Synchronization Analysis of Master-Slave Probabilistic Boolean Networks

In this paper, we analyze the synchronization problem of master-slave probabilistic Boolean networks (PBNs). The master Boolean network (BN) is a deterministic BN, while the slave BN is determined by a series of possible logical functions with certain probability at each discrete time point. In this paper, we firstly define the synchronization of master-slave PBNs with probability one, and then we investigate synchronization with probability one. By resorting to new approach called semi-tensor product (STP), the master-slave PBNs are expressed in equivalent algebraic forms. Based on the algebraic form, some necessary and sufficient criteria are derived to guarantee synchronization with probability one. Further, we study the synchronization of master-slave PBNs in probability. Synchronization in probability implies that for any initial states, the master BN can be synchronized by the slave BN with certain probability, while synchronization with probability one implies that master BN can be synchronized by the slave BN with probability one. Based on the equivalent algebraic form, some efficient conditions are derived to guarantee synchronization in probability. Finally, several numerical examples are presented to show the effectiveness of the main results.

problems like determining fixed points and observability of Boolean control networks have already been proved to be NP-hard. Hence, the computational complexity is intrinsic and also independent of the models adopted to describe BNs.
It is a curious phenomenon of some real-world systems that they can evolve in perfect synchronization. Synchronization is an important property, which makes two coupled systems oscillate in typical collective behavior. In recent years, synchronization problem of dynamic systems have drawn great attention, sucn as synchronization of complex networks [22][23][24] , consensus in multi-agent systems 25,26 , synchronization of Kauffman networks 27 , cooperation of networks [28][29][30][31] and so on. Since BNs can provide general features of living organism, and well illustrate genetic regulatory networks, the synchronization problem has been extended to BNs. The researches on synchronization of BNs can provide lots of useful information on the evolution of biological systems whose corresponding subsystem influences with each other. For example, investigation on synchronized BNs is beneficial to better understand synchronization between two coupled lasers 32 . Hence, studying the synchronization problem of BN is of both theoretical and practical importance. In the past few years, Some necessary and sufficient criteria of complete synchronization for two deterministic BNs has been obtained 33  In 9,13,37 , the target state of nodes in BNs is predicted by deterministic Boolean functions. Deterministic BNs always follow a static transition mechanism supervised by binary logical functions, and ultimately reach a limit set, from which the system cannot move. However, the stochastic feature of genetic regulation and micro array data used to infer the structure of networks may have errors because of external noise in the complex measurement processes. Hence, the stochastic factor is an important feature, and BNs with stochastic factor is more practical and favorable to such situations, resulting in the development of probabilistic Boolean networks (PBNs). In 38 , Shmulevich et al. firstly proposed PBNs model, which deals with the problem of uncertainty. A PBN can be regarded as a collection of BNs, in which the state of each node chooses its transition rule according to some probabilistic rules at discrete time point. And the transition rule for updating each node is randomly chosen among several possible rules with a given probability distribution. Hence, a PBN allows the model to have more flexibility, which is the basic idea of PBNs.
Recently, PBNs have been widely applied to infer functional connectivity between brain regions and to investigate the connectivity abnormality in Parkinson's Disease 39 . Some fundamental and interesting results on PBNs have been obtained, such as optimal control problem in context-sensitive PBNs 40 , controllability of PBNs with forbidden states 17 , steady-state probability distribution of PBNs 41 . Due to its rule-based and uncertainties properties, PBNs seem more practical to model genetic regulatory networks than usual deterministic BNs. And phenomenon of coupling is very common in real world systems. Hence, it is meaningful and challenging to study the synchronization problem of PBNs, and there has been no result investigating on synchronization of PBNs, to the best our knowledge. Thus, motivated by the above discussions, in this paper, we aim to investigate the synchronization problem of PBNs coupled in the master-slave configuration, in which the master BN is a deterministic BN while the slave BN is a PBN. In this paper, we firstly investigate synchronization of master-slave PBNs with probability one, then investigate synchronization in probability. New approaches based on STP are proposed to derive necessary and sufficient conditions for synchronization.
Notation: The following standard notations will be used in this paper. Throughout this paper,  n m × denotes the set of real matrices of order n × m, and  + denotes the positive integers. 1 n denotes the n-dimensional column vector with all entries being 1, and I k is the identity matrix of order k. k j δ is the j-th column of identity matrix I k , and Δ k denotes the set of all k columns of I k . In particular, when k = 2, we use Firstly, we introduce a bijective correspondence between Boolean vectors X ∈  and vectors x ∈ Δ , which is defined by the relationship: Then, we introduce semi-tensor product (STP) "" between matrices (and in particular, vectors) as follows 10 : given two matrices, ( ) Definition 1 An mn × mn matrix W [m,n] is called a swap matrix, if it is constructed in following way: label its columns by (11,12, …, 1n, …, m1, m2, …, mn) and similarly label its rows by (11,21, …, m1, …, 1n, 2n, …, mn). Then its element in the position ((I, J), (i, j)) is assigned as w I i and J j otherwise  [3,2] and obtain the matrix W [3,2] as following: By resorting to STP and the bijective correspondence between n  and Δ 2n , we can acquire an algebraic representation of logical functions. To do so, we have to identify the Boolean vectors 1 and 0 with the vectors 2 1 δ and 2 2 δ . That is to say, we consider a Boolean variable X ∈  as a vector x ∈ Δ , thus a Boolean function of n variables f : n   → is equivalent with a map f : ( )

Matrix expression of master-slave probabilistic Boolean networks (PBNs). Recall that two
BNs coupled in master-slave configuration, and each network has n nodes, which can be described as: i i  i } can be regarded as switching signals; t = 0, 1, 2, …, and here we simply denote = N : 2 n . We simply denote x t x t x t n 1 ( ) = ( ( ), …, ( )) and y t y t y t n 1 ( ) ( ) = ( ), …, ( ) to be the states of the master BN and the slave BN at time instant t, respectively. Moreover, we can observe that the state evolution of the master-slave BNs depends on the following initial states: The master-slave BNs (9) becomes a master-salve PBNs if the probability of g i being g i j is p i j , denoted In this section, we assume that the slave PBN is independent, that is g 1 , g 2 , …, g n are independent from each other, i.e.
, and y t y t , which is a bijective mapping pointed by D. Cheng 9,10 . For each logical functions , ∈ , f i n [1 ] i , we can find its corresponding structure matrix F i . Thus, using Lemma 1, for the master logical functions, we can obtain its algebraic form: ∈ , , we can find its structure matrix  L i ij . Thus, using Lemma 1, for the slave logical functions, we have  y t L x t y t 1 12 Thus, for master-slave PBNs (9), we obtain the following equivalent algebraic expression: In fact, the master-slave PBNs (9) can be regarded as a whole system. Let z t x t y t N 2  ( ) = ( ) ( ) ∈ Δ be the state of the whole system. Then for the master-slave PBNs (9), we can obtain following dynamics of the whole system: ( )  Hence, the overall expected value of z(t + 1) satisfies:

Remark 2
According to Eq. (14), we know that the state z(t + 1) is updated by the logical function Ψ i with a certain probability, i.e. p i . And actually, z(t + 1) has  number of choices to update its states. Unlike deterministic BNs, PBNs do not have accurate state evolutional process, and all the possible state evolutional processes exist with some certain probabilities.
Remark 3 According to Eq. (14) and Eq. (15), we can obtain that the pq-th entry of matrix L i is equal  (15). However, due to the coupling property between master BN and slave BN (slave BN is also affected by master BN), the slave BN is also a PBN. Thus, in order to investigate synchronization for this kind of system, we only need to check whether z(t) (state of the whole system) can reach the set of synchronized states with probability one. Hence, similar methods for synchronization of deterministic BNs can be used to investigate synchronization of these systems.
Example 4 Consider the following master-slave PBNs: x t x t y t y t where the switching signal on logical function g 2 is  t : Here, the probabilities of g 2 being g 2 1 and g 2 2 are  g g 0 4 . Denote x t x t x t 1 2 ( ) = ( ) ( ) and y t y t y t 1 2 ( ) = ( ) ( ). By resorting to STP and Lemma 1, we can obtain its equivalent algebraic form as follows: ( ) The state transition digraph of system (16) is shown in Fig. 1. Hence, we can obtain that the overall expected value of z(t + 1) satisfies: Synchronization of master-slave PBNs with probability one. In the following sebsection, we firstly define the definition of synchronization of the master-slave PBNs (9) with probability one as follows. Definition 2 Consider the master-slave PBNs (9). System (9) is said to be synchronized with probability one if for any initial state x 0

Remark 5
If the master-slave PBNs (9) can be synchronized with probability one, then there must exist an integer k such that for t ≥ k, By this meaning, the slave BN has only one deterministic trajectory after finite steps, which is exactly the same as the trajectory of master BN, i.e.
be the index set of Ξ .

Remark 6
In 33,34 , Li et al. have investigated the complete synchronization of BNs coupled in drive-response configuration. In those models, the drive BN and response BN are both deterministic BN, which implies that the trajectory of drive BN will coincide with that of response BN after finite steps. Since the stochastic factor is an important feature in real world, BNs with stochastic factor is more practical and favorable. Here, we consider that the master BN is a deterministic BN, while the slave BN is a probabilistic BN. Due to the fact that the master BN is a deterministic BN which means there will be only one trajectory, the slave BN must have only one trajectory coinciding with master BN after finite steps. Thus, the main difference between synchronization with probability one and general synchronization is that there will be some possible trajectories at the beginning of a period time but only one deterministic trajectory after finite steps.
According to Eq. (9), we observe that the master BN is a deterministic BN. Thus, the trajectory will enter into a cycle after finite steps starting from any state.
be the transient period of system and T > 0 be the smallest positive number satisfying F F k k T 0 0 = + . Thus, we can obtain the following proposition.
Proposition 1 Starting from any state, the trajectory of master BN (9) will enter into a cycle after k 0 steps.
Example 5 Consider the following master BN with 3 nodes: x t x t x t x t x t x t x t 1 1 1 24 where L follows immediately as L = δ 8 [3,7,8,8,1,5,6,6]. Thus, it is easy to check that k 0 = 2 and L 2 = L 7 , i.e. T = 5, which implies that the trajectory of BN will enter a cycle after 2 steps. The dynamic graph of system (24) is shown in Fig. 2, from which we can see that each state will enter a cycle with length 5 after 2 step.
Based on Proposition 1, we can obtain the following necessary and sufficient condition for synchronization of master-slave PBNs (9) with probability one.
Theorem 1 Consider the master-slave PBNs (9). System (9) can be synchronized with probability one if and only if the following conditions hold: Proof. According to Eq. (15), we can obtain that  (9) can be synchronized with probability one, then there must exist an integer k, such that for t ≥ k satisfying Since the trajectory of master BN will enter into a cycle after k 0 iterations, we only need to consider whether the limit set of slave BN can be coincided with that of master BN or not. For any initial state x(0), based on Proposition 1, the trajectory of master BN will reach a cycle: i.e. i T = i 0 . Since the master-slave PBNs (9) can be synchronized with probability one from any initial states x(0), y(0), we can obtain that the trajectory of slave BN also reach the same cycle: δ δ δ , , …, Thus, it is equivalent to . Since the initial state z(0) is arbitrary, we can derive that ( ) ⊆ Ξ, ∈ , The necessity is proved. (Sufficiency) Assuming that conditions (25) and (26) hold, we prove that under these conditions the master BN can be synchronized by the slave BN with probability one. Suppose that z 0 . If (25) holds, after k 0 steps, we have ( ) ∈ Ξ = , …, . Since the set Ξ is synchronized set, one has ( ), …, ( + ) is a cycle. By this meaning, the trajectory of system (14) enter into a cycle. This together with (29) yields that the master BN can be synchronized by the slave BN with probability one, as the index j is arbitrary. This completes the proof. Remark 7 According to Theorem 1, we observe that condition (25) guarantees that the master BN can be synchronized by slave BN for states in limit set with probability one. And condition (26) guarantees that the slave BN has the same cycles or fixed points with probability one after k 0 steps. Thus, condition (26) is a necessary condition to guarantee synchronization. Even for some systems satisfying condition (25), it can not reach synchronization.
Remark 8 According to Proposition 1, we can conclude that the trajectory of master BN will enter into a cycle after k 0 steps. To investigate the synchronization with probability one, we only need to consider the following time sequence k 0 , k 0 + 1, …, k 0 + T, because the matrix F satisfies F F k k T 0 0 = + . Since the set Ξ is the set of synchronized states, condition (25) implies that the slave BN can reach synchronization with probability one at time sequence k 0 , k 0 + 1, …, k 0 + T, but can not guarantee synchronization after time k 0 + T. However, due to the fact that F F k k T 0 0 = + , the slave BN also need to guarantee a periodic trajectory with the same length as the trajectory of master BN if the slave BN wants to reach synchronization. Thus, condition (26) guarantees that the periodic trajectory of slave BN coincides with that of master BN.
According to Theorem 1, we can easily obtain following corollary to check whether a given master-slave PBN can be synchronized with probability one or not.
Corollary 1 Consider the master-slave PBNs (9). System (9) can be synchronized with probability one if and only if the following conditions hold: where matrix Ω k is the matrix obtained from L k by deleting the rows with index ∈ = ( − ) + ≠ ∈ , Theorem 2 Consider the master-slave PBNs (9). The master-slave PBNs can be synchronized with probability one, if following two conditions hold:

Proof. For any initial states x(0), y(0), according to Eq. (15) and after k iterations, we have
Suppose that condition (1) holds, then we have Ez(k) ∈ Ξ , which implies that Then for the next step, we have Ez k L z k 1 ( + ) = ( ). The facts that z(k) ∈ Ξ and condition (2) holds means that Ez(k + 1) ∈ Ξ , which further implies that Thus using mathematical iteration, we obtain that τ ( + ) ∈ Ξ =  z k { } 1 r ,  τ ∈ + . By this meaning, for any initial states x(0), y(0), we have Thus, it implies that the master-slave PBNs (9) can be synchronized with probability one.
Corollary 2 Consider the master-slave PBNs (9). The master-slave PBNs can be synchronized with probability one, if following two conditions hold: (1) there exists a positive number 0 < k ≤ k 0 + T, such that ϒ ∈ × ℒ k N N 2, where matrix ϒ k is the matrix obtained from L k by deleting the rows with index ∈ = ( − ) + ≠ ∈ , where matrix Λ is the matrix obtained from L by deleting the column and rows with index ∈ = ( − ) + ≠ ∈ , Synchronization of master-slave PBNs (9) in probability. In the above section, we have investigated synchronization of master-slave PBNs (9) with probability one. Since the master BN is a deterministic BN, synchronization with probability one implies that the slave BN has deterministic trajectories coinciding with trajectories of master BN after finite steps. As we can see, this condition is relative strict in some real-world systems. If the slave BN has some trajectories coinciding with trajectories of master BN with some certain probability, what happens? Thus in following section, we will investigate synchronization of master-slave PBNs (9) in probability, which implies that the master BN can be synchronized by the slave BN with some certain probability. Now, we firstly define the definition of synchronization in probability as follows. Definition 3 Consider the master-slave PBNs (9). System (9) is said to be synchronized in probability if for any initial state x 0 j ( ) ∈ Δ, j n The main concern of synchronization in probability is that whether there exists one possible trajectory coinciding with the trajectory of master BN or not. The main difference between synchronization with probability one and synchronization in probability is that whether there exists one deterministic trajectory or one possible trajectory which coincides with the trajectory of master BN.
Here, we still let . Based on Theorem 1, we have the following algebraic criterion for synchronization in probability.
Theorem 3 Consider the master-slave PBNs (9). System (9) can be synchronized in probability if and only if the following conditions hold: where ℒ k is the matrix obtained from L k by substituting zeros in the rows with index  (31) and (32) hold, we prove that under these conditions, the master BN can be synchronized in probability by the slave BN. It should be noted that for any initial state, the trajectory of master BN will enter into some cycle after k 0 steps, i.e. x k x k T 0 0 ( ) = ( + ). Moreover, the master BN is a deterministic BN. Thus, we only need to check whether the master BN can be synchronized in probability by slave BN at the limit states: ( ), …, ( + ) x k . According to Eq. (27), we have . Hence, the master-slave PBNs (9) can be synchronized in probability.
(Necessity) If the master-slave PBNs (9) can be synchronized in probability, we prove that conditions (31) and (32) hold. Note that the master BN is a deterministic BN. Hence, it has exact trajectories. According to Proposition 1, we know that the trajectory will enter into certain cycle after k 0 steps. Due to the fact that 0 < α < 1, there must also exist some positive number in the rows with index , the probability can not be equal to 1. By this meaning, we have following equations: where k  is the matrix obtained from L k by substituting zeros in the rows with index ∈ = ( − ) + ≠ ∈ , [ 1 ]}. Now, we prove condition (32) holds, provided the master-slave PBNs (9) can be synchronized in probability. Note that for any initial state x(0), we can always find k 0 such that x k x k T 0 0 ( ) = ( + ). Thus, if the master-slave PBNs (9) can be synchronized in probability, it implies that . Thus, we have Let k 0  and k T 0  + be the matrices obtained from L k 0 and L k T 0 + by substituting zeros in the rows with index ∈ = ( − ) + ≠ ∈ , , it implies that for each column of matrices L k 0 and L k T 0 + , the index ∈ = ( − ) + ∈ , [1 ]} S must be the same. By this meaning, we derive Col k T }. This completes the proof. Remark 10 Due to the fact that the trajectory of master BN will enter into a cycle after k 0 steps, we also only need to consider the time sequence k 0 , k 0 + 1, …, k 0 + T. Since the set α δ α Θ = * ∈ , , < < Thus, we can conclude that at the time sequence k 0 , k 0 + 1, …, k 0 + T, the master-slave PBNs can reach synchronization in probability. Moreover, since k T k k T 0 0 0 Λ = Γ + Γ , + , condition (32) implies that the slave BN can generate one possible periodic trajectory with the same length as the trajectory of master BN. So, condition (32) guarantees that the master-slave PBNs can reach synchronization in probability after time k 0 + T.
Theorem 4 Consider the master-slave PBNs (9). The master-slave PBNs (9) can be synchronized in probability, if following two conditions hold: (1) there exists a positive number 0 < k ≤ k 0 + T, such that Col(ℒ k ) ⊆ Θ , where ℒ k is the matrix obtained from L k by substituting zeros in the rows with index , where  is the matrix obtained from L by substituting zeros in the rows with index ∈ = ( − ) + ≠ ∈ , Proof. Suppose that there exists a positive number 0 < k ≤ k 0 + T, such that Col( k  ) ⊆ Θ , where k  is the matrix obtained from L k by substituting zeros in the rows with index , p 2 , …, p μ ∈ , 0 < a 1 , a 2 , …, a μ . Considering the next step t = k + 1, we have Thus, using mathematical iteration, we can obtain that Hence, the master-slave PBNs (9) can be synchronized in probability.

Numerical Simulation
In this section, we present two numerical examples to demonstrate the applications of our main results.
Example 6 Let us consider the following two PBNs with 2 nodes coupled in the master-slave configuration: x t x t x t x t x t x t y t g x t x t y t y t y t g x t x t y t y t Our objective is to check whether these master-slave PBNs (40) can be synchronized with probability one or not. Denote x t x t x t 1 2 ( ) = ( ) ( ) and y t y t y t 1 2 ( ) = ( ) ( ). By resorting to STP and Lemma 1, we can obtain its algebraic form of system (40) as follows:  The state transition digraph of system (40) is shown in Fig. 3. δ ( ) = and L 3 = L 4 , which implies that conditions (25) and (26) hold. Thus, this master-slave PBNs (40) can be synchronized with probability one. From Fig. 3, we observe that all the possible trajectories of system (40) starting from any initial state z(0) ∈ 4  will eventually enter into the synchronized state (0, 0, 0, 0) at the third time step, and it will never escape.
Example 7 Now, we present another example to illustrate synchronization of master-slave PBNs in probability. Let us consider the following two PBNs with 2 nodes coupled in the master-slave configuration: x t x t y t y t y t g x t x t y t y t   x t x t y t x t y t x t x t y t x t y t g x t x t y t y t y t x t y t x t x t y t y t x t y y t y t y t Here, we let the probabilities be  { } . Thus, we can obtain the following possible model index of matrix , which is listed as follows: Hence, the slave BN has 2 possible BNs to be chosen. One of the possible BN has the probability 0.4, while the other possible BN has the probability 0.6.
In order to check whether these two PBNs (48) can be synchronized in probability or not, we need to use Theorem 3. Denote x t x t x t 1 2 ( ) = ( ) ( ) and y t y t y t 1 2 ( ) = ( ) ( ). By resorting to STP, we can obtain the following equivalent algebraic form of system (48):  To better illustrate dynamic of the master-slave PBNs, the state transition digraph of system (48) is shown in Fig. 4.
Since In  The red dash line implies that state transfer to its next states with probability 0.4, while the black solid line implies that state transfer to its next states with probability 0.6.

Conclusions
In this paper, both synchronization of master-slave PBNs with probability one and synchronization in probability have been investigated. One restriction in this paper is that master BN is a deterministic BN, while slave BN is a probabilistic BN. Slave BN is determined by a series of possible logical functions with certain probability at each time point. The definitions of synchronization with probability one and synchronization in probability are firstly presented in this paper. Due to the fact that the master BN is a deterministic BN while the slave BN is a probabilistic BN, this paper considers two different cases: synchronization with probability one and synchronization in probability. The main concern of synchronization in probability is that whether there exists one possible trajectory coinciding with the trajectory of master BN or not. The main difference between synchronization with probability one and synchronization in probability is that whether there exists one deterministic trajectory or one possible trajectory which coincides with the trajectory of master BN. Based on STP and its equivalent algebraic form, several necessary and sufficient conditions for two types of synchronization are derived. According to obtained necessary and sufficient conditions, we derive some effective conditions to judge whether some given master-slave PBNs can be synchronized with probability one or not. And then, some effective conditions   are also obtained to judge whether some given master-slave PBNs can be synchronized in probability or not. Moreover, the main results are well illustrated by numerical examples.
Unfortunately, determining whether the master-slave PBN can be synchronized or not is still NP-hard. Some interesting and meaningful topics that deserve further research include the following: to investigate synchronization problem with different (or time-varying) delays, to investigate the feedback controller based on switching signals, and so on.