Finite-Time Projective Lag Synchronization and Identification between Multiple Weights Markovian Jumping Complex Networks with Stochastic Perturbations

Two nonidentical dimension Markovian jumping complex networks with stochastic perturbations are taken as objects. *e network models under two conditions including single weight and double weights are established, respectively, to study the problem of synchronization and identification. A finite-time projection lag synchronization method is proposed and the unknown parameters of the network are identified. First of all, based on Itô’s formula and the stability theory of finite-time, a credible finite-time adaptive controller is presented to guarantee the synchronization of two nonidentical dimension Markovian jumping complex networks with stochastic perturbations under both conditions. Meanwhile, in order to identify the uncertain parameters of the network with stochastic perturbations accurately, some corresponding sufficient conditions are given. Finally, numerical simulations under two working conditions are given to demonstrate the effectiveness and feasibility of the main theory result.


Introduction
Complex network, a form of system structure and function, is an essential abstraction of the interaction of the complex system and has attracted substantial number of interest from researchers in varied realm [1,2]. e World Wide Web, power network, and neural network as well as social network all belong to the complex network of our life. Synchronization is a momentous nonlinear phenomenon of nature. As a most valuable topic in many dynamic behaviors of the complex network, it has attracted a growing number of concerns in the investigation [3,4]. If the dynamic behavior of the coupling node in the network evolves over time and eventually reaches the same state, then it will be called synchronization. At present, a series of related research results have been established, including complete synchronization, exponential synchronization, asymptotic synchronization, and generalized synchronization. In [5], a method of complete synchronization of complex networks is proposed by constructing adaptive control technology. A sufficient condition for exponential synchronization of a complex network with time-varying delays is given in [6]. Among these research studies, the dimensions of networks are identical. However, in some practical situations this hypothesis is inappropriate and there are still a large number of networks formed with nonidentical dimensions. Projective synchronization, an important synchronization phenomenon, has received adequate attention which can be used as an appropriate synchronization scheme for the nonidentical dimension complex networks.
Moreover, the abovementioned network synchronization assumes that the synchronization time tends to be infinite [7,8]. However, in the engineering field, it is often needed to achieve synchronization as quickly as possible, which means finite-time synchronization. In order to accomplish faster synchronization in complex dynamical

Preliminaries
Firstly, some significantly mathematical notations are introduced as follows, which will be applied throughout this paper. Let matrix A T (or x T ) means the transpose of the A (or x). Denote ‖x‖ as the 2-norm. Indicate I n ∈ R n×n as the ndimensional identity matrix. Let ⊗ represent the Kronecker product of the matrix. λ max (·) is the largest eigenvalue of a matrix.

Assumption 1.
e uniform Lipschitz condition is satisfied by the noise intensity function σ i (t, e i (t)) and the constant ρ i ≥ 0 is existed to make the following formula hold: High voltage network Low voltage network 2 Complexity trance σ T i t, e i (t), r(t) σ i t, e i (t), r(t) ≤ ρ i (r(t))e T i (t)e i (t). (3) Lemma 1 (see [23]). As for any vectors x, y ∈ R n , the following matrix inequality will be hold, where the P ∈ R n×n is a positive definite matrix: 2x T y ≤ x T Px + y T P − 1 y .
As well as n i�1 a n c ≥ n i�1 a n where ∀a 1 , a 2 , . . ., a n ∈ R n are any vectors and γ is a real number satisfying 0 < γ < 2.

Finite-Time-Generalized Matrix Projective Synchronization with Single-Weight Networks
e single-weight Markovian jumping complex network with random perturbations is taken as the research object and its parameter identification and finite-time synchronization are studied. e corresponding synchronization criteria and updating rules are obtained as well as the reliability and validity of the method are illustrated by numerical simulation. is is a preparation for further research on synchronization of complex networks with double weights.

Network Models.
e drive-coupled complex network is described as follows, and its number of dynamic nodes is N: where x i (t) � (x i1 (t), x i2 (t), . . . , x in (t)) T ∈ R n are the state vectors of the ith node; f i1 : R n ⟶ R n×l is a continuous matrix function and f i2 : R n ⟶ R n is a continuous vector function, respectively; α i ∈ R l are the unidentified node dynamic parameters; Γ 1 ∈ R n×n represents the inner coupling matrix; the coupling configuration matrix is expressed by A(r(t)) � (a ij r(t)) N×N ∈ R n×n , which represents the topological structure and the coupling strength of the network at time t in mode r(t).
{r(t, t ≥ 0)} is defined as a right-continuous Markovian process in probability space, describing the switching state between different parameters of time t. It takes values from the finite space S � 1, 2, . . . , m { } with generator Π � (π pq ) m×m (p, q ∈ S). e transition probability from pth mode at time t to the qth mode at time t + Δt will be defined as follows: where Δt > 0: As the transition rate π pq ≥ 0 which can be represented from mode p at time t to mode q at time t + Δt, erefore, the elements of matrix A(r(t)) will be defined as follows. If it exists as a link from node j to node i (j ≠ i), then a ij ≠ 0; otherwise, a ij � 0. e diagonal elements of matrix A(r(t)) can be determined by a ii � − N j�1,j ≠ i a ij . Describe the response-coupled network as follows: where y i (t) � (y i1 (t), y i2 (t), . . . , y im (t)) T ∈ R m are the state vectors of node i; g i 1 : R m ⟶ R m×o is a continuous matrix function and g i 2 : R m ⟶ R m is a continuous vector function; the node dynamic parameter vectors which need to be identified are β i ∈ R o ; Γ 2 ∈ R m×m represents the inner coupling matrix; the coupling configuration matrix is defined by B(r(t)) � (b ij r(t)) M×M ∈ R m×m as well as in network (9); u i (t) is the finite-time nonlinear feedback controller which needs to be devised; σ i (t, e i (t), and r(t)) are the noisy intensity function which mainly describe the impact of environmental fluctuations or imprecise design of coupling strength on network synchronization; and e i (t) is the error of system synchronization. e error of system synchronization can be defined as follows: Complexity 3 where M ∈ R m×n is the scaling matrix (i.e., generalized matrix) consisting of constant; e i (t) � (e i1 (t), e i2 (t), . . . , e im (t)) T ; and τ is the time delay.
Definition 2. e driving network and response network can be realized synchronization through the designed finite-time nonlinear controllers. As for scaling matrix M ij � (m ij ) ∈ R m×n , if the condition can be satisfied that each row of the element cannot be 0 at a time, it will hold that

Main Result.
In this section, the synchronization and identification of unknown parameters for Markovian jumping complex networks with nonidentical dimensions and stochastic perturbations are investigated by utilizing the control theory of finite-time. e following theorem and remarks are the main results.
First, from (9) and (13), the synchronization error dynamic system can be obtained as below: Theorem 1. With Assumption 1 holding, the single-weight Markovian jumping complex network (9) and (13) can be realized synchronization in a finite-time. Meanwhile, with the following controllers and undated laws, the value of uncertain parameters can be simultaneously identified: where k i (r) > 0, 0 < c < 2, ψ > 0, θ i and λ i are any positive constant, and sign(·) represents symbolic function.
Remark 1. When Markovian jumping complex network (9) and (13) achieve projection lag synchronization in a finite time, the uncertain parameter vectors α i and β i of the system are identified. If the following inequalities hold, where D(q) is a positive definite matrix; C(p) is a symmetrical matrix; B(r) � (B(r)B T (r))/2; B(r) � B(r) ⊗ Γ 2 ; Ξ(r) � diag k 1 (r), k 2 (r), . . . , k N (r) ; Θ(r) � diag ρ 1 (r), ρ 2 (r), . . . , ρ N (r); and r ∈ S. erefore, the synchronization between single-weight Markovian jumping complex network (9) and (13) with stochastic perturbations can be achieved global asymptotic stability in a finite-time under this condition: where Proof. e Lyapunov function is defined as below: By Definition 1, we can obtain the differential operator LV as follows: 4 Complexity Substitute (16) and feedback controllers (17) into (21), resulting in According to Lemma 3, we obtain that From Assumption 1, the conclusion is drawn:

Complexity
From Lemma 2, the system error of synchronization e i (t) will converge to a steady state with a finite-time, which can be estimated by where (9) and (13) can be realized synchronization with finite time t 1 . Also, the uncertain system parameter vectors α i and β i are adapted to the true value of parameters. e proof is completed.
e proposed approach is applicable to the finitetime-generalized matrix projection lag synchronization of any two single-weight Markovian jumping complex networks with uncertain parameters, stochastic perturbations, and different initial conditions. In addition, the Markovian jumping complex network can be identical or nonidentical dimensions.

Remark 3.
e synchronization and identification speed can be adjusted through choosing the constants k i (r), θ i (r), and λ i (r), adequately.
Remark 4. Inequality (18) in eorem 1 is only just sufficient but not a necessary condition for single-weight Markovian jumping complex networks (9) and (13) to realize projection lag synchronization and parameter identification.

Simulation
Results. Some numerical simulations are performed to prove the feasibility and effectiveness of the abovementioned result.
e Chen system is taken as the node dynamic of the drive complex network: e identifying value of the parameter vector α i is e four-dimensional hyperchaotic Lüchaotic system will be the node dynamic of the response complex network: And the identifying value of the parameter vector β i is e size of network is taken as N � 10. e coupling configuration matrices of the network are separately given, as shown in Figures 2 and 3. Figure 2 is the topological structure of the driving system with mode 1 and mode 2. Figure 3 is the response system. Among them, the structure of network topology changing with two modes is shown with the dotted line.
In order to judge the quality of the network synchronization, we define the system error value as follows: When lim t⟶∞ E(t) � 0 and ∀t > t 1 , systems (9) and (13) will implement global synchronization within the finite time t 1 .
Complexity 7 e other relevant parameters are as below: e switching of two different modes in the complex system can be shown at  Figure 5. From that, one can see that the error converge to zero which means that the two networks are in the synchronization state. Figure 6 is the system error values E(t). e conclusion of Figure 6 matches that of Figure 5. Figures 7  and 8 give the identification of parameters α i (i � 1, 2, 3) and β i (i � 1, 2, 3, 4). It is clear that the estimated parameters are approached to the value (35, 3, 28) T and (36, 3, 20, 1) T . ey all attest that the control scheme can realize the identification and synchronization of two single-weight Markonian jumping complex networks, effectively.

Finite-Time-Generalized Matrix Projective Synchronization with Double-Weight Networks
In order to build a practical Markovian jumping complex network model with double-weights, this paper introduces a method of network splitting, which divides the different coupling configuration into different subnetworks. Each network after splitting has its own nature and structure. In this paper, different side voltage levels in the power network are defined as different properties, which are divided into two subnetworks (as shown in Figure 9). en, the finitetime generalized function lag projection synchronization is studied. e synchronization criteria are given and the values of the related unknown parameters are identified further.

Network Models.
In this section, the driving system is defined as follows and its dynamic node number is N: where Γ i (i � 1, 2) are the inner coupling matrices; A(r(t)) � (a ij r(t)) N×N and B(r(t)) � (b ij r(t)) N×N represent the coupling configuration matrices which are represented as different coupling relationships. Its elements satisfy the following. If there exists a link from 8 Complexity . e diagonal elements of matrix A(r(t))(B(r(t))) are defined as e response coupled network can be described as follows: where Γ i (i � 3, 4) represent the inner coupling matrices; M(r(t)) � (m ij r(t)) N×N and N(r(t)) � (n ij r(t)) N×N are Complexity the coupling configuration matrices, same as A(r(t))(B(r(t))) in (33); u i (t) is the finite-time nonlinear feedback controller which needs to be designed; and σ i (t, e i (t), r(t)) is the noisy intensity function and e i (t) is the synchronization error. e system synchronization error is defined as follows: where M(t) ∈ R m×n is the scaling matrix consisting of functions (i.e., generalized function matrix).
Definition 3. e driving network (33) and response network (35) can be realized synchronization with the designed finite-time nonlinear controllers. As for scaling matrix M(t) � (m ij (t)) ∈ R m×n , if the condition can be satisfied that the each row of element cannot be 0 at a time, it will hold that 10 Complexity

LV(x(t), t, p) ≤ e T (t) Q(r) + P(r) +(Θ(r) − Ξ(r)) ⊗ I m e(t)
From Lemma 2, the system error of synchronization e i (t) will converge to a steady state with a finite-time which can be estimated by where Hence, networks (33) and (35) could be realized synchronization with finite time t 1 . As well as, the unknown parameters vector α i and β i of the system are adapted to the true value with themselves. e proof is completed.
e proposed approach is applicable to the finitetime-generalized matrix projection lag synchronization of any two double-weights Markovian jumping complex networks with uncertain parameters, stochastic perturbations, and different initial conditions. In addition, the dimensions of the Markovian jumping complex network can be identical or nonidentical.

Remark 7.
e synchronization and identification speed can be adjusted by choosing constants k i (r), h i (r), and s i (r), adequately.
Remark 8. Inequality (40) in eorem 2 is only just the sufficient but not necessary condition for double-weights Markovian jumping complex networks (33) and (35) to realize projection lag synchronization and parameter identification.

Simulation Analysis I.
e Chen system is taken as the node dynamic of drive complex network: e identifying value of the parameter vector α i is e four-dimensional Lüchaotic system will be the node dynamic of the response complex network: e identifying value of the parameter vector β i is e size of network is taken as N � 10. And the coupling configuration matrices of the network are separately given as Figures 10 and 11. Figure 10 is the topological structure of the driving system with mode 1 and mode 2. And Figure 11 is the response systems. e complex coupling network is divided into two simple coupling structures according to the theory of network splitting. Among the two subgraphs, the structure of network topology change with two modes is shown with the dotted line. e switching of the system mode is shown in Figure 12.
e other relevant parameters are as below:    Figure 16 is the system error values E(t). e conclusion of Figure 16 matches that of Figure 13. e all attest that the control scheme can realize the identification and synchronization of two double-weights Markonian jumping complex networks, effectively.

Simulation Analysis II.
In order to further verify that the synchronization criterion can be applied to the synchronization research of double-weights Markovian jumping complex networks, different driving systems are selected for simulation analysis.
e Liu system is taken as the node dynamic of the drive complex network. Among them, k � 1 and h � 4: e identifying value of the parameter vector α i is α i � (α i1 , α i2 , α i3 ) T � (10, 40, 2.5) T . e four-dimensional hyperchaotic Lüchaotic system will be the node dynamic of the response complex network.
Other relevant parameters are the same as those in Section 4.3.1. t 1 � 17.806 is calculated. e time-varying curves of the system synchronization errors e i (t)(1 ≤ i ≤ 4) are demonstrated in Figure 17. From that, one can see that the error converge to zero. Figure 18 is the identification of unknown parameters α i (i � 1, 2, 3). It is clear that the estimated parameters are approached to the value (10, 40, 2.5) T .
Under the same research background and system parameters, the control scheme proposed in this paper is compared with the control scheme without finite time to demonstrate the better performance of the proposed control scheme. Figures 19 and 20 are the simulation results under the control scheme without finite time. Figure 19 Figure 20 indicates that estimated parameter α i converge to their truth values (10, 40, 2.5) T . By comparing with the proposed control scheme in Figures 17 and 18, it is obvious that the proposed control scheme can achieve synchronization and identification more effectively and cost less time. Figure 21 is the synchronization error value E(t) with time-varying under the control scheme with finite time and without finite time.
ese illustrate that the synchronization and identification of the drive and response networks have been effectively achieved in finite time.

Conclusions
In this paper, the models of nonidentical dimension Markovian jumping complex networks with stochastic perturbations under the conditions of single weight and double weights are established, firstly. en, by means of Itô's formula and finite-time convergence theory, feedback controllers and some update laws are given under corresponding conditions to realize the synchronization, and the value of uncertain parameters in networks can be identified successfully. Finally, the simulation results are shown and the effectiveness and feasibility of the proposed method and the method can be applied to various complex dynamic networks.

Data Availability
No data were used to support this study.