Finite-Time Synchronization for a Class of Multiweighted Complex Networks with Markovian Switching and Time-Varying Delay

In this paper, we investigate the finite-time synchronization control for a class of nonlinear coupled multiweighted complex networks (NCMWCNs) with Markovian switching and time-varying delay analytically and quantitatively. The value of this study lies in four aspects: First, it designs the finite-time synchronization controller to make the NCMWCNs with Markovian switching and time-varying delay achieve global synchronization in finite time. Second, it derives two kinds of finite-time estimation approaches by analyzing the impact of the nonlinearity of nonlinear coupled function on synchronization dynamics and synchronization convergence time. Third, it presents the relationship between Markovian switching parameters and synchronization problems of subsystems and the overall system. Fourth, it provides some numerical examples to demonstrate the effectiveness of the theoretical results.


Introduction
In the past several decades, since the pioneer work of Watts and Strogatz [1], complex networks have received extensive concern in various fields of science and engineering [2][3][4][5]. As a matter of fact, complex networks are composed of lots of interconnected nodes, in which each node is a fundamental unit and may have specific dynamics [6][7][8][9][10][11], and exist everywhere in the real world, such as sensor networks, smartphone networks, industrial control system networks, neural networks, and communication networks [12][13][14][15][16].
erefore, the research of complex networks can help us to further comprehend the functions and effects of the realworld networks.
It is worthy to note that the applications of complex networks heavily depend on the dynamics of them, for example, passivity dynamics [17,18], synchronization dynamics [19][20][21][22], and stabilization dynamics [23,24], and hence, exploring dynamics of complex networks is necessary and significant. Especially, synchronization, as one of the most collective dynamics of complex networks, has gained widespread concern, and many valuable theoretical results have been obtained [23][24][25][26][27][28][29][30][31]. And numerous studies have been conducted to establish some feasible control methods such as event-triggered sliding mode control [32], adaptive sliding mode fault-tolerant control [33], and finite-time control [34,35], to make the addressed systems get the desired dynamics behaviors.
Furthermore, in some practical engineering fields, the desired dynamic behavior is often required to be achieved in finite time interval [36,37]. erefore, recently, more researchers have increasingly drawn attention to finite-time synchronization dynamics control problem for complex networks [38][39][40][41][42][43][44]. In [38], an asynchronous switching feedback controller based on the derived sufficient conditions was designed to realize finite-time synchronization for a class of uncertain coupled switched networks. In [39], the authors gave some sufficient conditions which ensure finite-time synchronization for a class of switched coupled neural networks with discontinuous or continuous activations. In [40], under the framework of Filippov solution, the authors studied finite-time synchronization for two classes of coupled Markovian discontinuous neural networks with mixed delays. It should be noted that in [38][39][40][41][42]44], the proposed complex networks were entangled by linear coupling. In fact, many physical networks are entangled by nonlinear coupling. For example, in Kuramoto oscillator network, there exist nonlinear interactions among different oscillators [45]. In electrical gird dynamical networks, different electrical elements are coupled by nonlinear interactions [46]. Obviously, linear function is a special case of nonlinear function, and nonlinear coupled complex networks are more general and their synchronization dynamics may be more complicated and unpredictable [2,6,25,[47][48][49][50][51]. In [47], under finite-time pinning control, the authors investigated the finite-time cluster synchronization problem of nonlinearly coupled and discontinuous Lur'e networks. In [48], the authors derived some sufficient conditions to ensure finite-time synchronization of the considered linear coupled and nonlinear coupled complex networks. Unfortunately, there are merely a few results on finite-time synchronization control of nonlinear coupled complex networks. And the impact of the nonlinearity of the coupled function on synchronization dynamics cannot be reflected by the derived settling finite time t * [47,48]. is is because that the finite-time control method cannot process how nonlinearity of nonlinear coupled function impacts synchronization dynamics of the addressed complex networks. So, there was no function relationship between t * and the nonlinearity of the coupled function. Actually, every element attribute in a dynamical system may impact synchronization dynamics and synchronization convergence time, which means that nonlinearity of the coupled function, as one of the coupling function attributes, may affect synchronization dynamics and synchronization convergence time of the considered nonlinear coupled complex networks. erefore, it is important to further design more reasonable and feasible finite-time control method to deal with the existing issue.
It should be mentioned that most of existing research results on synchronization control problems of complex networks focused on synchronization dynamics of complex networks with single weight (for example, [42,[44][45][46][47][49][50][51][52] and references therein). But in reality, for example, people can contact each other by mail, telephone, MSN, e-mail, and so on; suppose every piece of contact information is of different weight, so human connection network is a complex network with multiweights, where the nodes are connected by more than one weight [53]. Indeed, this is a wellestablished way of introducing environmental noise into dynamic networks. Recently, synchronization dynamics of complex networks with multiweights have increasingly attracted interests [17, 18, 31, 34, 41-43, 48, 52-58]. Of them, Huang et al. [17] established several finite-time passivity criteria for several classes of linear coupled MWCNs (LCMWNCs). Qiu et al. [41] proposed finite-time synchronization control methods to make the addressed LCMWNCs. He et al. [52] considered four classes of LCMWNCs and investigated the global synchronization of them. Particularly, in [48,55], the authors studied the finitetime synchronization control of nonlinear coupled MWCNs (NCMWCNs) and analyzed how the nonlinearity of nonlinear coupled function impacts synchronization dynamics and synchronization convergence time.
Motivated by the above discussion and analysis, in this paper, we will focus on the finite-time synchronization control for a class of NCMWCNs with Markovian switching and time-varying delay and propose that the finite-time estimation approaches based on the designed controllers can reflect how nonlinearity of nonlinear coupled function impacts synchronization dynamics and synchronization convergence time of the addressed network. e rest of this paper is organized as follows. In Section 2, we present the model derivations and preliminaries. In Section 3, we provide the main results of the present paper. In Section 4, we give some numerical results to show the complicated dynamics of the system. In Section 5, we provide a brief discussion and summarize our main results.

Problem Description and Preliminaries
For any vector x(t) ∈ R n and matrix A ∈ R n×m , we denote the following: where λ max (A T A) denotes the maximal eigenvalue of A T A and T represents the matrix transposition. diag(·) represents a block diagonal matrix. e number N represents a positive integer. e Kronecker product of matrices A ∈ R m×n and B ∈ R M×N is a matrix in R mM×nN denoted as A ⊗ B. I n ∈ R n×n means an n-dimensional identity matrix. Let (Ω, F, F t t≥0 , P) be a complete probability space with a filtration F t t≥0 satisfying the usual conditions (i.e., the filtration contains all P-null and is right continuous). E(x) means the expectation of the random variable x. Consider the following NCMWCNs with Markovian switching and time-varying delay: where . . , f(y in (t))] T ∈ R n denotes the activation function of the i th node y i (t), h(·): R n ⟶ R n represents nonlinear coupled function, A ∈ R n×n , a k > 0 and a k > 0(k � 1, 2, . . . , m) are coupled strength, G k (r(t)) � (g k ij (r(t))) N×N stands for the k th outer-coupled weight matrix, Γ k > 0 and Γ k > 0 are the inner-coupled matrix, τ(t) is the coupled time-varying delay, and r(t) is a right-continuous Markov chain with known transition rate on the probability space (Ω, F, F t t≥0 , P) taking values in a finite 2 Complexity where Δt > 0, lim Δt⟶0 (o(Δt)/Δt) � 0, δ pq > 0 (∀p ≠ q) is the transition rate from mode p at time t to mode q at time t + Δt, and δ pp � − s q�1,p ≠ q δ pq < 0. For notation simplicity, we denote G k (r(t)), g k ij (r(t)), and r(t) as G r k , g k,r ij , and r, respectively.

Remark 2.
In [48,55], the authors investigated finite-time synchronization control of NCMWCNs and analyzed how the nonlinearity of nonlinear coupled function impacts synchronization dynamics and synchronization convergence time. However, in [48], the derived settling finite-time t * based on the designed finite-time control scheme cannot reflect how the nonlinearity of nonlinear coupled function affects synchronization dynamics in finite time and synchronization convergence time. In [55], though the derived settling finite time t * can reflect the impact of the nonlinearity of nonlinear coupled function on synchronization dynamics in finite time and synchronization convergence time of the proposed NCMWCNs with switching topology, the designed finite-time control method cannot process time-varying delay. at is, in [55], if time-varying delay is considered into the addressed NCMWCNs with switching topology, the derived finite-time control method is invalid.
In the present paper, we will establish the finite-time control method which can effectively process time-varying delay of the addressed network. Furthermore, we will show that the impact of synchronization dynamics in finite time and synchronization convergence time can be reflected by the obtained settling finite time t * .
Definition 2 (see [47,48]). e nonlinearity of h(·) is defined as follows: where x, y ∈ R n . In order to study the finite-time synchronization control for NCMWCNs with Markovian switching and time-varying delay, the following assumptions are needed.

Remark 5.
In the finite-time controller u i (t, r) (6), there is a sign function sgn. It is well known that traditional finite-time control techniques are based on sliding mode controllers, which utilize sign function and give rise to the phenomenon of chattering. How can we avoid this phenomenon in (6)? From [6][7][8][9], chatting will occur when the control in the addressed system adopts switching function. In [6], although the sign function in the switching control term was used, the switching control term can be softened to be a smooth signal by using low-pass filter technique. In [7], Tang addressed that some "smooth" functions must be used instead of the sign function in order to eliminate chatting of the sliding mode control system. Despite there is the sign function in the controller (6), the phenomenon of chatting cannot occur because the switching control term β(χ(h)) η k,r i Q (α− 1)/2 diag(sgn(e i (t)))|e i (t)| α is a smooth function when e i (t) > 0 and e i (t) < 0. e analysis is as follows: Assume that e i (t) satisfies ‖lim △⟶0 (e i (t + △) − e i (t))‖ � C i and C i > 0; then, ‖(de i (t)/ dt)‖ � ‖lim △⟶0 e i (t + △) − e i (t)/ △‖ � ‖lim △⟶0 C i /△‖ � +∞. Because functions F(e i (t)) and H(e j (t − τ(t))) satisfy Assumption 1, these two functions are bounded. us, combining (5), we can know Obviously, this is wrong. In order to make ‖(de i (t)/dt)‖ bounded, we have ‖( is a smooth function. erefore, e i (t) α (in the case of e i (t) > 0 and sgn(e i (t)) � 1) and − e i (t) α (in the case of e i (t) < 0 and sgn(e i (t)) � − 1) are smooth functions. us, when e i (t) > 0 and e i (t) < 0, we can conclude that function. Simply speaking, in the finite-time controller u i (t, r) (6), there is no phenomenon of chattering.

Main Results
In this section, we will focus on the sufficient conditions for ensuring that network (2) with controller (6) is finite-time synchronized. Furthermore, based on the designed controller (6), we will refine more feasible controllers. (6) can achieve synchronization within finite-time t * if Assumptions 1 and 2 and the following conditions hold:

Theorem 1. Network (2) with controller
(ii) e following inequalities are satisfied: Proof. Construct a Lyapunov functional for network (2) with controller (6) as follows: (2), we can obtain the closed-loop system of error system (5) as follows: en, from Lemma 1, we can compute the derivative LV(e(t), t, p) along the trajectory of closed-loop system (17) as follows: Complexity 5 From Assumption 2, we have By Lemma 3 and Assumption 1, we obtain where 6 Complexity Combining Lemma 2 and inequalities (25) and (26), we can obtain where η � min η 1 , . . . , η s . Substituting (19)- (24) and (27) into (18), we can get where 8 Complexity us, taking the expectation on both sides of (28) and using condition (II) of eorem 1, we have where r is a right-continuous Markov chain with known transition rate δ pq which is located in formula (3) above.

Remark 7. How to eliminate the synchronization error e(t)?
Actually, if the synchronization error e(t) is eliminated, then e(t) � 0. According to the proof of eorem 1, we can use the following four steps to derive the synchronization error e(t): Step 2: by using generalised Itô formula, LV(e(t), t, p) is derived (iii) Step 3: some inequality techniques are utilized to make hold, where r � 1, 2, . . . , s, β(χ(h)) > 0, m > 0, and η > 0. (iv) Step 4: according to finite-time stability theory given in Lemma 4, the settling finite time t * can be obtained.

Remark 8.
In the design procedure of controller (6), there exist some constraints which include ι k,r i > 0, η k,r i > 0, θ and k i e i (s)ds � 0 and u i (t, r) � 0. It should be noted that the constraints above in controller (6) are necessary. If not, eorem 1 may not hold. For example, in order to use Lemma 4 to derive the settling finite time t * , there must be 0 < α < 1. If the parameter α does not satisfy 0 < α < 1, it is obvious that t * located in inequality (36) cannot be derived by Lemma 4. In addition, according to eorem 1, the control parameters of controller (6) can be chosen and designed. For instance, for the given network (2), by choosing the control parameters ι k,r i > 0, η k,r i > 0, θ r > 0, Q � diag(q 1 , q 2 , . . . , q n ) > 0, F k i ∈ R n×n > 0, 0 ≤ ρ < 1, and r � 1, 2, . . . , s, it is easy to realize conditions (I) and (II) of eorem 1. Furthermore, it can also be seen that for network (2), there exist many solutions of controller (6) designed by eorem 1 and these solutions can make conditions (I)-(III) hold.

Remark 11.
e nonlinearity of nonlinear coupling function may impact synchronization dynamics and synchronization convergence time of the considered nonlinear coupled complex network [2,48,55]. As discussed in Section 1, the nonlinearity of nonlinear coupled function h(·) in network (2) will make χ(h) become more complex and lead to the following two questions: (i) If the nonlinearity of h(·) is more serious, does synchronization dynamics in finite time for network (2) become poorer or does synchronization convergence time of network (2) become longer? (ii) How can we use the settling finite time t * to reflect the impact of the nonlinearity of h(·) on finite-time synchronization dynamics and synchronization convergence time of network (2)?
It is a pity that controller (6) designed by eorem 1 cannot answer the two questions above. Next, we give the answer to the two questions above in Corollaries 1 and 2, respectively. 10 Complexity − τ(t))), and G r k > 0 or G r k < 0, with the increase of χ(h) of nonlinear coupled function h(·) in network (2), synchronization dynamics of network (2) with controller (6) within finite time t * C1 becomes poorer and synchronization convergence time of network (2) with controller (6) becomes longer.
us, under eorem 1, we can analyze the impact of χ(h) of h(·) on finite-time synchronization dynamics and synchronization convergence time of network (2) with controller (6).

Proof.
e proof is similar to that of Corollary 1, and hence, we omit it here. □ Remark 12. From Corollaries 1 and 2, we can observe that with the increase of the nonlinearity χ(h) of nonlinear coupled function h(·) in network (2), synchronization dynamic of network (2) in Corollaries 1 and 2 become poorer and better, respectively. Meanwhile, if χ(h) increases, synchronization convergence time and the derived settling finite time t * C1 and t * C2 of network (2) with controller (6) in Corollaries 1 and 2, respectively, become longer and shorter. It is obvious that compared with t * , t * C1 and t * C2 can more effectively reflect and accurately describe the impact of χ(h) of h(·) on finite-time synchronization dynamics and synchronization convergence time of network (2) with controller (6).
Besides this, from the proofs of Corollaries 1 and 2, it can also be obtained that the impact of the nonlinearity of nonlinear coupled function h(·) on synchronization dynamics and synchronization convergence time of network (2) with controller (6) is not only related to χ(h) of h(·) but also closely connected with the synchronization state s(t) and the initial conditions. Remark 13. Due to some factors such as limited communications and environmental changes, parameter switching in a dynamic system is usually inevitable [63,64]. Parameter switching may add some interesting dynamic behaviors to a dynamic system, which reveals that it is essential to investigate dynamic problems of systems with switching parameters. During the past decades, many researchers began to explore dynamic problems of some classes of systems with switching parameters [10,[65][66][67][68][69][70][71][72][73]. For instance, based on the Lyapunov function method and inequality technology, Mao et al. [69] studied the stability problem of switched continuous-time systems with all subsystems unstable. In [70], Xu et al. derived a sufficient condition to ensure global synchronization of a class of complex networks with switched adjacent matrices. Regrettably, up to now, although a great deal of valuable and meaningful results on dynamic problems for some classes of systems with switching parameters have been obtained [10,[63][64][65][66][67][68][69][70][71][72][73], to the best of our knowledge, there are still no literature studies to discuss the following two problems: (i) If the overall system consists of a subsystem achieving synchronization within finite time t 1 and subsystems not achieving synchronization, can the overall system achieve synchronization? (ii) If the overall system consists of a subsystem achieving synchronization within finite time t 1 and subsystems achieving synchronization within finite time t 2 , can the overall system achieve finite-time synchronization? If yes, what is the finite time?
In the next two remarks, we will focus on the two questions above.

Remark 14.
In network (2), the switching signal r(t) satisfying the rule given by formula (3) has s finite state, which shows that overall network (2) has s subsystems. Assume that the pth subsystem with controller (6) can achieve synchronization within finite time t * and the qth subsystem without controller (6) cannot achieve synchronization, where p � 1 and q � 2, . . . , s. In this situation, can overall network (2) achieve the synchronization?
Markovian switching usually models random abrupt variations which are often caused by random failures and repairs of the components [59]. is means that network (2) with Markovian switching parameter g k ij (r(t)) which consists of s subsystems actually originates from a subsystem of network (2). For example, due to noise perturbations and some other elements [59], the parameter g k ij (r(t 1 )) of the 1st subsystem is randomly switched to g k ij (r(t 2 )). us, the 2nd subsystem is produced. By the similar rule, the 3rd. . . and the sth subsystem are also produced. erefore, the overall network (2) with a Markovian switching parameter actually is one dynamical system and the number of the solution y ij (t) of the overall network (2) is N × n, instead of s × N × n, where i � 1, 2, . . . , N and j � 1, 2, . . . , n.
Moreover, the solution y ij (t) of the overall network (2) with or without controller (6) is a smooth function.
From (4)-(6), one can obtain that e i (t) is also a smooth function. Combining (16) with (17), one can obtain that E[LV(e(t), t, r)] is also a smooth function. erefore, though LV(e(t), t, r) is divided into by the switching sequence S, because E[LV(e(t), t, r)] is a smooth function, then must be a smooth function. It is clear that en, we can conclude that although network (2) is composed by s subsystems which are coupled by Markovian switching parameter g k ij (r(t)), the overall network (2) is actually a dynamical system. is reflects that in network (2), there does not exist independent subsystems.
And if subsystems in network (2) are independent, every subsystem may exhibit different synchronization dynamical behaviors. All these testify that if every subsystem of network (2) is independent, it may be realized that the pth subsystem with controller (6) can achieve synchronization within finite time t * and the qth subsystem without controller (6) cannot achieve synchronization, where p � 1 and q � 2, . . . , s. Otherwise, if every subsystem of network (2) is coupled by Markovian switching parameter g k ij (r(t)), the above addressed synchronization dynamic behaviors of subsystems in network (2) cannot be realized. erefore, every subsystem of overall network (2) is independent.
Similar to the proof of eorem 1, we can get LV q (e(t), t) < 0 and LV q (e(t), t) > 0, where LV q (e(t), t) and LV q (e(t), t) are with respect to the pth independent subsystem with controller (6) and the qth independent subsystem without controller (6), respectively. Here, LV q (e(t), t) and LV q (e(t), t) replace LV p (e(t), t, p) and LV q (e(t), t). Because in the independent pth subsystem with controller (6) and the independent qth subsystem without controller (6), there is no Markovian switching phenomenon; thus, if LV q (e(t), t) ≤ 0 and LV q (e(t), t) > 0, then E[LV(e(t), t, r)] ≤ 0, the overall network (2) can achieve global synchronization within infinite time interval or finite time interval.
Remark 15. If the pth subsystem with controller (6) can achieve synchronization within finite time t * (1) and the qth subsystem with controller (6) can achieve synchronization within finite time t * (2) , can overall network (2) achieve synchronization? If yes, what type of finite-time synchronization is the obtained finite-time synchronization? By the similar analysis of Remark 14, one can get the following

Numerical Examples
In order to illustrate the effectiveness of the obtained results, this section gives four numerical examples. And synchronization total error of the addressed network is defined as e(t) � N i�1 n j�1 e ij (t), where e ij (t) is synchronization error of the addressed network. For a given rate transition matrix, the right-continuous Markov chain r(t) can be generated. As an example, we adopt S � 1, 2, 3 { } and the rate transition matrix as follows: Example 1. Consider the following network with sixty coupling nodes: where i � 1, 2, . . . , 60, e coupling matrix G r k is as follows: By using the steps in Remark 14, we can design controller (6): (i) Let υ p � 1 and Q � I 2 , where p � 1, 2, 3. (ii) Combining Assumptions 1 and 2, f(·), and h(·), we have L � L � 1.2, ψ � I 2 , ψ p � I 120 , and τ M ≐ ρ � 0.1. (iii) From inequality (15) of condition (II) of eorem 1, we obtain that θ p � 1 and F k � 1.5 ⊗ I 120 .
us, the design of controller (6) is completed and the finite time t * is successfully estimated.
Actually, from Definition 1 and the derived t * in condition (III) of eorem 1, it can be seen that finite time t * heavily depends on the initial state value y(0) of network (58). is shows that the initial value of the addressed system can affect the simulation result.
In the simulation results of Figure 1, the trajectories marked by red and blue are with respect to the initial conditions y(0) and y (1) (0), respectively. From Figure 1, we can see that within finite time t * � 2.74 and t * (1) � 3.04, synchronization errors e ij (t) and e (1) ij (t) and synchronization total errors e(t) and e (1) (t) gradually tend to zero, where i � 1, 2, . . . , 60 and j � 1, 2. is means that network (58) with designed controller (6) can achieve global synchronization in finite time t * and t * (1) , respectively. Moreover, compared with the simulation results in Figure 1, we can find that under the same control rule, if the initial state value of network (58) becomes larger, synchronization convergence time of network (58) becomes longer. e obtained t * and t * (1) can also reflect the results.
Example 4. In network (60) above, s � 3. erefore, network (52) includes three subsystems. Assume that the 1st subsystem with controller (6) can achieve synchronization in finite time t * (1) and the other two subsystems without controller (6) cannot achieve synchronization, where r � 1.

Complexity
According to the analysis of Remark 14, we know the following: (a) If LV p (e(t), t) ≤ 0 and LV q (e(t), t) > 0 can make E[LV(e((t)), t, r)] ≤ 0 hold, where p � 1, q � 2, 3, and 1 , t s+1 , 1 overall network (60) can achieve synchronization (b) In order to make every subsystem of network (60) have different synchronization dynamic behaviors, every subsystem is an independent subsystem (c) In every subsystem, there is no Markovian switching phenomenon (d) If s subsystems are coupled by Markovian switching parameter g k ij (r(t)), the overall network (60) is actually one dynamical system (e) Although there exists Markovian switching parameter g k ij (r(t)), synchronization error e ij (t) of the error system of the overall network (60) is a smooth function, where i � 1, 2, . . . , 18 and j � 1, 2 Actually, according to the principles above, it is very difficult to design controller (6) to make addressed overall network (60) achieve synchronization. Next, we will focus on the synchronization of the overall network (60) by the following steps: Step 1: combining LV p (e(t), t) ≤ 0 and the proof of eorem 1, we have where p � 1 and us, we can design Ξ k 1 , θ 1 , and F k to make the 1st subsystem of overall network (60) achieve global synchronization.

Concluding Remarks
In this paper, we mainly focus on the impact of the nonlinearity of nonlinear coupled function on finite-time synchronization dynamics and synchronization convergence time for a class of NCMWCNs with Markovian switching and time-varying delay. In order to make the addressed network achieve global synchronization in finite time, we design a kind of finite-time synchronization controller. And based on the finite-time synchronization controller, we derive two kinds of finite-time estimation approaches and find that the impact of synchronization dynamics on finite time and synchronization convergence time can be reflected by the obtained settling finite time t * . e proposed finitetime estimation methods can reflect how the nonlinearity of nonlinear coupled function impacts synchronization dynamics and synchronization convergence time of the addressed network. Furthermore, we investigate the relationship between Markovian switching parameters and synchronization problems of subsystems and the overall system.
It is worthy to note that the obtained finite-time estimation methods heavily depend on the initial conditions of the NCMWCNs with Markovian switching and timevarying delay. is shows that if the initial conditions of the addressed system are not accurately obtained, the proposed finite-time synchronization control methods are invalid.
Moreover, in NCMWCNs (2), nonlinear coupled function h(·) must satisfy the Lipschitz condition. Compared with sector-bounded nonlinearity condition, Lipschitz condition is a special case [74]. If h(·) satisfies sector-bounded nonlinearity condition, how can we design finite-time and fixed-time synchronization controllers of NCMWCNs with Markovian switching and timevarying delay? Furthermore, the fixed-time control can effectively overcome the faultiness [75,76]. erefore, it is necessary to explore fixed-time synchronization control and fixed-time synchronization dynamic for some classes of NCMWCNs with Markovian switching and timevarying delay. How can we analyze the impact of nonlinearity of nonlinear coupled function on finite-time and fixed-time synchronization dynamics and synchronization convergence time? In addition, if h(·) is a discontinuous right-hand function, how can we investigate the related finite-time and fixed-time problems of nonlinear coupled delayed multiweighted complex networks with Markovian switching? ese are desirable in future studies.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.