Pinning Stabilization of Probabilistic Boolean Networks With Time Delays

In this article, the stabilization issues for probabilistic Boolean Networks (PBNs) with time delays are discussed. This article’s objective is designing an efficient algorithm to choose suitable nodes to be pinning controlled for PBNs with time delays. By using the semi-tensor product (STP) of matrices, a PBN with time delays can be converted into a discrete-time linear system, and the transition matrix also can be obtained. Then, the necessary and sufficient conditions in the form of algebraic expression are given for the existence and solvability of the pinning feedback controllers with minimum pinning nodes for PBNs with time delays. Besides, three algorithms are proposed for designing and solving minimum pinning controllers.


I. INTRODUCTION
Boolean Networks (BNs), which were first proposed by Kauffman in 1969 [1], are a kind of logical dynamical models to describe gene regulatory networks (GRNs) [2]. As we all know, in a gene regulatory network, each gene can be expressed (1) or not expressed (0), which corresponds to binary state variables. A BN is a deterministic model to simulate the evolution of binary state variables. What's more, Boolean Networks have been widely studied in state estimation [3], logical networks [4], neural networks [5], etc. Recently, the STP of matrices was introduced by Cheng's team. With the help of STP, a BN can be transformed into a discrete-time linear system. Moreover, a logic function can be represented by an algebraic form with STP. This new matrix product was introduced to the study of BNs in many fields, such as the controllability, event-triggered control, ect., which have been studied in [6]- [11].
To better handle of biological system uncertainty, Shmulevich etc. in [12] generalized the concept of BNs for application to probabilistic Boolean Networks (PBNs). In general, the PBNs can be seen as a kind of randomly switched BNs in given sets of BNs. Every BN is chosen with an definite probability. Many interesting results have been obtained for The associate editor coordinating the review of this manuscript and approving it for publication was Tingwen Huang. PBNs and probabilistic Boolean control networks (PBCNs), such as stability and stabilization [13], optimal control [4], controllability [14], and pinning control [15], etc.
Stability and stabilization are two important problems in BNs. For example, the apoptotic pathway can be activated to allow an organism to clear damaged or unwanted cells by combining with tumor necrosis factor (TNF) to death receptor tumor necrosis factor receptor 1 (TNFR1) [16]. Without TNF, cells can be bistable in two different states: survival and initiation of apoptosis [17]. However, the decision on one state or the other mainly depends on the initial conditions of random variation in each cell, and it can be seen as a stability problem in PBNs. Meanwhile, time delays are unavoidable in many real world systems, such as biological, physiological systems, and economic, and so on [18]- [20]. For GRNs, the direction of gene evolution is uncertain due to the possibility of gene mutation. Hence, PBNs with time delays can be better to simulate the real biological systems and GRNs in some cases. Thus, in this article, we will discuss the stabilization of PBNs with time delays.
In [21], BNs realize stabilization via state feedback control. Different from feedback control, only a small part of nodes are selected to be pinning controlled, which reduce the cost of the control effectively. A natural question in pinning control is how to select the nodes to be pinned. In [22], an algorithm is proposed to solve the minimum number of pinning controllers. Recently, minimizing controlled nodes to realize stabilization of BNs has been investigated deeply in [23]. Moreover, the stochastic networks can realize stabilization via minimum pinning controlled nodes, which has been studied in [24]. In [25], BNs with time delays realize stabilization via the pinning control. Thus, for a PBNs with time delays, how to stabilize via the pinning control and how to solve the minimum number pinning nodes to stabilize system are worth considering. Inspired by above works, the stabilization of PBNs with time delays via pinning control is investigated in this article.
The difficulties of this article are mainly two folds. 1) How to select pinning nodes for a PBN with time delays? Since a PBN with time delays is a dynamics system with random variables and multiple time delays, which make the pinning control problem for PBNs with time delays more complicated and challenging than that of BNs. 2) How to solve the minimum number of pinned nodes for a PBN with time delays through the algebraic method? In [22] and [26], the stabilization issue of BNs via minimum pinned nodes is solved through the graph theory method, rather than the algebraic method. Thus, it is a challenge to obtain corresponding results via the algebraic method. To overcome these difficult problems, inspired by the work of [24] and [25], we will take three steps to solve these difficulties. (i) Changing columns of the structure matrix to obtain the desired structure matrix. Thus, we propose a new algorithm to obtain the desired structure matrix. (ii) Selecting the pinning nodes via the columns of new structure matrix directly. The existence of the pinning feedback controllers for PBNs with time delays is considered, and the corresponding necessary and sufficient conditions in the form of algebraic expression are given. (iii) Choosing the minimal number pinning nodes by an efficient way. Moreover, an effective algorithm is proposed to calculate the minimum number of pinning controllers. Notations where δ k h is the kth column of the identity matrix I h . D := {1, 0}. 1 n and 0 n denote the column vector of length n, where all of the elements are equal to 1 and 0 respectively. M r×h stands for the set of all r × h matrices and M k r×h stands for the set of matrix A where A ij = (a 1 , a 2 , . . . , a k ) T , 1 ≤ i ≤ r and 1 ≤ j ≤ h. We denote Row f (W )(Col f (W )) stands for the f th row(column) of matrix W r×h and Row(W )(Col(W )) is the set of rows(columns) of matrix W r×h . A matrix W ∈ M r×h is called a logical matrix if its columns Col(W ) ⊂ r . Moreover, we define the set of r ×h logical matrices as L r×h . W = [δ k 1 r , δ k 2 r , . . . , δ k h r ] is denoted by W := δ r [k 1 , k 2 , . . . , k h ]. p = (k 1 , k 2 , . . . , k h ) T is called a h-dimensional probabilistic vector if k r ≥ 0, r = 1, 2, . . . , h and h r=1 k r = 1. we define the set of h-dimensional probabilistic vectors as P h . For a probabilistic vector p = (k 1 , k 2 , . . . , k h ) T , we denote a operator p = {δ r k | k r > 0, r = 1, 2, . . . , h}. For a matrix W ∈ M r×h , if its columns Col(W ) ⊂ P r , then this matrix is called a probabilistic matrix. Moreover, we define the set of r × h probabilistic matrices as P r×h .

II. PRELIMINARIES
A. STP OF MATRICES Definition 1: [27] For matrices W ∈ M r×h and Q ∈ M s×t . Then, the STP of W and Q is Here ⊗ is the Kronecker product of matrices and q is the least common multiple of h and s (q = lcm{h, s}).
Remark 1: Since STP is a generalization of the general matrix products, this notation can be omitted in the following discussion if no confusion arises.
Lemma 1: [27] 1) Let X ∈ R t be a row vector and a matrix A ∈ M m×n , we have A X = X (I t ⊗ A); 2) Let X ∈ R t be a column vector and a matrix A ∈ M m×n , we have X A = (I t ⊗ A) X . Definition 2: [27] Define a matrix: then for column vectors a ∈ R r and b ∈ R h , we have Q [h,r] b a = a b. Lemma 2: [27] Define a logical matrix n = diag(δ 1 2 n , δ 2 2 n , . . . , δ 2 n 2 n ) = δ 2 2n [1, 2 n + 2, . . . , 2 2 n ], and let X ∈ 2 n . Then, X X = n X .

B. ALGEBRAIC REPRESENTATIONS OF PROBABILISTIC BOOLEAN NETWORKS WITH TIME DELAYS
Letting True = 1 ∼ δ 1 2 , False = 0 ∼ δ 2 2 . Then we can express the logical function by using STP of matrices.
Lemma 3: [27] A logical function h(L 1 , . . . , L r ) with logical arguments L 1 , . . . , L r ∈ 2 can be expressed in a multi-linear form as where M h ∈ L 2×2 r is unique. Moreover, we define the matrix M h as the structure matrix of h.
A PBN with time delays is described as where x h (t) ∈ D is the state of node h at time t, h = 1, 2, . . . , r, and τ is a positive integer.
In system (2), f h is randomly selected from a given finite set of Boolean Based on the above discussion, the evolution of the state expectation can be obtained as follows . As a result, the following eauqtion holds where M =M 1 * . . . * M r ∈ P 2 r ×2 r(τ +1) and * is Khatri-Rao product. Definition 3: A probabilistic Boolean networks with time delays is globally stable with probability one to a statē x = δ q 2 r (1 ≤ q ≤ 2 r ), for any sequence of initial states Based on the discussion of remark 1 of [28], for obtaining the condition of the globally stable to a state with probability one of PBNs (2), we only need to study the globally stable of system (4).

III. MAIN RESULTS
In this section, the stabilization of PBCNs with time delays is considered by designing pinning controllers. We need to design a algorithm to get some suitable controlled nodes. In the end, we discuss how to stabilize system (2) to a given state δ q 2 r by minimum pinned nodes. Suppose that the first l nodes are pinned and state feedback controllers are as follows, Then system (2) with controllers becomes the following system where F s is a logical function of variables u s (t) and f s , u s (t) is the state feedback controller of Hence, systems (6) can be transformed to be Therefore, we can get the following results where
then change the sth column of M to δ q 2 r . 12: end if 13: end for 14: M = M 15: return M Theorem 1: Suppose that the structure matrix of (4) is M , and M is changed to M by the above algorithm. Then, the PBN with delays is globally stabilized to δ q 2 r with probaility one.
Remark 3: The above Theorem 2 provides a sufficient and necessary condition for the solvability of the pinning feedback controllers for PBNs with time delays, which generalizes the results of [25]. In other words, if the transition matrix is a determined matrix, Theorem 2 is degenerated to be Proposition 3.2 of [25]. Once the system of equation (12) is solvable, then L s and H s , s = 1, 2, . . . , l, can be obtained according to the proof of Theorem 2.
Note Thus, it holds that M j , as shown at the bottom of the next page.
For all α j i , i ∈ {k 1 , k 2 , . . . , k l }, they can be classified into 7 parts as follows (a) α Suppose there is a new matrix B j ∈ P 2×2 r(τ +1) . The elements in B j not mentioned in the following discussion are the same as the corresponding elements in matrix M j .
For . . x(t − τ ) satisfying B = H (I 2 r(τ +1) ⊗M j ) r . Therefore, the one pinned node is j and the pinning controller with delays can be solved from (12) by using Theorem 2.
Next, the above results can be further generalized as follows where the matrix M satisfies the following conditions: if δ Then, the above theorem can be further generalized as follows. .
From Theorem 3 and 4, the existence of minimum pinning nodes are discussed, and the necessary and sufficient conditions are obtained about exact number of pinning controllers. Next, the following Algorithm 3 is given to stabilize system (4) to the objective state δ q 2 r via minimum pinning controllers, which is based on these two theorems, and the minimum number of pinning controllers is solved by traversal.
In the following Algorithm 3, we denote the set (N , ), where N stands for the minimum number and stands for the set of pinning controllers. And t of i t stands for the number of pinning controllers. Since there are finite nodes totally, if t adds to r, then the corresponding results will be returned.
Hence, we haveM 1 ,M 2 , and M , as shown at the bottom of the page.
Thus, the feedback controllers can be designed as follows Then, we can use Algorithm 3 to calculate the minimum number of pinning controllers. It can be found that the minimum number of pinning controllers is 2 for (21).

V. CONCLUSION
In this article, for PBNs with time delays, the stabilization issue has been discussed. With the help of STP, the transition matrix of a PBN with time delays can be obtained, and the model is converted into a discrete-time linear system. Then, the necessary and sufficient conditions in the form of the algebraic expression for the pinning feedback controllers' existence and solvability are given. Moreover, the existence of minimum pinning nodes is discussed and the corresponding algorithm is designed. In the future, we will extend the results of this article to PBNs with more communication constraints, such as impulsive effects, stochastic perturbations, etc.