Finite-time synchronization and identiﬁcation of the Markovian switching delayed network with multiple weights

This paper focuses on ﬁnite-time synchronization and parameters identiﬁcation in the Markovian switching complex delayed network with multiple weights. Considering the control cost required for network synchronization, ﬁnite-time control technique and pinning control strategy are adopted. Based on these methods, when ﬁnite-time synchronization of the network is achieved, the unknown parameters of the system can also be identiﬁed in ﬁnite time. In addition, in order to solve the problem of the network performance change caused by network topology switching, an Optimal Nodes Selection Control Strategy (ONSCS) is proposed. On the one hand, ﬁnite-time synchronization and identiﬁcation of the network can be realized faster, on the other hand, it also further reduces the energy consumption and control cost of the network. Finally, two sets of comparative numerical simulations are given to prove the superiority and applicability of the proposed ONSCS.

synchronization can be divided into outer synchronization and inner synchronization [12,13]. The outer synchronization means that the behaviour of corresponding nodes between two or more networks tends to be consistent. In [14], in order to study the outer synchronization problem of a class of Markovian switched neural networks, a unified theoretical model framework is proposed. The inner synchronization means that the behaviour of each node in a network tends to be consistent. In [15], a universal pinning controller with different power parameter ranges is designed to achieve inner synchronization of the system. At present, various control methods have been proposed to achieve inner synchronization of systems. Among them, pinning control strategy is one of the most effective methods to achieve synchronization of the Markovian switching networks, but pinning control strategy also faces two problems, one is the selection of controlled nodes; the other is the design of the controller [16,17]. Compared with the research on the latter, the research on the former is ignored by scholars. However, the problem of controlled nodes selection faced by pinning control strategy will be studied in this paper, and an Optimal Nodes Selection Control Strategy (ONSCS) is proposed to achieve synchronization of Markovian switching complex network.
The above discussed are basically single weight complex network models, but single weight Markovian switching network models can no longer meet the requirements of real networks  [18,19]. In fact, power grid, social networks, communication networks and other real networks should be modelled by Markovian switching network models with multiple weights [20]. This is mainly because the multi-weighted network models can not only describe the real network more accurately, but also better reflect the relationship between the networks. In the follow, cyber-physical power system with multi-layer coupling characteristics is taken as an example to illustrate the necessity of studying multiple weights system. The cyber-physical power system is an important link in the development of smart grid, it fully reflected the interaction mechanism between the physical process of power grid operation, information transmission process and information processing process. The model frame diagram and simplified diagram of the cyber-physical power system are shown in Figure 1, it can be seen from the figure that cyber-physical power system is a two-layer system coupled by a power network and an information network. For the system, both power transmission and information transmission are indispensable. From the above example, the advantages of the multi-weight network models in practical engineering are further reflected. In [21], the stochastic multiple weighted coupled Markovian switching networks are considered. The finite-time synchronization for a class of non-linear coupled multi-weighted Markovian switching complex networks with time-varying delay is studied in [22]. In addition, the parameters of the real systems may be unknown or uncertain, and the synchronization process of the multi-weighted Markovian switching network may be affected or even disrupted by these unknown parameters [23,24]. Therefore, it is valuable to research the multi-weighted Markovian switching network models with unknown parameters. In the real environment, due to signal transmission speed restrictions and network traffic congestion, time delays will inevitably occur in the networks. In fact, the time delays will affect the quality of the network, and even make the system oscillate, diverge and unstable. In [25], the influence of time-varying delays on neural network synchronization is considered by authors. In [26], the neural network models with multiple time-varying delays and infinite distributed delays are established. On the other hand, stochastic disturbances caused by some uncertain factors will also affect the stability of the networks [27,28]. Thus, the effects of time delays and stochastic disturbances on network should be considered [29,30]. In addition, there are very few research results about the multi-weighted Markovian switching dynamic network model with stochastic disturbances and time delays.
In many practical engineering fields, it is often more practical to realize the synchronization of systems in a finite time rather than an infinite time [31,32]. This is because finite time control technology not only allows engineers to know the specific time of system synchronization, but also saves synchronization time and reduces control costs. In addition, in the synchronization process of the system, the finite-time control method has also shown better robustness [33,34]. Based on finite-time technology and intermittent control, synchronization of complex dynamical networks with Markovian switching topologies is realized in [35]. In [36], synchronization of two classes of Markovian jump complex networks is achieved in a finite time via feedback control. Thus, the finite-time synchronization of multi-weighted Markovian switching complex delayed network model with stochastic perturbations will be investigated in this paper [37,38].
Based on the above discussion, the main contributions of this paper are as follows. In order to make the studied network model closer to the actual system, the multi-weighted Markovian switching complex delayed network model with stochastic perturbations and unknown system parameters is established. Then, considering the control cost and synchronization time problems, the pinning control method and finite time control techniques are adopted in the synchronization process. Finally, based on the controlled nodes selection problem faced by pinning control strategy, the ONSCS is proposed to achieve network synchronization. To the best of our knowledge, there are rarely studies on the multi-weighted Markovian switching complex delayed network model with stochastic perturbations; On the other hand, the selection of controlled nodes in the synchronization of Markovian switching networks is always ignored. Thus, the research has value in terms of theory and practice.
The rest of the paper is arranged as follows. In Section 2, some important conditions and network models are given. The sufficient conditions and theoretical analysis process for finite-time synchronization and identification via pinning control are given in Section 3. The sufficient conditions and theoretical analysis process for finite-time synchronization and identification via the ONSCS are given in Section 4. In Section 5, two sets of simulations are given to prove the validity of the above analysis. Finally, Section 6 concludes the paper.

MODELS AND PRELIMINARIES
In order to complete the theoretical analysis and proof in Sections 3 and 4, the mathematical model of the network is established, and some necessary assumptions and lemmas are also given in this section. Consider the following uncertain dynamical system: where x(t ) ∈ R n is the state vector; ∈ R m is the system parameter vector; f ∶ R n → R n is a non-linear vector function on x(t ). Uncertain dynamical system (1) can also be rewritten asẋ where f 1 ∶ R n → R n and f 2 ∶ R n → R n×m are continuous vector function and matrix function on x(t ), respectively. Define a right-continuous Markovian process {r (t ), t ≥ 0} in the complete probability space (Ω,  , { t } t ≥0 ,  ), which takes values in the finite state space  = {1, 2, … , m} with generator Π = ( pq ) m×m (p, q ∈  ). Define the transition probability (from the pth mode at time t to the qth mode at time t + Δt ) in the following form: = 0(Δt > 0), and pq ≥ 0 is the transition rate from mode p at time t to mode q at time t + Δt that satisfies Consider the Markovian switching dynamical delayed network model with multiple weights and stochastic perturbations, if the network has N nodes, then it can be described aṡ … , x in (t )) T ∈ R n is the state vector of node i; f i1 and f i2 have the same meanings as f 1 and f 2 in Equation (2), respectively;̂i ∈ R m is an unknown parameter vector of system; time delays satisfy condition > 0. Γ 1 ∈ R n×n and Γ 2 ∈ R n×n are inner coupling matrices; c > 0 is the coupling strength. A(r (t )) = a i j (r (t )) ∈ R N ×N and B(r (t )) = b i j (r (t )) ∈ R N ×N are the coupling configuration matrices, which represent the topological structure of the network mode r (t ) at time t − . The matrices A(r (t )) and B(r (t )) can be defined as follows: if there exists a connection between node i and node j (i ≠ j ), then a i j (r (t )) ≠ 0 (b i j (r (t )) ≠ 0); otherwise a i j (r (t )) = 0 (b i j (r (t )) = 0). The diagonal elements of matrices A and B are defined by a ii (r (t )) = − ∑ N j =1, j ≠i a i j (r (t )); b ii (r (t )) = − ∑ N j =1, j ≠i b i j (r (t )). Moreover, the noise intensity function is expressed as i (t , x i (t ), r (t )).
The isolated node of the network (5) is given bẏ Here, s(t ) can be regarded as a particular solution of the formula (6). As mentioned in Section 1, the pinning control strategy is adopted to achieve system (5) synchronization. Under this control method, a small fraction ∈ (0, 1) of the nodes in the network (5) are controlled. Here, the first l = [[ N ]] nodes are selected as controlled nodes, where [[ N ]] represents to keep the integer part. Then, the pinning controlled network can be described by where u i (t ) ∈ R n is the controller to be designed.
then the synchronization of complex dynamical network (5) will be realized in the finite time t * . Here, e i (t ) = (e i1 (t ), e i2 (t ), … , e in (t )) T represents state error of ith node in the system (5); E (⋅) represents the mathematical expectation.
Remark 1. The unknown parameter vector̂i of the system is identified, if there exists a constant t * > 0, such that and for any t ≥ t * , if we have ‖̃i‖ = ‖̂i − i ‖ ≡ 0, then the system parameter vector is achieved in the finite time t * .
In the next follow, some assumptions and lemmas needed for the proof will be given.

Assumption 1.
For the non-linear function f (⋅) ∶ R n → R n and ∀x(t ), y(t ) ∈ R n , there exists a non-negative constant and a symmetric positive matrix P such that (10) where is the system parameter vector. Assumption 2. The noise intensity function i (t , e i (t ), r (t )) satisfies the uniform Lipschitz condition and there exists a normal number i , such that
Lemma 1. ( [40]) Suppose that d 1 , d 2 , … , d n are positive numbers, c (0 < c < 1) is a positive constant, then we have Lemma 2. For any two vectors of the same dimension ∈ R n and ∈ R n , if there is a positive constant a and satisfies ‖ ‖ ≤ a, then the following inequality holds: where A is an n-dimension vector composed of a, i.e. A = (a, … , a) T ∈ R n .
Let ‖ ‖ ≤ a hold, then we have

FINITE-TIME SYNCHRONIZATION AND IDENTIFICATION OF MARKOVIAN SWITCHING NETWORK VIA PINNING CONTROL
As mentioned in Section 1, the pinning control is one of the most effective methods to achieve inner synchronization of network. This is mainly because the pinning control does not need to control all the nodes of the system, which reduces the control cost of the actual engineering. In this section, the sufficient conditions for network synchronization and identification are first given. Next, on the basis of pinning control strategy, the theoretical analysis process of system synchronization and identification will be given.
According to (6) and (5), the following synchronization error system can be obtained: In the succeeding theorem, an effective controller and updated law will be designed, so that the errors e i (t ) can gradually converge to zero and identify the unknown parameter vector. Theorem 1. Let Assumptions 1-4 hold. Based on the following designed controller and updated law, the synchronization error system (17) will tend to zero in the finite time.

is a positive definite matrix of proper dimension; D(p) is arbitrary symmetrical matrix; I is an identity
Then the synchronization and parameters identification of the system (5) will be realized in a finite time Proof. We design the Lyapunov-Krasovskii functional as According to the differential operator  ([41]), we can get Let Assumptions 1 and 2 hold, then we can get where O(q) is a positive definite matrix of suitable dimensions.
In the view of controllers (18) and Assumption 4, we can get By Lemma 1, we obtain Let Assumption 3 hold, and for Λ = ( , … , ) T ∈ R n we have Then, by Lemma 2 Combining (25) and (27), we obtain According to Lemma 1, Assumption 3, and h According to Lemma 1, we can get where D(p) = D T (p)(p ∈  ) is a symmetric matrix of suitable dimension, and has ∑ m q=1 pq D(p) = 0; e(t ) = (e 1 (t ), … , e N (t )) T ∈ R nN .
According to the sufficient conditions (19) of Theorem 1 and denote = (1 + )∕2, then we take the expectation on both sides of Equation (30) Assume that there exist a positive constant ℏ satis- Integrate both sides of inequality (32), and by solving the integral equation we have Then we can get If there exists V (t , e(t ), r (t )) ≡ 0 for any t 0 , then V (t ) will converge to zero in a finite time t * . Therefore, when t 0 = 0, the finite time t * can be estimated by where Thus, based on the traditional pinning control method and some inequality techniques, the synchronization and parameters identification of the network model (5) established in Section 2 can be achieved in a finite time t * .
Remark 2. In Theorem 1, the controlled nodes of the Markovian switching network are the same in the network synchronization and identification process, these controlled nodes may consume more control costs and energy consumption. On the basis of Theorem 1, a new controlled nodes selection strategy is proposed in Section 4 to reduce control costs. Remark 4. The speed of network synchronization and parameters identification depends on the selection of control parameters i (r (t )), k(r (t )) and h i (r (t )) in the controller (18). In addition, the inequalities (19) are sufficient conditions rather than necessary conditions to realize system synchronization and identification.

FINITE-TIME SYNCHRONIZATION AND IDENTIFICATION OF MARKOVIAN SWITCHING NETWORK VIA THE ONSCS
In the previous section, through theoretical analysis and proof, we verified that the finite-time pinning synchronization and parameters identification of multi-weighted Markovian switching complex delayed network with stochastic disturbances (5) can be realized under Theorem 1. However, compared with general complex systems, the parameters and topological structure of Markovian switching complex networks will change randomly, which will affect the performance and dynamic behaviour of the system more seriously. Therefore, in order to solve these problems, we need to adopt the fast and stable pin-ning control strategy, which can achieve network synchronization in a shorter time by controlling fewer nodes. Below, the new control strategy will be introduced to achieve this goal.
Step 2: In Markovian switching networks, the selection of the optimal controlled nodes is affected by the changes of system parameters and topology. Therefore, a selection method of optimal controlled nodes is introduced. In any time interval (t −1 , t ], the set l * can be defined as follows: where l * is a set of control nodes selected when the network topology changes; Υ 1 and Υ 2 represent the sets of nodes with fixed and variable connection modes, respectively; is defined as the screening factor and satisfies ≥ 0.7; The definitions of 1 i (t ) and 2 i (t ) are as follows: where and ′ represent the number of elements in the Υ 1 and Υ 2 sets, respectively, and satisfy + ′ = N .
Step 3: In any time interval (t −1 , t ], the network (5) average synchronization error is described as where i (t ) represents the average synchronization error of node i in interval (t −1 , t ].
Step 4: In each time interval, the optimal controlled nodes are obtained by comparing the average synchronization error of each node, and the selection strategy of controlled nodes in each time interval can be described as follows: if the number of controlled nodes is recorded as   Figure 2 is a schematic diagram of the optimal controlled nodes selection for a five-node system. We selected the nodes with the largest synchronization errors (controlled nodes 3, 5) at time t 0 , and select the nodes at time t 1 (controlled nodes 3, 4) through condition (37). According to Equation (39), compared the average synchronization errors of nodes 3, 4 and 5 to obtain the optimal controlled nodes (optimal controlled nodes 4, 5) in the interval (t 0 , t 1 ]. The selection of nodes at time t 2 also satisfied Equation (37) (controlled nodes 2, 5), compared the average synchronization errors of nodes 2, 4 and 5 to get the optimal controlled nodes (optimal controlled nodes 2, 4) in the interval (t 1 , t 2 ]. By analogy, the optimal controlled nodes for each time interval can be obtained.
Then pinning control network model in the time interval (t −1 , t ] can be described as other, (40) where x i (t ) and u i (t ) are the state variables and pinning controller of the ith node at time t , respectively.
The error equation of the ith node at time t is described as Then error system in the time interval (t −1 , t ] can be obtained other. (42) Obviously, the sum of the synchronization error system (42) = 1, 2, … Δ(Δ → ∞) converges to zero in a finite time, then the network (5) can achieve the pinning synchronization via the ONSCS. Therefore, under the action of an effective controller and adaptive laws, the finite-time synchronization and parameters identification of Markovian switching delayed network with multiple weights via the ONSCS can be solved.

Theorem 2. Design the controller and update laws as follows:
where h (1+ )∕2 i (r (t )) ≥ 1∕h ′ , control parameters i (r (t )), k(r (t )) and h i (r (t )) are positive constants, r ∈ , and if the following inequality can be satisfied: where other parameters and V (0, e(0), r (0)) have the same meaning as the parameters in formula (20).
Proof. The Lyapunov-Krasovskii function in the entire time interval is described as (21), then the Lyapunov-Krasovskii function constructed in the interval (t −1 , t ] is as follows: (e i (t )) T e i (t ) According to the differential operator  ( [41]), we can get (e i (t )) T e i (t ) In the view of controller (43), the proof process is similar to the proof of Theorem 1, so we can get where O(q) is a positive definite matrix of suitable dimension. According to Lemma 1, we can get − (e (t )) T (Ξ (p) ⊗ I n )e (t ) + (e (t )) TK e (t ) + (e (t )) T ( I N ⊗ P )e (t ) where D(p) = D T (p)(p ∈  ) is a symmetric matrix of suitable dimension, and has ∑ m q=1 pq D(p) = 0; e (t ) = (e 1 (t ), … , e N (t )) T ∈ R 3N .
According to the sufficient conditions (43) of Theorem 2, and denote = (1 + )∕2, then we take the expectation on both sides of Equation (49) Assume that there exist a positive constant ℏ satis- Integrate both sides of inequality (51) in the time interval (t −1 , t ], and by solving the integral equation we have Then, in the time interval (t −1 , t ], we have Similar to inequality (53), in the time interval (t −2 , t −1 ] we have Similarly, the corresponding inequalities in the time interval (t −3 , t −2 ], (t −4 , t −3 ], …, (t 1 , t 2 ], (t 0 , t 1 ] can also be obtained, and the + 1 inequalities can be added to get Then, when = 1, 2, … , Δ satisfies Δ → ∞, the following inequality holds: (56) If there exists V (t , e(t ), r (t )) ≡ 0 for any t 0 , then V (t ) will converge to zero in a finite time t * . Therefore, when t 0 = 0, the finite time t * can be estimated by □ Hence, the synchronization and parameters identification of Markovian switching delayed network (5) with multiple weights can be achieved via the ONSCS in a finite time t * . Remark 6. In Theorem 2, the synchronization process is divided into Δ time intervals, and l controlled nodes are selected for each time interval based on the ONSCS; The proof process of each time interval is the same as that of Theorem 1, finally each time interval is added up to obtain the whole synchronization process. Therefore, Theorem 1 is the basis of Theorem 2, and Theorem 2 is an extension of Theorem 1.
Remark 7. Under the ONSCS in this section, the finite-time synchronization of the multi-weighted Markovian switching delayed network (5) can be realized, and at the same time the unknown parameter vector̂i(t ) of the network is also identified in the finite time t * .
Remark 8. The inequalities (44) in Theorem 2 are sufficient conditions rather than necessary conditions to realize system synchronization and parameters identification via the ONSCS. In addition, the ONSCS mentioned can also be applied to other more general Markovian switching complex network models.
Remark 9. As we all know, when the network coupling strength c is sufficiently large, the pinning synchronization of the complex network can be achieved with a small number of controlled nodes. However, through the ONSCS in this section, not only the control requirements of the synchronization are improved, but also the synchronization control and parameters identification of the multi-weighted Markovian switching delayed network is achieved with weak coupling strength and fewer controlled nodes.

ILLUSTRATIVE EXAMPLES
In this section, two sets of comparative simulations will be given to prove the superiority and wide applicability of the ONSCS. First, the 30-node chaotic system is used to illustrate the superiority of the proposed ONSCS; Secondly, in order to prove the wide applicability of the proposed ONSCS, the 100node chaotic system is given; Finally, some important simulation results are obtained.

FIGURE 3
The trajectories of the chaotic system in Example 1 Example 1. In this simulation, the node dynamics of complex network is described by the chaotic system, and Figure 3 shows the trajectory of the chaotic system.
(58) The system (58) can also be rewritten as: wherêi is the system unknown parameter vector, and the true value iŝi = (̂i 1 ,̂i 2 ,̂i 3 ) T = (1, 20, 5.46) T . In order to prove the superiority of the ONSCS mentioned in Section 4, the network model, the topology switching mode and all parameters are kept consistent in the two simulations. The multi-weighted Markovian switching complex network is composed of 30 nodes, and the coupling configuration matrices of the network in two modes are given as Figure 4. Moreover, Figure 5 shows the system mode switching process. Example 1.1. In this simulation, the synchronization and parameters identification of the multi-weighted Markovian switching complex delayed network with stochastic disturbances will be realized via traditional pinning control method in a finite time. According to Theorem 1, the first three nodes of the network (5) are selected as the controlled nodes, i.e. The other network parameters will be given below: In addition, the simulation start time is determined to be t 0 = 0, and time delay  (35), we can get t * ≤ 11.0142 by simple calculation.
In Figure 6, the curves of network synchronization errors e i (t ) versus time are shown, from which one can see that e i (t ) have been converged to zero at about t ≈ 4. The identification process curves of system unknown parameter vec-tor̂i are given in Figure 7, we can clearly observe that the  (1; 20; 5.46) T at about t ≈ 10. In summary, the synchronization and identification of Markovian switching network can be achieved at about t ≈ 10, which further proved that synchronization and identification of the system can be achieved through traditional pinning control strategy. Example 1.2. In this simulation, the finite-time synchronization and parameters identification of the multi-weighted Markovian switching complex delayed network with stochastic disturbances will be realized via the ONSCS. According to ONSCS in Section 4, the controller (43) is applied at time t of each time interval (t −1 , t ], and the node with the largest average synchronization error in Equation (39) is selected as the controlled node, that is to say, only one controlled node is selected in the example.
The value of the control parameters in Theorem 2 is the same as that in Theorem 1, especially, Ξ(1) = diag(90,  the network is taken as c = 0.01. According to (45), we can get t * ≤ 9.4283 by simple calculation. In Figure 8, the curves of network synchronization errors e i (t ) versus time are shown, from which one can see that e i (t ) have been converged to zero at about t ≈ 3. The identification process curves of system unknown parameter vec-tor̂i are given in Figure 9, we can clearly observe that the unknown parameterŝi (i = 1, 2, 3) converge to their truth values (1; 20; 5.46) T at about t ≈ 7. Figure 10 shows the number of controlled nodes at time t of the time interval (t −1 , t ]. In summary, the synchronization and identification of Markovian switching network can be achieved at about t ≈ 7, which further proved that synchronization and identification of the system can be achieved through traditional pinning control strategy.
By comparing the results of Examples 1.1 and 1.2, the following conclusions can be drawn. On the one hand, under the traditional pinning control strategy and the proposed ONSCS, the synchronization and parameters identification of the network can be realized in a finite time. On the other hand, finite-time synchronization and identification of multi-weighted Markovian switching complex delayed network with stochastic perturbations can be achieved more quickly and stably under the ONSCS. In addition, based on the proposed ONSCS, the synchronization and identification of the network can be achieved under the condition of weak coupling strength and one controlled node, the energy consumption and control cost of the system are reduced. This also further illustrates the superiority of the proposed method.

Example 2.
In this simulation, the node dynamics of complex network is described by the Lorenz system, and Figure 11 shows the trajectory of the Lorenz system.
(60) The system (60) can also be rewritten as wherêi is the system unknown parameter vector, and the true value iŝi = (̂i 1 ,̂i 2 ,̂i 2 ) T = (10, 8∕3, 28) T . In order to prove the broad applicability of the ONSCS mentioned in Section 4, the above network model was adopted in this simulation. The multi-weighted Markovian switching complex network is composed of 100 nodes, and the network topology is switched in two modes, Figure 12 shows the system mode switching process.
The other system parameters are given as below: In addition, the simulation start time is determined to be t 0 = 0, and time delay In Figure 13, the curves of network synchronization errors e i (t ) versus time are shown, from which one can see that e i (t ) have been converged to zero at about t ≈ 16. The identification process curves of system unknown parameter vec-tor̂i are given in Figure 14, we can clearly observe that the unknown parameterŝi (i = 1, 2, 3) converge to their truth val- In summary, the synchronization and identification of Markovian switching network can be achieved at about t ≈ 25, which proved once again that synchronization and identification of the system can be achieved through traditional pinning control strategy.
Example 2.2. In this simulation, the finite-time synchronization and parameters identification of multi-weighted Markovian switching complex delayed network with stochastic disturbances will be realized via the ONSCS. According to the ONSCS mentioned in Section 4, the controller (43) is applied at time t of each time interval (t −1 , t ], and the node with the largest average synchronization error in Equation (39) is selected as the controlled node, that is to say, only one controlled node is selected in the example.
The value of the control parameters in Theorem 2 is the same as that in Theorem 1, especially, Ξ(1) = diag(90,  the network is taken as c = 0.01. According to (45), we can get t * ≤ 8.2217 by simple calculation. In Figure 15, the curves of network synchronization errors e i (t ) versus time are shown, from which one can see that e i (t ) have been converged to zero at about t ≈ 5. The identification process curves of system unknown parameter vector̂i are given in Figure 16, we can clearly observe that the unknown parameterŝi (i = 1, 2, 3) converge to their truth values (10; 8∕3; 28) T at about t ≈ 6. In summary, the synchronization and identification of Markovian switching network can be achieved at about t ≈ 6, which proved once again that synchronization and identification of the system can be achieved through traditional pinning control strategy.
By comparing the results of Examples 2.1 and 2.2, the following conclusions can be drawn. On the one hand, the conclusion in Examples 1 is verified again. On the other hand, it also shows that the proposed ONSCS can achieve faster synchronization and parameters identification under larger networks scale. In addition, an interesting thing has also been dis-covered, in the 100-node network, comparing the actual simulation time for network synchronization and identification by the traditional pinning control method and the proposed ONSCS, the actual simulation time required by the latter is much shorter than that of the former. This also further illustrates the wide applicability and superiority of the proposed method.
Except for the above conclusions, by comparing Examples 1 and 2, the following conclusions can be drawn. On the one hand, according to the results of Examples 1.1 and 2.1, we can find that the synchronization and parameters identification time of the network gradually increases with the increase of network nodes; On the other hand, according to the results of Examples 1.2 and 2.2, we can find that the synchronization and parameters identification time of the network will not change significantly with the increase of network nodes. According to the simulation results in this section, the correctness of the theoretical analysis in Sections 3 and 4 is proved, and the superiority and wide applicability of the proposed ONSCS are also better proved.

CONCLUSIONS
This paper proposed a new node selection method to improve the efficiency of synchronization and identification of multiweighted Markovian switching complex delayed network. First, by designing a finite-time adaptive pinning controller, unknown parameters are also identified while network synchronization is achieved. Secondly, considering that the existence of Markovian switching will change the performance of the network, and in order to reduce energy consumption and control costs, the ONSCS is proposed, which can select more important nodes for control in Markovian switching network. Finally, in order to prove the superiority and applicability of the proposed method, two sets of comparative simulation experiments are given. Moreover, by the ONSCS, the synchronization and parameters identification of the system can be realized more quickly and stably under the condition of weak coupling strength c = 0.01 and one controlled node, and the synchronization and parameters identification time of the network will not increase significantly with the increase of the network scale. In the future, we will continue to study the following aspects. Based on the model proposed in this manuscript, it is very interesting to study the non-linear coupled Markovian switching complex network models with multiple weights and time-varying delays. On the other hand, it is also challenging to extend the ONSCS proposed in this manuscript to the other complex networks synchronization.