,4,2,4,1,1,1,1,2,2,2,2,1,1,1,1,2,2,2,2. Thus, one can obtain its equivalent augmented system of (19): wτ+1=Lξτwτ, where matrix L∈Ω=F1∗F2,F1∈Λ1,F2∈Λ2=L1,L2,…,L234. Due to the high computational complexity of the set Ξ, the detailed information for matrices L1,L2,…,L234 is omitted.
A straightforward computation shows thatS1≠18×8,S2=18×8,S3=18×8,S4≠18×8. By referring to Theorems 1, 3, 5, and 7, system (19) achieves SCSC and TCSC, while it cannot achieve FCSC or LCSC.
WhenF2=δ22,2,2,2,2,1,2,2,1,1,1,2,1,2,1,1,2,1,1,2,1,2,2,2,2,2,1,1,1,2,2,2, if F1=F11, then L=δ86,6,6,6,8,7,8,8,5,5,5,6,7,8,7,7,2,5,1,6,3,8,4,8,6,6,5,5,7,8,8,8, C=18×8; if F1=F12, then L=δ86,6,6,6,8,7,8,8,5,5,5,6,7,8,7,7,2,1,1,2,3,4,4,4,2,6,1,5,3,8,4,8, C=18×8; if F1=F13, then L=δ82,6,2,6,4,7,4,8,5,5,5,6,7,8,7,7,2,1,1,2,3,4,4,4,2,2,1,1,3,4,4,4, C=18×8; if F1=F14, then L=δ82,2,2,2,4,3,4,4,1,5,1,6,3,8,3,7,2,1,1,2,3,4,4,4,2,2,1,1,3,4,4,4, C=18×8. Thus, S2=S3=18×8. By referring to Theorems 3 and 5, system (19) achieves SCSC and TCSC. This implies that, for any w0,wd∈Δ2ν, and any logical functions f1, f2, there is a logical functions f3 satisfying that state xd can be reached from initial state w0. According to Definitions 3 and 4, one can conclude that system (19) achieves SCSC and TCSC, respectively.
WhenF2=δ22,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,2,1,1,1,2,2,2,1,2,2,2,1,2, that is, w3τ+1=ξ¯1τ∧w2τ∧ξ2τ∨w3τ∧w¯2τ, if F1=F11, then L=δ86,6,6,6,8,8,8,8,6,6,6,6,8,8,8,8,1,5,1,6,3,7,3,8,6,6,5,6,8,8,7,8, C≠18×8; if F1=F12, then L=δ86,6,6,6,8,8,8,8,6,6,6,6,8,8,8,8,1,1,1,2,3,3,3,4,2,6,1,6,4,8,3,8, C≠18×8; if F1=F13, then L=δ82,6,2,6,4,8,4,8,6,6,6,6,8,8,8,8,1,1,1,2,3,3,3,4,2,2,1,2,4,4,3,4, C≠18×8; if F1=F14, then L=δ82,2,2,2,4,4,4,4,2,6,2,6,4,8,4,8,1,1,1,2,3,3,3,4,2,2,1,2,4,4,3,4, C≠18×8. Thus, S4≠18×8. According to Theorem 7, system (19) cannot achieve LCSC. In this case, there exists a logical function f3, such that system (19) is not controllable for any logical functions f1, f2. By Definition 5, system (19) cannot achieve LCSC.


## 5. Conclusion

In this paper, SC of BCNs has been systematically addressed, where only partial information of nodes’ connections is available. By referring to its equivalent algebraic representation of BCNs, four kinds of SC have been established, based on different issues of nodes’ connections. Then, certain criteria for different types of SC have been proposed, by defining its corresponding SC matrix. Finally, a Boolean model of the lactose operon inEscherichia coli has been simulated to show the effectiveness of the main theoretical results. One interesting research issue for future works includes SC of probabilistic BCNs with time-variant information of nodes’ connections.



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*Source: 9442358-2021-06-30.xml*