Open AccessArticle

One-Bit Function Perturbation Impact on Robust Set Stability of Boolean Networks with Disturbances

Lei Deng

^*,

Xiujun Cao

and

Jianli Zhao

Research Center of Semi-Tensor Product of Matrices: Theory and Applications, School of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China

Author to whom correspondence should be addressed.

Mathematics 2024, 12(14), 2258; https://doi.org/10.3390/math12142258

Submission received: 29 May 2024 / Revised: 17 July 2024 / Accepted: 17 July 2024 / Published: 19 July 2024

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Abstract

This paper investigates the one-bit function perturbation (OBFP) impact on the robust set stability of Boolean networks with disturbances (DBNs). Firstly, the dynamics of these networks are converted into the algebraic forms utilizing the semi-tensor product (STP) method. Secondly, OBFP’s impact on the robust set stability of DBNs is divided into two situations. Then, by constructing a state set and defining an index vector, several necessary and sufficient conditions to guarantee that a DBN under OBFP can stay robust set stable unchanged are provided. Finally, a biological example is proposed to demonstrate the effectiveness of the obtained theoretical results.

Keywords:

boolean networks; disturbances; one-bit function perturbation; robust set stability; semi-tensor product of matrices

MSC:

93D09

1. Introduction

A Boolean network is a typical binary logical dynamic system, which was first introduced by Kauffman for modeling and analyzing gene regulatory networks (GRNs) [1]. Now, it has attracted extensive attention and has become an effective model of many other complex networks, including wireless sensor networks, neural networks and networked evolutionary games, etc. [2,3,4]. With the development of practical research problems, BNs have been generalized in many aspects [5,6]. It turns out that when modeling BNs, the undesired signals (or, equivalently, the disturbances) are unavoidable due to internal or external environmental perturbations. For instance, in gene regulatory networks, the disturbances may be gene mutation, the duplication or deletion of fragments in genetic recombination, and external environmental stimuli, etc. [7]. Based on the

H_{\infty}

complimentary sensitivity function, the design method of a decentralized control scheme was provided in reference [8], which can improve the performance of the system in terms of disturbance rejection and tracking. Considering the model subject to disturbance, the concept of BNs with disturbances (DBNs) was presented. In fact, DBNs serve as models of complex networks, which describe complex biological phenomena and have been widely investigated [9,10,11].

In classical control theory, there is a fundamental and meaningful issue, that is, the robust set stability of system [12]. The robust set stability of system means that all the initial states can eventually converge to an attractor subset of a given set under arbitrary disturbance sequence. Recently, the semi-tensor product (STP) method has been introduced, which establishes an algebraic state space representation to analyze the robust set stability of DBNs [13]. Specifically, Zhong et al. presented a robust reachable sets method for investigating the robust set stability of DBNs [14]. The robust set stability of probabilistic DBNs was studied in [15], and some efficient criteria for robust set stability were proposed based on the largest robust invariant set. Moreover, STP was also used to study various control problems of DBNs, such as stabilization [16], controllability [17], detectability [18], synchronization [19] and output tracking [20], etc. In addition to DBNs, the STP method has been also successfully applied to many other logical networks, including switched BNs [21], mix-valued logical networks [22], game theory [23], etc.

It is noted that gene mutation is universal in GRNs. These mutations have a significant impact on organisms. For instance, sickle cell anemia is caused by gene mutation in hemoglobin [24]. When modeling GRNs with BNs, function perturbation is introduced to describe gene mutation [25]. Recently, the effect of function perturbation on the dynamics of logical networks has been widely investigated. Under STP framework, reference [16] studied OBFP impact on the stability of BNs, while references [5,26] first migrated this issue to probabilistic BNs and switched BNs, respectively. Additionally, the effect of OBFP on the robust stability of DBNs was discussed in [27]. The effect of function perturbation on other fundamental problems of logical networks, such as optimal control [28], observability [29], detectability [30], has been investigated. It should be pointed out that function perturbation may change the attractors of the original DBNs. In fact, the robust set stability of DBNs is closely related to the attractors of DBNs, which may also be influenced due to function perturbation. There is a question that comes up naturally: how will the OBFP influence the robust set stability of DBNs? From the perspective of network models, most existing works do not consider disturbances. When the system contains disturbances, the evolution trajectory of each state is not unique, which will make the considered problem more complicated. In terms of the research issue, many references studied the stability problem of the system under OBFP. If the target set is a set rather than a fixed point, the robust set stability must consider the relationship between the trajectories of perturbed state and the target set. However, robust stability does not need to consider this aspect. To our best knowledge, there exist few results about the robust set stability for DBNs subject to function perturbation at present.

In this paper, we aim to propose some criteria to detect the robust set stability for DBNs affected by OBFP. The main contributions are as follows. (i) The concept of OBFP in DBNs is first proposed. Based on the possible relationship between the perturbed state and the largest invariant set, the robustness analyses for the set stability of DBNs are divided into two situations. For the case in which the perturbed state does not belong to the largest invariant set, a necessary and sufficient condition that DBNs stay robust set stable unchanged is derived. The condition to verify whether the DBN is still robust set stable is established, when the perturbed state belongs to the largest invariant set. (ii) Our results can be seen as an extension of reference [27]. When the given set becomes a single point, our results will degenerate into the results of robust stability of DBNs. However, the methods proposed in this article are easier to understand and detect than previous techniques.

The rest of this paper is organized as follows. Section 2 shows some notations and definitions about semi-tensor product, and presents the algebraic state space representation of a BN with disturbances. Section 3 presents the main results. Section 4 provides a biological example to describe the validity of the proposed method, and Section 5 is a brief conclusion.

2. Preliminaries

This section gives some necessary symbols and definitions, as well as algebraic representations of BNs with disturbances.

2.1. Semi-Tensor Product of Matrices

We first give some notations in Table 1.

Some other notations can be explained as follows. If

Col (A) \subset Δ_{m}

, then

A \in M_{m \times n}

is called a logical matrix. If

M = [δ_{n}^{i_{1}}, δ_{n}^{i_{2}}, \dots, δ_{n}^{i_{m}}]

, then briefly denote it as

M = δ_{n} [i_{1} i_{2} \dots i_{m}]

. A matrix

A \in M_{m \times n}

is called a Boolean matrix, if all its entries are either 0 or 1. Assume

A = (a_{i j}), B = (b_{i j}) \in B_{m \times n}

, then

A +_{B} B = (a_{i j} \lor b_{i j})

, where “∨” represents the logical operator “or”.

(B) \sum_{i = 1}^{n} L_{i} : = L_{1} +_{B} L_{2} +_{B} \dots +_{B} L_{n}

A^{(k)} = A^{(k - 1)} \times_{B} A

, where k is a positive integer.

Definition 1

([13]). The semi-tensor product of two matrices

A \in M_{m \times n}

and

B \in M_{p \times t}

is defined as

A ⋉ B = (A \otimes I_{\frac{α}{n}}) (B \otimes I_{\frac{α}{p}}),

where

α = l c m (n, p)

is the least common multiple of n and p, and ⊗ is the Kronecker product.

Semi-tensor product is a generalization of ordinary matrix product, and it retains almost all the basic properties of ordinary matrix product. Throughout this paper, we omit the symbol “

⋉

”.

Next, we give an example to illustrate how to calculate the semi-tensor product of two matrices by Definition 1.

Example 1.

Let

A = [\begin{matrix} 1 & 2 & 3 & 1 \\ 2 & 3 & 1 & 2 \\ 3 & 2 & 1 & 0 \end{matrix}] and B = [\begin{matrix} 1 & - 2 \\ 2 & - 1 \end{matrix}],

then

A ⋉ B = A (B \otimes I_{2}) = [\begin{matrix} 3 & 4 & - 3 & - 5 \\ 4 & 7 & - 5 & - 8 \\ 5 & 2 & - 7 & - 4 \end{matrix}] .

For a logical variable

x \in D

, we define its vector form as

{(x, 1 - x)}^{T}

, then there is an equivalence relationship between

D

and Δ. It is easy to see that if

x_{i}

is the vector form of logical variable

X_{i}

, then there is a one-to-one correspondence between

X = {(X_{1}, X_{2}, \dots, X_{n})}^{T} \in D^{n}

and

x = ⋉_{i = 1}^{n} x_{i} \in Δ_{2^{n}}

. We call x the vector form of X.

Lemma 1

([13]). Consider a logical mapping

f : D^{n} \to D

. There exists a unique matrix

M_{f} \in L_{2 \times 2^{n}}

, called the structure matrix of f such that

\begin{matrix} f (X_{1}, X_{2}, \dots, X_{n}) = M_{f} ⋉_{i = 1}^{n} x_{i}, \end{matrix}

where

x_{i} \in Δ

is the vector form of

X_{i} \in D

i = 1, 2, \dots, n

2.2. Algebraic Representations of BNs with Disturbances

A BN with finite disturbance inputs can be described as

\begin{matrix} X_{i} (t + 1) = f_{i} (X (t), ξ (t)), i = 1, 2, \dots, n, \end{matrix}

(1)

where

X (t) = (X_{1} (t), X_{2} (t), \dots, X_{n} (t)) \in D^{n}

ξ (t) = (ξ_{1} (t), ξ_{2} (t), \dots,

\dots, ξ_{q} (t)) \in D^{q}

are the state and disturbance, respectively,

f_{i} : D^{n + q} \to D

are Boolean functions.

Since the number of variables in (1) is finite, and each variable just takes values from a finite set

D

, we can see that (1) is a standard finite-valued discrete dynamical system.

Denote the vector forms of

X_{i}

and

ξ_{i}

x_{i}

and

ε_{i}

, respectively, that is,

x_{i} = {(X_{i}, 1 - X_{i})}^{T}

ε_{i} = {(ξ_{i}, 1 - ξ_{i})}^{T}

. By Lemma 1, system (1) can be expressed as

\begin{matrix} x_{i} (t + 1) = M_{f_{i}} ε (t) x (t), i = 1, 2, \dots, n, \end{matrix}

(2)

where

x (t) = ⋉_{i = 1}^{n} x_{i} \in Δ_{2^{n}}

ε (t) = ⋉_{i = 1}^{q} ε_{i} \in Δ_{2^{q}}

, and

M_{f_{i}} \in L_{2 \times 2^{q + n}}

Multiplying the n equations in (2) together yields

\begin{matrix} x (t + 1) = L ε (t) x (t), \end{matrix}

(3)

where

L = M_{f_{1}} * M_{f_{2}} * \dots * M_{f_{n}} \in L_{2^{n} \times 2^{q + n}}

, and ∗ is the Khatri–Rao product of matrices. System (3) is called the algebraic form of BN (1), and the matrix

L : = [L_{1} L_{2} \dots L_{2^{q}}] \in L_{2^{n} \times 2^{q + n}}

is called the state transition matrix of system (3).

For initial state

x (0) \in Δ_{2^{n}}

, under disturbance sequence

ε : = {ε (t), t \in [0, τ]} \subseteq Δ_{2^{q}}

, the state of system (3) at time

τ + 1

is indicated as

x (τ + 1; x (0), ε)

. According to the algebraic form (3), the concept for robust set stability of DBNs is reviewed below [14].

Definition 2

([14]). System (3) is said to be robustly stable to the nonempty set

W \subseteq Δ_{2^{n}}

, if for any initial state

x (0) \in Δ_{2^{n}}

, there exists

τ \in Z_{+}

such that

x (t; x (0), ε) \in W

for any

t ⩾ τ

and arbitrary disturbance sequence ε.

In the following, a numerical example is provided to check whether the system is robustly stable to set

W

via Definition 2.

Example 2.

Consider a DBN as follows:

\begin{matrix} x (t + 1) = L ε (t) x (t), \end{matrix}

(4)

where the state transition matrix of system (4) is

L = [L_{1} L_{2}]

L_{1} = δ_{4} [2 1 1 3]

and

L_{2} = δ_{4} [1 1 2 2]

. In the following, we check whether system (4) will robustly stable to set

W = {δ_{4}^{1}, δ_{4}^{2}}

. For

x (0) = δ_{4}^{1}

, through simple calculations, it holds that

\begin{matrix} \begin{matrix} x (1; x (0) = δ_{4}^{1}, ε (0) = δ_{2}^{1}) = L ε (0) x (0) = L δ_{2}^{1} δ_{4}^{1} = L_{1} δ_{4}^{1} = δ_{4}^{2}, \\ x (1; x (0) = δ_{4}^{1}, ε (0) = δ_{2}^{2}) = L ε (0) x (0) = L δ_{2}^{2} δ_{4}^{1} = L_{2} δ_{4}^{1} = δ_{4}^{1}, \end{matrix} \end{matrix}

(5)

which means that state

δ_{4}^{1}

can reach set

W

in one step under arbitrary disturbance input ε. Similarly, for

x (0) = δ_{4}^{2}

, we have

\begin{matrix} \begin{matrix} x (1; x (0) = δ_{4}^{2}, ε (0) = δ_{2}^{1}) = L ε (0) x (0) = L δ_{2}^{1} δ_{4}^{2} = L_{1} δ_{4}^{2} = δ_{4}^{1}, \\ x (1; x (0) = δ_{4}^{2}, ε (0) = δ_{2}^{2}) = L ε (0) x (0) = L δ_{2}^{2} δ_{4}^{2} = L_{2} δ_{4}^{2} = δ_{4}^{1} . \end{matrix} \end{matrix}

(6)

Combining with Equations (5) and (6), we can obtain that the states in set

W

will always remain in set

W

under arbitrary disturbance sequence ε. Thus,

W

is a robust invariant set. For

x (0) = δ_{4}^{3}

, skipping some redundant calculations, it can be obtained that

x (1; x (0) = δ_{4}^{3}, ε) \in W

under arbitrary disturbance input ε.

Next, we will judge whether

x (0) = δ_{4}^{4}

can be robustly stabilized to the set

W

. By simple calculations, we obtain

x (1; x (0) = δ_{4}^{4}, ε (0) = δ_{2}^{1}) = δ_{4}^{3}

and

x (1; x (0) = δ_{4}^{4},

ε (0) = δ_{2}^{2}) = δ_{4}^{2}

, which means that state

δ_{4}^{4}

can reach set

{δ_{4}^{2}, δ_{4}^{3}}

in one step. According to the above discussion, we obtain that both state

δ_{4}^{2}

and state

δ_{4}^{3}

can robustly reach set

W

in the next step, which means that state

δ_{4}^{4}

can reach set

W

at the second step under arbitrary disturbance sequence

ε

. It follows that the trajectory of the system (4) starting from any initial state

x (0) \in Δ_{4}

will robustly reach set

W

and then stay there forever under arbitrary disturbance sequence

ε

. Based on Definition 2, system (4) is robustly stable to set

W

As is well known, the largest robust invariant subset of a given set plays an important role in robust set stability. The following criterion is reviewed for the robust set stability of DBNs.

Lemma 2

([22]). System (3) is robustly stable to

W

if and only if it can be robustly stable to the largest invariant subset of

W

There are many methods to calculate the largest robust invariant subset; thus, we will not elaborate on them further. For more details, please refer to [10,22]. This paper assumes that the largest robust invariant subset contained in

W

I (W)

When using DBNs to model gene regulatory networks, gene mutation can be regarded as the function perturbation. The definition of one-bit function perturbation is proposed as below.

Definition 3.

The one-bit function perturbation on system (3) is that only one column in the state transition matrix L is changed.

In order to study the robust set stability of DBNs subject to OBFP, we give a natural assumption as follows.

Assumption 1.

Before OBFP occurs, system (3) is robustly stable to set

W

3. Main Results

In this section, we devote to discussing the impact of OBFP on the robust set stability of DBNs. First, we analyze the change in state transition matrix when the OBFP occur.

Given

l \in {1, \dots, 2^{n + q}}

. Assume that after OBFP occurs,

{Col}_{l} (L)

is perturbed from

δ_{2^{n}}^{γ}

δ_{2^{n}}^{γ^{*}}

, where

γ \neq γ^{*}

. The state

δ_{2^{n}}^{γ^{*}}

is called the perturbed state. Obviously, the state transition matrix L of (3) is changed into a new matrix

\hat{L} \in L_{2^{n} \times 2^{n + q}}

, where

\begin{matrix} {Col}_{i} (\hat{L}) = \begin{matrix} \{\begin{matrix} δ_{2^{n}}^{γ^{*}}, if i = l; \\ {Col}_{i} (L), otherwise . \end{matrix} \end{matrix} \end{matrix}

(7)

Hence, system (3) under OBFP becomes the following form:

x (t + 1) = \hat{L} ε (t) x (t) .

(8)

Similarly, for initial state

x (0) \in Δ_{2^{n}}

, under disturbance sequence

ε : = {ε (t), t \in [0, τ]} \subseteq Δ_{2^{q}}

, the state of system (8) at time

τ + 1

is indicated as

\hat{x} (τ + 1; x (0), ε)

Lemma 3

([13]). For any integer

1 \leq i \leq 2^{n + q}

, there exist unique positive integers

i_{1} \in [1, 2^{q}]

and

i_{2} \in [1, 2^{n}]

such that

\begin{matrix} δ_{2^{n + q}}^{i} = δ_{2^{q}}^{i_{1}} ⋉ δ_{2^{n}}^{i_{2}}, \end{matrix}

(9)

where

i = (i_{1} - 1) 2^{n} + i_{2}

Set

l = (k^{*} - 1) 2^{n} + φ^{*}

. It follows from Definition 3 that OBFP only affects state

δ_{2^{n}}^{φ^{*}}

when

ε = δ_{2^{n}}^{k^{*}}

. Therefore, for any

x = δ_{2^{n}}^{φ}

and any

ε = δ_{2^{q}}^{k}

, if

φ \neq φ^{*}

, it follows that

\begin{matrix} L ε x = δ_{2^{n}}^{α_{φ}^{k}} = \hat{L} ε x . \end{matrix}

(10)

φ = φ^{*}

and

k = k^{*}

, one has

\begin{matrix} L ε x = δ_{2^{n}}^{γ} \neq δ_{2^{n}}^{γ^{*}} = \hat{L} ε x . \end{matrix}

(11)

The possible relationship between the perturbed state

δ_{2^{n}}^{γ^{*}}

and the largest robust invariant set

I (W)

can be divided into two cases: Case 1:

δ_{2^{n}}^{γ^{*}} \notin I (W)

; Case 2:

δ_{2^{n}}^{γ^{*}} \in I (W)

. Next, we analyze how OBFP affects the robust set stability of system (3) based on the above two cases.

Firstly, let

P = (B) \sum_{k = 1}^{2^{q}} L_{k}

and construct a state set:

Ω_{φ^{*}} = {δ_{2^{n}}^{θ} : {[Q]}_{φ^{*}, θ} > 0} \cup {δ_{2^{n}}^{φ^{*}}},

(12)

where

Q = (B) \sum_{i = 1}^{2^{n}} P^{(i)}

. Thus the set

Ω_{φ^{*}}

contains all the states that can reach state

δ_{2^{n}}^{φ^{*}}

under some disturbance sequence before OBFP, and includes

δ_{2^{n}}^{φ^{*}}

itself. Then, construct a matrix

\hat{Q} = (B) \sum_{i = 2^{n}}^{2^{n + 1}} {\hat{P}}^{(i)}

, where

\hat{P} = (B) \sum_{k = 1}^{2^{q}} {\hat{L}}_{k}

Denote the index vector of a given set

W

J_{W} \in B_{2^{n} \times 1}

, where

\begin{matrix} {(J_{W})}_{i} = \begin{matrix} \{\begin{matrix} 1, if δ_{2^{n}}^{i} \notin W, \\ 0, if δ_{2^{n}}^{i} \notin W, \end{matrix} \end{matrix} \end{matrix}

and

{(J_{W})}_{i}

is the i-th element of

J_{W}

In the light of the set

Ω_{φ^{*}}

, matrix

\hat{Q}

and index vector

J_{W}

, we provide several criteria to detect whether a BN with arbitrary disturbance is still robustly stable to the set

W

after OBFP.

Theorem 1.

Suppose Assumption 1 holds and

δ_{2^{n}}^{γ^{*}} \notin I (W)

. System (3) under OBFP is still robustly stable to the set

W

, if and only if one of the following two conditions holds

(i): $δ_{2^{n}}^{γ^{*}} \notin Ω_{φ^{*}}$ ,
(ii): $δ_{2^{n}}^{γ^{*}} \in Ω_{φ^{*}}$ , $J_{Δ_{2^{n}} ∖ W}^{T} {Col}_{φ^{*}} (\hat{Q}) = 0$ .

Proof.

(Necessity) We prove the necessity by contradiction. Assume that

δ_{2^{n}}^{γ^{*}} \in Ω_{φ^{*}}

and

J_{Δ_{2^{n}} ∖ W}^{T} {Col}_{φ^{*}} (\hat{Q}) \neq 0

. It follows from Assumption 1 that for any initial state

x (0) \in Δ_{2^{n}}

, it can robustly reach set

I (W)

in finite steps before OBFP occurs. This together with

δ_{2^{n}}^{γ^{*}} \in Ω_{φ^{*}}

means that there exists at least one path from

δ_{2^{n}}^{γ^{*}}

I (W)

include

δ_{2^{n}}^{φ^{*}}

before OBFP. Without loss of generality, we suppose that

δ_{2^{n}}^{γ^{*}}

can be steered to

δ_{2^{n}}^{φ^{*}}

at the

s_{1}

th step and

δ_{2^{n}}^{γ^{*}}

can be steered to

I (W)

at the sth step, where

s_{1} < s

. Then, the path from

δ_{2^{n}}^{γ^{*}}

I (W)

can be described as

δ_{2^{n}}^{γ^{*}} \overset{ε (0)}{\to} \dots \overset{ε (s_{1} - 1)}{\to} δ_{2^{n}}^{φ^{*}} \to \dots \overset{ε (s - 1)}{\to} I (W),

(13)

where the corresponding disturbance sequence is

δ_{2^{q}}^{k_{0}}, \dots, δ_{2^{q}}^{k_{s_{1} - 1}}, \dots, δ_{2^{q}}^{k_{s - 1}}

Based on Equation (11), one has

δ_{2^{n}}^{γ^{*}} = \hat{L} δ_{2^{q}}^{k^{*}} δ_{2^{n}}^{φ^{*}}

, that is,

δ_{2^{n}}^{φ^{*}}

can reach

δ_{2^{n}}^{γ^{*}}

in one step under disturbance input

ε = δ_{2^{q}}^{k^{*}}

after OBFP. By selecting disturbance sequence

ε_{1} = : {δ_{2^{q}}^{k^{*}}, δ_{2^{q}}^{k_{0}}, \dots, δ_{2^{q}}^{k_{s_{1} - 1}}}

, we can obtain the following path

δ_{2^{n}}^{φ^{*}} \to δ_{2^{n}}^{γ^{*}} \to \dots \to δ_{2^{n}}^{φ^{*}} .

(14)

There forms a new cycle (14) for system (3) after OBFP.

Since

J_{Δ_{2^{n}} ∖ W}^{T} {Col}_{φ^{*}} (\hat{Q}) \neq 0

, we know that there is a positive integer

τ \geq 2^{n}

satisfying

x (τ; δ_{2^{n}}^{φ^{*}}, ε) \notin W

in the above cycle (14). That is, the state

δ_{2^{n}}^{φ^{*}}

could not be stabilized set

W

in finite time after OBFP, which is a contradiction to the fact that system (3) is still robustly stable to set

W

(Sufficiency) Suppose condition (i) holds firstly. By following Assumpution 1, for any initial state

x (0) \in Δ_{2^{n}}

, it can robustly reach set

I (W)

in finite steps before OBFP occurs. Hence, for initial state

x (0) = δ_{2^{n}}^{θ}

, the paths from

δ_{2^{n}}^{θ}

to set

I (W)

have the following two situations.

(i): $δ_{2^{n}}^{θ} \notin Ω_{φ^{*}}$ , which implies that set $I (W)$ is reachable from $δ_{2^{n}}^{θ}$ and there exists no path from $δ_{2^{n}}^{θ}$ to set $I (W)$ containing $δ_{2^{n}}^{φ^{*}}$ , simultaneously.
(ii): $δ_{2^{n}}^{θ} \in Ω_{φ^{*}}$ , which implies that set $I (W)$ is reachable from $δ_{2^{n}}^{θ}$ and there exists at least one path from $δ_{2^{n}}^{θ}$ to set $I (W)$ containing $δ_{2^{n}}^{φ^{*}}$ , simultaneously.

First, we discuss

δ_{2^{n}}^{θ} \notin Ω_{φ^{*}}

. Without loss of generality, we suppose that

I (W)

is reachable from

δ_{2^{n}}^{θ}

at the

λ

th step. The path from

δ_{2^{n}}^{θ}

to set

I (W)

is described as

\begin{matrix} δ_{2^{n}}^{θ} \to \dots \to x (t) \to \dots \to I (W), \end{matrix}

(15)

where

ε : = {ε (t) = δ_{2^{q}}^{k_{t}}, t \in [0, λ - 1]} \subseteq Δ_{2^{q}}

, and the

{x (1), \dots, x (λ - 1)}

is a sequence of states in the path from

δ_{2^{n}}^{θ}

I (W)

. Apparently,

x (t) \neq δ_{2^{n}}^{φ^{*}}, t \in [1, λ - 1]

After OBFP, it follows from (10) that

\begin{matrix} \hat{x} (λ; δ_{2^{n}}^{θ}, ε) & = \hat{L} ε (λ - 1) x (λ - 1) = \hat{L} ε (λ - 1) \hat{L} ε (λ - 2) x (λ - 2) \\ = \dots \\ = ⋉_{t = λ - 1}^{0} (\hat{L} ε (t)) δ_{2^{n}}^{θ} = L ε (λ - 1) x (λ - 1) \\ = L ε (λ - 1) L ε (λ - 2) x (λ - 2) \\ = \dots \\ = ⋉_{t = λ - 1}^{0} (L ε (t)) δ_{2^{n}}^{θ} = x (λ; δ_{2^{n}}^{θ}, ε) \\ \in I (W) . \end{matrix}

Thus, OBFP has no effect on the path (15). Due to the arbitrariness of path (15), it holds

I (W)

is robustly reachable from any state

δ_{2^{n}}^{θ} \in Δ_{2^{n}}

In the following, we consider

δ_{2^{n}}^{θ} \in Ω_{φ^{*}}

. Select an arbitrary path from

δ_{2^{n}}^{θ}

I (W)

\begin{matrix} δ_{2^{n}}^{θ} & \to \dots \to x (t_{1}) \to \dots \to δ_{2^{n}}^{φ^{*}} \to δ_{2^{n}}^{η} \to \\ \dots \to x (t_{2}) \to \dots \to I (W), \end{matrix}

(16)

where

ε : = {ε (t) = δ_{2^{q}}^{i_{t}} : t \in [0, τ_{1} + τ_{2} - 1]} \subseteq Δ_{2^{q}}

τ_{1} + τ_{2}

indicates the number of steps from

δ_{2^{n}}^{θ}

I (W)

. Here, the corresponding state sequence in the path from

δ_{2^{n}}^{θ}

δ_{2^{n}}^{η}

{x (t_{1}) = δ_{2^{n}}^{i_{t_{1}}} : t_{1} \in [1, τ_{1} - 1]}

, and the corresponding state sequence from

δ_{2^{n}}^{η}

I (W)

{x (t_{2}) = δ_{2^{n}}^{i_{t_{2}}} : t_{2} \in [τ_{1} + 1, τ_{1} + τ_{2} - 1]}

. Clearly,

x (τ_{1} - 1) = δ_{2^{n}}^{φ^{*}}, x (τ_{1}) = δ_{2^{n}}^{η}

and

x (t) \neq δ_{2^{n}}^{φ^{*}}, t \in {1, . . ., τ_{1} + τ_{2} - 1} ∖ {τ_{1} - 1}

In path (16), the relationship between the state

δ_{2^{n}}^{η}

and perturbed state

δ_{2^{n}}^{γ^{*}}

can be divided into two situations: (i)

δ_{2^{n}}^{η} \neq δ_{2^{n}}^{γ^{*}}

; (ii)

δ_{2^{n}}^{η} = δ_{2^{n}}^{γ^{*}}

. For

δ_{2^{n}}^{η} \neq δ_{2^{n}}^{γ^{*}}

, similar to the analysis of path (15), OBFP has no effect on the path (16), and every state

δ_{2^{n}}^{θ} \in Δ_{2^{n}}

can still robustly reach to

I (W)

after OBFP. For

δ_{2^{n}}^{η} = δ_{2^{n}}^{γ^{*}}

, when OBFP occurs, it holds that

\begin{matrix} \{\begin{matrix} \hat{x} (τ_{1} - 1; δ_{2^{n}}^{θ}, ε) & = ⋉_{t = τ_{1} - 2}^{0} (\hat{L} δ_{2^{q}}^{i_{t}}) δ_{2^{n}}^{θ} \\ = ⋉_{t = τ_{1} - 2}^{0} (L δ_{2^{q}}^{i_{t}}) δ_{2^{n}}^{θ} = δ_{2^{n}}^{φ^{*}}, \\ \hat{x} (1; δ_{2^{n}}^{φ^{*}}, ε) & = \hat{L} δ_{2^{q}}^{k^{*}} δ_{2^{n}}^{φ^{*}} = δ_{2^{n}}^{γ^{*}} . \end{matrix} \end{matrix}

(17)

Due to

δ_{2^{n}}^{γ^{*}} \notin Ω_{φ^{*}}

, we know that

δ_{2^{n}}^{φ^{*}}

is not reachable from

δ_{2^{n}}^{γ^{*}}

. Combined with Assumption 1, we derive that

\begin{matrix} \hat{x} (τ_{3} - 1; δ_{2^{n}}^{γ^{*}}, ε) & = ⋉_{t = τ_{3} - 1}^{0} (\hat{L} δ_{2^{q}}^{j_{t}}) δ_{2^{n}}^{γ^{*}} \\ = ⋉_{t = τ_{3} - 1}^{0} (L δ_{2^{q}}^{j_{t}}) δ_{2^{n}}^{γ^{*}} \\ \in I (W), \end{matrix}

(18)

where

ε : = {ε (t) = δ_{2^{q}}^{j_{t}} : t \in [0, τ_{3} - 1]} \subseteq Δ_{2^{q}}

and

τ_{3}

indicates the number of steps from

δ_{2^{n}}^{γ^{*}}

I_{s} (W)

. Combining with (17) and (18), it can be concluded that

\begin{matrix} \hat{x} (τ_{1} + τ_{3}; δ_{2^{n}}^{θ}, ε) \\ = ⋉_{t = τ_{3} - 1}^{0} (\hat{L} δ_{2^{q}}^{j_{t}}) \hat{L} δ_{2^{q}}^{k^{*}} ⋉_{t = τ_{1} - 2}^{0} (\hat{L} δ_{2^{q}}^{i_{t}}) δ_{2^{n}}^{θ} \\ = ⋉_{t = τ_{1} + τ_{3} - 1}^{0} (\hat{L} ε (t)) δ_{2^{n}}^{θ} \\ \in I (W), \end{matrix}

which shows that path (16) can be rewritten as

\begin{matrix} δ_{2^{n}}^{θ} & \to & \dots \to x (t_{1}) \to \dots \\ \to & δ_{2^{n}}^{φ^{*}} \to ∖ δ_{2^{n}}^{γ} \to \dots \to x (t_{2}) \to \dots \to I (W) \\ ↓ \\ δ_{2^{n}}^{γ^{*}} \to \dots \to x (t_{3}) \to \dots \to I (W), \end{matrix}

(19)

where

ε : = {ε (t) = δ_{2^{q}}^{i_{t}} : t \in [0, τ_{1} - 2]} \cup {ε (t) = δ_{2^{q}}^{k^{*}} : t = τ_{1} - 1} \cup {ε (t) = δ_{2^{q}}^{j_{t - τ_{1}}} : t \in [τ_{1}, τ_{1} + τ_{3} - 1]}

τ_{1} + τ_{3}

denotes the number of steps from

δ_{2^{n}}^{θ}

to set

I (W)

. Here,

{x (t_{3}) : t_{3} \in [τ_{1} + 1, τ_{1} + τ_{3} - 1]}

is a sequence of states in the path from

δ_{2^{n}}^{γ^{*}}

to set

I (W)

. Due to the arbitrariness of path (19), we obtain that

δ_{2^{n}}^{θ}

can robustly reach to

I (W)

δ_{2^{n}}^{η} = δ_{2^{n}}^{γ^{*}}

. On the basis of the above analysis, for

δ_{2^{n}}^{θ} \in Ω_{φ^{*}}

, it holds that set

I (W)

is still robustly reachable from every state

δ_{2^{n}}^{θ} \in Δ_{2^{n}}

after OBFP occurs.

To sum up, if condition (i) holds, system (3) is still robustly stable to the set

I (W)

Next, we suppose that condition (ii) holds. For any state

δ_{2^{n}}^{θ} \in Δ_{2^{n}}

, we just discuss the situation:

δ_{2^{n}}^{θ} \in Ω_{φ^{*}}

and

δ_{2^{n}}^{η} = δ_{2^{n}}^{γ^{*}}

. The analysis of other situations is similar to the proof in condition (i). If

δ_{2^{n}}^{θ} \in Ω_{φ^{*}}

and

δ_{2^{n}}^{η} = δ_{2^{n}}^{γ^{*}}

, without loss of generality, the path from

δ_{2^{n}}^{θ}

δ_{2^{n}}^{γ^{*}}

can be described as

δ_{2^{n}}^{θ} \to \dots \to x (t_{1}) \to \dots δ_{2^{n}}^{φ^{*}} \to δ_{2^{n}}^{γ^{*}},

(20)

where

ε : = {ε (t) = δ_{2^{q}}^{l_{t}} : t \in [0, l_{1} - 2]} \cup {ε (t) = δ_{2^{q}}^{k^{*}} : t = l_{1} - 1}

l_{1}

denotes the number of steps from

δ_{2^{n}}^{θ}

δ_{2^{n}}^{γ^{*}}

. This together with

δ_{2^{n}}^{γ^{*}} \in Ω_{φ^{*}}

shows that a new cycle as (14) is formed for system (3) after OBFP. Denote the cycle (14) by

Λ = {δ_{2^{n}}^{φ^{*}}, δ_{2^{n}}^{γ^{*}}, δ_{2^{n}}^{γ_{1}^{*}} \dots δ_{2^{n}}^{γ_{s}^{*}}}

Since

J_{Δ_{2^{n}} ∖ W}^{T} {Col}_{φ^{*}} (\hat{Q}) = 0

, we have

Λ \subseteq W

, which means state

δ_{2^{n}}^{θ} \in Δ_{2^{n}}

can robustly reach to set

W

after OBFP occurs. Due to the arbitrariness of

δ_{2^{n}}^{θ}

, by Definition 2, system (3) is still robustly stable to the set

W

. □

Below we discuss when OBFP is the case 2, that is,

δ_{2^{n}}^{γ^{*}} \in I (W)

. The following theorem can be drawn.

Theorem 2.

Suppose Assumption 1 holds and

δ_{2^{n}}^{γ^{*}} \in I (W)

. Then, system (3) under OBFP is still robustly stable to the set

W

Proof.

The relationship between the perturbed state

δ_{2^{n}}^{γ^{*}}

and the set

Ω_{φ^{*}}

can be divided into two situations: (1)

δ_{2^{n}}^{γ^{*}} \notin Ω_{φ^{*}}

and (2)

δ_{2^{n}}^{γ^{*}} \in Ω_{φ^{*}}

. Below, we demonstrate that OBFP has no effect on the robust set stability of system (3) in the above two situations.

δ_{2^{n}}^{γ^{*}} \notin Ω_{φ^{*}}

, we can obtain from the proof of the sufficiency of Theorem 1 that the path from

δ_{2^{n}}^{θ}

to set

I (W)

can be described as (15). Moreover, OBFP does not affect the path (15). Combining with Assumption 1, we can conclude that every

δ_{2^{n}}^{θ} \in Δ_{2^{n}}

can be stabilized to set

W

under arbitrary disturbance sequence.

δ_{2^{n}}^{γ^{*}} \in Ω_{φ^{*}}

, the path from

δ_{2^{n}}^{θ}

to set

I (W)

can be described as (16). When

ε (τ_{1} - 1) \neq δ_{2^{q}}^{k^{*}}

, the path (16) is not affected by OBFP. When

ε (τ_{1} - 1) = δ_{2^{q}}^{k^{*}}

, after OBFP occurs, the path (16) changes to be

δ_{2^{n}}^{θ} \to \dots \to x (t_{1}) \to \dots δ_{2^{n}}^{φ^{*}} \to δ_{2^{n}}^{γ^{*}} .

(21)

Since

δ_{2^{n}}^{γ^{*}} \in I (W)

, it follows from the property of the largest robust invariant set that

δ_{2^{n}}^{θ}

can be stabilized to set

I (W)

and stay in set

I (W)

forever under arbitrary disturbance sequence. Thus, every

δ_{2^{n}}^{θ} \in Δ_{2^{n}}

can robustly reach to

I (W)

after OBFP occurs.

Due to the arbitrariness of

δ_{2^{n}}^{θ}

, one has that system (3) is still robustly stable to the set

W

under Assumption 1 and Case 2 of OBFP. □

Remark 1.

In [27], OBFP impact is only considered for the robust stability problem, where the target set is a single point set. The system is robust stable only when the trajectories of perturbed state cannot form a cycle. If the trajectory of perturbed state does not form a cycle, then the system can maintain robust set stability. However, even if a new cycle is formed, the system may still be robust set stable. As long as the new cycle is in the target set, the robust set stability of the system can be achieved. It is a difficult work to determine whether the new cycle is in the target set, which is the main technical difficulty [16] (see Theorem 1). In addition, this article needs to consider whether the perturbation state belongs to the target set, which is also different from References [16,27] (see Theorem 2).

Since both the robust synchronization and the robust output tracking can be transformed into the robust set stability problems, the results of this paper can be extended to the OBFP impact on the robust synchronization and robust output tracking of DBNs, which indicates that the research content of this work is more general than [16,27].

4. Illustrative Example

In this section, an illustrative example is given to show the impact of OBFP on robust set stability for DBNs.

Example 3.

Consider the biological example: a reduced E. coli lactose operon network [21]. The six genes, termed

l a c

mRNA, the high-concentration lactose, medium-concentration lactose, the extracellular glucose, the extracellular glucose, the high extracellular glucose and the medium extracellular glucose are denoted by state variable

X_{1}

, state variable

X_{2}

, state variable

X_{3}

, input variable

U_{1}

, input variable

U_{2}

and input variable

U_{3}

, respectively. An environmental signal is denoted by disturbance input ξ. The network model is

\begin{matrix} \begin{matrix} \{\begin{matrix} X_{1} (t + 1) = \neg U_{1} (t) \land (X_{2} (t) \lor X_{3} (t)), \\ X_{2} (t + 1) = \neg U_{1} (t) \land U_{2} (t) \land X_{1} (t) \land ξ (t), \\ X_{3} (t + 1) = \neg U_{1} (t) \land (U_{2} (t) \lor (U_{3} (t) \land X_{1} (t))) . \end{matrix} \end{matrix} \end{matrix}

(22)

Denote

1 \sim δ_{2}^{1}

and

0 \sim δ_{2}^{2}

, the algebraic form of (22) is represented as

\begin{matrix} x (t + 1) = M u (t) ε (t) x (t), \end{matrix}

(23)

where

x (t)

u (t)

ε (t)

are the vector forms of

(X_{1} (t), X_{2} (t), X_{3} (t)

(U_{1} (t), U_{2} (t), U_{3} (t)

ξ (t)

respectively, and

M = δ_{8} [8 8 8 8 \dots 4 4 4 8]

A feasible state feedback controller is pre-given as

u (t) = δ_{8} [5 8 5 5 5 8 5 5] x (t) .

(24)

By adding the control (24) into (23), we derive the following closed-loop system:

x (t + 1) = L ε (t) x (t),

(25)

where

L = [L_{1}, L_{2}]

with

L_{1} = δ_{8} [1 4 1 5 3 4 3 7]

and

L_{2} = δ_{8} [3 4 3 7 3 4 3 7]

Given a set

W = {δ_{8}^{1}, δ_{8}^{2}, δ_{8}^{3}}

. By simple calculation, it can be obtained that

I (W) = {δ_{8}^{1}, δ_{8}^{3}}

P = L_{1} +_{B} L_{2} = [\begin{matrix} 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}],

(26)

and

Q = (B) \sum_{i = 1}^{8} P^{(i)} = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] .

(27)

Apparently, DBN (25) is robustly stable to set

{δ_{8}^{1}, δ_{8}^{3}}

. The state trajectory graph of DBN (25) before OBFP is shown as Figure 1.

In the following, the impact of OBFP on DBNs will be discussed.

(1) After OBFP,

{Col}_{4} (L)

is perturbed from

δ_{8}^{5}

δ_{8}^{8}

. Thus, the perturbed state is

δ_{8}^{8}

. The state transition matrix of system (25) is changed to be

\hat{L} = δ_{8} [1 4 1 8 3 4 3 7 3 4 3 7 3 4 3 7]

. By following Lemma 3, we obtain

k^{*} = 1

and

φ^{*} = 4

According to Equation (27), one has

Ω_{4} = {δ_{8}^{2}, δ_{8}^{4}, δ_{8}^{6}}

. Since

δ_{8}^{8} \notin I (W)

and

δ_{8}^{8} \notin Ω_{4}

, it follows from Theorem 1 that DBN (25) is still robustly stable to the set

W

after OBFP. The corresponding state trajectory graph of dynamics can be described by Figure 2.

(2) After OBFP,

{Col}_{14} (L)

is perturbed from

δ_{8}^{4}

δ_{8}^{3}

, that is the perturbed state is

δ_{8}^{3}

. From Lemma 3, we obtain

k^{*} = 2

and

φ^{*} = 6

. The state transition matrix of system (28) is changed to be

\hat{L} = δ_{8} [1 4 1 5 3 4 3 7 3 4 3 7 3 3 3 7]

Since

δ_{8}^{3} \in I (W)

, we know that OBFP is the Case 2. Therefore, we can conclude from Theorem 2 that DBN (25) under OBFP is still robustly stable to set

W

. The state trajectory graph of dynamics is described in Figure 3.

5. Conclusions

We have investigated robust set stability about DBNs affected by OBFP. Based on the algebraic representation of a DBN, we have provided several necessary and sufficient conditions to guarantee that a DBN under OBFP can stay robust set stable unchanged. In the actual GRNs, gene mutations often occur in a stochastic manner and occur at multiple bits simultaneously. Hence, future work can study multi-bit stochastic function perturbations impact on the behavior of DBNs.

Author Contributions

Conceptualization, L.D. and X.C.; methodology, L.D. and X.C.; writing—original draft preparation, L.D.; writing—review and editing, L.D.; funding acquisition, J.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China under grant 62103176, and the Natural Science Foundation of Shandong Province under grants ZR2019BF023 and ZR2022MA030.

Data Availability Statement

The data presented for this study are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The state trajectory of DBN (25) before function perturbation.

Figure 2. State trajectory graph of dynamics after function perturbation in Case 1.

Figure 3. State trajectory graph of dynamics after function perturbation in Case 2.

Table 1. Notations.

Notations	Definitions
$Z_{+}$	the set of positive integers
$M_{m \times n}$	the set of $m \times n$ real matrices
$A^{T}$	the transpose of matrix A
$D$	$\{0, 1\}$
$D^{n}$	$\underset{n}{\underset{︸}{D \times D \times \dots \times D}}$
${Col}_{i} (A)$	the i-th column of matrix A
$Col (A)$	the set of columns of matrix A
$δ_{n}^{i}$	${Col}_{i} (I_{n})$
$Δ_{n}$	${δ_{n}^{1}, δ_{n}^{2}, \dots, δ_{n}^{n}}$
$0_{n}$	${\underset{n}{\underset{︸}{(0, 0, \dots, 0)}}}^{T}$
$1_{n}$	${\underset{n}{\underset{︸}{(1, 1, \dots, 1)}}}^{T}$
$L_{m \times n}$	the set of $m \times n$ logical matrices
$B_{m \times n}$	the set of $m \times n$ Boolean matrices
${[A]}_{i, j}$	$(i, j)$ -element of matrix A

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Deng, L.; Cao, X.; Zhao, J. One-Bit Function Perturbation Impact on Robust Set Stability of Boolean Networks with Disturbances. Mathematics 2024, 12, 2258. https://doi.org/10.3390/math12142258

AMA Style

Deng L, Cao X, Zhao J. One-Bit Function Perturbation Impact on Robust Set Stability of Boolean Networks with Disturbances. Mathematics. 2024; 12(14):2258. https://doi.org/10.3390/math12142258

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Deng, Lei, Xiujun Cao, and Jianli Zhao. 2024. "One-Bit Function Perturbation Impact on Robust Set Stability of Boolean Networks with Disturbances" Mathematics 12, no. 14: 2258. https://doi.org/10.3390/math12142258

APA Style

Deng, L., Cao, X., & Zhao, J. (2024). One-Bit Function Perturbation Impact on Robust Set Stability of Boolean Networks with Disturbances. Mathematics, 12(14), 2258. https://doi.org/10.3390/math12142258

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One-Bit Function Perturbation Impact on Robust Set Stability of Boolean Networks with Disturbances

Abstract

1. Introduction

2. Preliminaries

2.1. Semi-Tensor Product of Matrices

2.2. Algebraic Representations of BNs with Disturbances

3. Main Results

4. Illustrative Example

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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