it has long been understood that many physical , social , biological , and technological networks are modeled by a graph with non - trivial topological features . in this model
, every node is an individual element of the whole system with certain pattern of connections , in which connections between each pair of nodes are neither entirely regular nor entirely random [ 13 ] .
these features do not occur in the mathematical models of networks that have been studied in the past , such as lattices or random graphs , but they do truly exist in nature . at present , derivatives of network science have been successfully applied to the analysis of metabolic and genetic regulatory networks , the design of robust and scalable communication networks both wired and wireless , the development of vaccination strategies for the control of disease , and a broad range of other practical issues .
secure communication [ 4 , 5 ] , parallel image processing , and chemical reaction implemented by coupled chaotic systems have been an active research field during the last two decades .
synchronization issues are fundamentally important for any dynamical networks , and there is no exception for complex networks . as a consequence , theory and methods for synchronization of different families of complex networks have been extensively studied by many researchers ( such as [ 715 ] , and references therein ) .
the results on complete synchronization , phase synchronization , space lag synchronization , and cluster reflect the ideas of [ 1719 ] .
the improvement on different regimes of synchronization of discrete complex networks is abstracted from papers authored by [ 9 , 20 ] .
some general cases of synchronization of complex networks with switching topology can be found in the literatures of [ 15 , 21 ] . from another angle ,
adaptive synchronization , impulsive synchronization scheme , and pining control synchronization have been considered by authors in [ 1012 , 22 , 2327 ] .  
in addition , systems with delays and multiplicative noises have been studied by considering h - infinity method in   and .
however , all previous studies have limitations on synchronizing a state delayed discrete complex network . as known , connections between each pair of nodes in a complex network always change .
the mutable topology can pose a significant threat on the global dynamical property of the whole networks .
additionally , the large time - varying delays may exist in switching topology which means data communication may occur in different sub - networks .
if the mutable topology and a large time - varying delay occur simultaneously in the discrete complex networks , it would be difficult to employ previous synchronization control schemes .
impulsive control has been successfully used to stabilize and synchronize dynamical systems , for examples , [ 3037 ] . and
impulsive control technique could be an efficient method when a discrete change behavior is needed .
, we proposed an impulsive synchronization scheme for a state delayed discrete complex network with switching topology .
for this control scheme , we consider that the impulsive control signal is designed to be input into all of nodes . meanwhile ,
every time instant of impulsive effects occur precisely at the time instant of switching happening . in other words , when the complex networks switch its state at every instant of time , there is no delay between the controller functioning . the paper is organized as follows .
section 2 presents some mathematical preliminaries needed in this work , and a generalized mathematical model for delayed discrete complex networks with switching topology . the main theorem for global synchronization of this type of discrete complex networks
4 , a small - world network with 3 sub - networks involving 30 nodes is constructed to illustrate the effectiveness of our result . section 5 concludes the paper .
first , we need to introduce some notations and definitions for the sake of exploring our main results .
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				 \begin{document}$${\mathbb{n}=\{0,1,2,\ldots\}}$$\end{document } , and , for certain positive integer  , we let \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathbb{z}_{-\tau}=\{-\tau,-\tau+1,\ldots,0\}}$$\end{document}. the family of n linearly coupled discrete complex networks , consisting of time delay with respect to its system state and the switched topology , can be described by 1\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } x_i(k+1)&=a x_i(k)+ b f(x_i(k))+ d f(x_i(k-\tau(k)))+i(k ) \\ & + \sum\limits_{j=1}^nc_{ij,\sigma(k)}\upgamma x_j(k-\tau(k ) ) , \quad i=1,2,\ldots , n , k\in { \mathbb{n } } \end{aligned}$$\end{document}2\documentclass[12pt]{minimal } 
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				 \begin{document}$$x_{ik_0}=\phi(\theta),\quad \theta \in { \mathbb{z}}_{-\tau},$$\end{document}where \documentclass[12pt]{minimal } 
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				 \begin{document}$${x_i(k)=(x_{i,1},x_{i,2},\ldots , x_{i , n})\ ; \in \mathbb{r}^n}$$\end{document } represents the state vector of the i - th node at every instant of time k and n denotes the number of nodes affiliated to each sub - network .
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				 \begin{document}$${{\bf a } \in \mathbb{r}^{n\times n},\;{\bf b } \in \mathbb{r}^{n\times n}}$$\end{document } and \documentclass[12pt]{minimal } 
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				 \begin{document}$${{\bf d } \in \mathbb{r}^{n\times n}}$$\end{document } are known real matrices .
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				 \begin{document}$${\bf f(x_i(k))}=({\bf f_1(x_{i,1}(k))},{\bf f_2(x_{i,2}(k))},\;\ldots,{\bf f_n(x_{i , n}(k))})^{t}$$\end{document } and \documentclass[12pt]{minimal } 
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				 \begin{document}$${{\bf f}(\bullet):\mathbb{r}^n \longrightarrow \mathbb{r}^n}$$\end{document } is a smooth nonlinear vector - valued functions .
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				 \begin{document}$${\bf i(k)}=(i_1(k),i_2(k),\;\ldots , i_n(k))^{t}$$\end{document } is a n - dimensional vector from external input .
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				 \begin{document}$${\bf s}=\{s_1,s_2,\ldots , s_r\}$$\end{document}. let the switching function be denoted by \documentclass[12pt]{minimal } 
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				 \begin{document}$${\sigma(k):\mathbb{n } \longrightarrow { \bf s}}$$\end{document } , which is the switching signal from sudden changing of system dynamic without jumps in the state x at any switching instant .
specifically , we consider that it is a piecewise constant function and continuous from the right , indicating certain active sub - system regime , at every instant of time k the index \documentclass[12pt]{minimal } 
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				 \begin{document}$$\sigma(k)=s_k \in { \bf s}$$\end{document } ; meanwhile , let the switching instants of  be denoted by \documentclass[12pt]{minimal } 
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				 \begin{document}$$k_{m , x}\;(m=1,2,\ldots)$$\end{document } and let k0,x :  =  0 .
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				 \begin{document}$${c_{s_k}=(c_{ij , s_k})\in \mathbb{z}^{n\times n}}$$\end{document } represents the outer coupling configuration symmetric matrix defined as follows : for each active sub - system regime sk , if there is a connection from node j to node i ( j    i ) , then \documentclass[12pt]{minimal } 
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				 \begin{document}$$c_{ij , s_k}=c_{ji , s_k}>0$$\end{document } ; otherwise \documentclass[12pt]{minimal } 
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				 \begin{document}$$c_{ij , s_k}=c_{ji , s_k}=0$$\end{document}. assume that 3\documentclass[12pt]{minimal } 
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				 \begin{document}$$c_{ii , s_k}=-\sum\limits_{j=1,j\neq i}^{n}c_{ij , s_k}=-\sum\limits_{j=1,j\neq i}^{n}c_{ji , s_k } , i\in n , s_k\in { \bf s}.$$\end{document } the notation \documentclass[12pt]{minimal } 
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				 \begin{document}$${\upgamma\in \mathbb{r}^{n\times n}}$$\end{document } represents the diagonal inner coupling matrix between two connected nodes .
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				 \begin{document}$${\tau(k)\in \mathbb{z}_{-\tau}}$$\end{document}. \documentclass[12pt]{minimal } 
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				 \begin{document}$${\phi(\bullet):\mathbb{z}_{-\tau}\longrightarrow \mathbb{r}^{n\times n}}$$\end{document } is smooth everywhere except at a finite number of points .
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				 \begin{document}$${\|\phi(\theta)\|_{\tau}=\sup_{\theta \in \mathbb{z}_{-\tau}}\{\|\phi(\theta)\|\}}$$\end{document}. in order to design an impulsive control scheme to synchronize system ( 1 ) , we consider the evolutionary state is abruptly jumping at every impulsive instant of time ku from its open - loop state , which can be formulated by 
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				 \begin{document}$$\updelta x_i(k_{m , u})=j_u x_{i}^{*}(k_{m , u}),\quad m=1,2,\ldots { \mathbb{n}}$$\end{document}where xi*(km , u ) stands for the primal state at time instant km , u without impulsive jump . as usual , every impulsive instant of time kl , u satisfies \documentclass[12pt]{minimal } 
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				 \begin{document}$$0=k_{0,u}<k_{1,u}<k_{2,u}<\cdots < k_{m , u}<k_{m+1,u}<\ldots$$\end{document } and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\lim\nolimits_{m\rightarrow \infty}k_{m , u}=\infty;\;j_u:\mathbb{r}^{n}\rightarrow \mathbb{r}^{n}(m=1,2,\ldots)}$$\end{document } represents the impulsive jump strength . therefore , at every impulsive instant of time km , u , the coupled states xi(k )    xj(k ) between connected node i and j can be described by 
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				 \begin{document}$$\begin{aligned } x_i(k_{m , u})-x_j(k_{m , u})&=x_{i}^{*}(k_{m , u})-x^{*}_{j}(k_{m , u } ) \\ & \quad+j_u[x_{i}^{*}(k_{m , u})-x^{*}_{j}(k_{m , u } ) ] .
\end{aligned}$$\end{document } intuitively , a family of impulsive controller can be designed as 
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				 \begin{document}$$u_i(k , x_i(k))=\sum\limits_{u=1}^{\infty}\delta(k - k_{m , u})j_u(x_{i}^{*}(k_{m , u } ) ) , \quad m=1,2,\ldots { \mathbb{n}},$$\end{document}where ui(k ,  xi(k ) ) represents a class of impulsive controller at each instant of time \documentclass[12pt]{minimal } 
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				 \begin{document}$$k_{m , u};\;\delta(\bullet)$$\end{document } denotes the dirac discrete - time function .
correspondingly , by virtue of impulsive controller , the closed - loop discrete complex networks can be derived by the following form : 
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				 \begin{document}$$\begin{aligned } x_i(k+1)&=a x_i(k)+ b f(x_i(k))+ d f(x_i(k-\tau(k)))+i(k ) \\ & \quad + \sum\limits_{j=1}^nc_{ij,\sigma(k)}\upgamma x_j(k-\tau(k))+u_i(k+1,x_i^*(k+1 ) ) , \\ & \quad i=1,2,\ldots , n. \end{aligned}$$\end{document}8\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } u_i(k+1,x_i^*(k+1))&=\sum\limits_{u=1}^{\infty}\delta(k+1-k_{m , u})j_u[a x_i(k ) \\ & \quad + b f(x_i(k))+ d f(x_i(k-\tau(k ) ) ) + i(k ) \\ & + \sum\limits_{j=1}^nc_{ij,\sigma(k)}\upgamma x_j(k-\tau(k ) ) ] , \quad m=1,2,\ldots , { \mathbb{n}}. \end{aligned}$$\end{document } for each nonlinear function \documentclass[12pt]{minimal } 
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				 \begin{document}$${\bf f_i(\bullet)}\;(i=1,2,\ldots , n)$$\end{document } , suppose that it is globally lipschitz continues function and satisfies 
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				 \begin{document}$$\|f_i(x_1)-f_i(x_2)\| \leq \hat l_i\|x_1-x_2\|,\quad i=1,2,\ldots , n , \forall x_1,x_2 \in { \mathbb{r}},$$\end{document}where \documentclass[12pt]{minimal } 
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				 \begin{document}$$\hat l_i$$\end{document } is certain positive constant .
the system of the impulsive controlled discrete complex networks ( 7 ) is said to be globally exponentially synchronized , if for any initial condition \documentclass[12pt]{minimal } 
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				 \begin{document}$${\phi ( \bullet):\mathbb{z}_{-\tau}\rightarrow\mathbb{r}^{n\times n}}$$\end{document } , and there exist two positive constants  and m0    1 such that 
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				 \begin{document}$$\|x_i(k)-x_j(k)\|\leq m_0 \text{e}^{-\lambda(k - k_0)},\quad 1\leq i \leq j\leq n$$\end{document}holds for all k  >  k0 .  
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				 \begin{document}$${{\bf w}=(w_{ij})_{n\times
n},\;{\bf p } \in \mathbb{r}^{n\times n},\;{\bf x}=(x_1,x_2,\;\ldots , x_n)^{t}}$$\end{document}and\documentclass[12pt]{minimal } 
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				 \begin{document}$${\bf y}=(y_1,y_2,\ldots , y_n)^{t}$$\end{document}with\documentclass[12pt]{minimal } 
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				 \begin{document}$${x_k,\;y_k \in \mathbb{r}^{n}\;(k=1,2,\ldots , n)}$$\end{document}. if w  =  w and each row sum of w is zero , then 
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				 \begin{document}$${\bf x}^{{\bf t}}({\bf w}\otimes { \bf p}){\bf y}=-\sum\limits_{i=1}^{n-1}\sum\limits_{j = i+1}^{n}w_{ij}(x_i - x_j)^{t}p(y_i - y_j).$$\end{document }
when the impulsive controller can be functioning simultaneously at the state of discrete complex networks switching signal , the equivalent impulsive controlled system is rewritten by using the matrix kronecker product ( see ) 12\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } x(k+1)&=(i_n\otimes a)x(k)+(i_n\otimes b)f(x(k ) ) \\ & \quad + ( i_n\otimes d)f(x(k-\tau ) ) + i(k)+(c_{\sigma ( k)}\otimes \upgamma)x(k-\tau ) , \\ & \quad k\neq k_{m , u } \end{aligned}$$\end{document}13\documentclass[12pt]{minimal } 
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				 \begin{document}$$x(k_{m , u})=[i_n\otimes ( i_n+j_u(k_{m , u}))]x(k_{m , u}-1),$$\end{document}for any \documentclass[12pt]{minimal } 
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				 \begin{document}$${k,\;m\in \mathbb{n}}$$\end{document}. under assumption 1 . , the impulsive controlled complex network ( 12 ) is exponentially synchronized if there exist certain positive integer m , positive scalars \documentclass[12pt]{minimal } 
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				 \begin{document}$$\varepsilon_{\sigma ( k)},\;p_{\sigma ( k)},\;q_{\sigma ( k)}$$\end{document } , and positive - definite matrices \documentclass[12pt]{minimal } 
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				 \begin{document}$${p_{\sigma ( k)}\in \mathbb{r}^{n\times n},\;q_{l,\sigma ( k)}\in \mathbb{r}^{n\times n}\;(l=1,2,\ldots 6)}$$\end{document } such that 
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				 \begin{document}$$p_{\sigma ( k_{m , x})}\le \mu p_{\sigma ( k_{m+1,x})}$$\end{document } , for any \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in [ k_{m , x},k_{m+1,x}-1]$$\end{document } in corresponding sub - state (km , x ) , 
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				 \begin{document}$$p_{\sigma(k_{m , x})}-\left[\frac{\lambda_{\rm max}(\uppi_{\sigma(k_{m , x } ) } ) } { \lambda_{\rm min}\left(p_{\sigma(k_{m , x})}^{-1}\right)}+ \mu q_{\sigma(k_{m , x } ) } \frac{\lambda_{\rm max}(\upomega_{\sigma(k_{m , x})})}{\lambda_{\rm min } \left(p_{\sigma(k_{m - m_{\tau},x})}^{-1}\right)}\right]\geq0,$$\end{document}where 
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				 \begin{document}$$\begin{aligned } \uppi_{\sigma(k_{m , x})}&=a^{t}p_{\sigma(k_{m , x})}a+l^{t}b^{t}p_{\sigma(k_{m , x})}bl \\ & \quad + l^{t}b^{t}p_{\sigma(k_{m , x})}^{t}q_{1,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}bl \\ & \quad + a^{t}q_{1,\sigma(k_{m , x})}a + a^{t}q_{2,\sigma(k_{m , x})}^{-1}a \\ & \quad -nc_{\sigma(k_{m , x})}a^{t}q_{3,\sigma(k_{m , x})}^{-1}ac_{\sigma(k_{m , x } ) } \\ & \quad -l^{t}nc_{\sigma(k_{m , x})}b^{t}q_{5,\sigma(k_{m , x})}bc_{\sigma(k_{m , x})}l \\ & \quad + l^{t}b^{t}q_{4,\sigma(k_{m , x})}bl \\
\upomega_{\sigma(k_{m , x})}&=l^{t}d^{t}p_{\sigma(k_{m , x } ) } dl - nc_{\sigma(k_{m , x})}^{2}\upgamma^{t}p_{\sigma(k_{m , x})}\upgamma \\ & \quad + l^{t}d^{t}p_{\sigma(k_{m , x})}^{t}q_{2,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}dl \\ & \quad -\upgamma^{t}p_{\sigma(k_{m , x})}^{t}q_{3,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}\upgamma \\ & \quad + l^{t}d^{t}p_{\sigma(k_{m , x})}^{t}q_{4,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}dl \\ & \quad -\upgamma^{t}p_{\sigma(k_{m , x})}^{t}q_{5,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}\upgamma \\ & \quad -l^{t}nc_{\sigma(k_{m , x})}d^{t}q_{6,\sigma(k_{m , x})}^{-1}dc_{\sigma(k_{m , x})}l \\ & \quad -\upgamma^{t}p_{\sigma(k_{m , x})}^{t}q_{6,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}\upgamma .
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				 \begin{document}$$\mu\lambda_{\rm max}^{2}(1+j_u(k_{m , x}))<\text{e}^{\varepsilon_{\sigma(k_{m , x } ) } ( k_{m+1,x}-k_{m , x})}$$\end{document}.\documentclass[12pt]{minimal } 
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				 \begin{document}$$q_{\sigma(k_{m , x})}\geq \text{e}^{\varepsilon_{\sigma(k_{m , x})}(k_{m+1,x}-k_{m , x}+1)+ \sum\nolimits_{i=0}^{m_{\tau}-1}\varepsilon_{k_{m - i , x}}(k_{m+1-i , x}-k_{m - i , x})}$$\end{document } , where \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_{\tau}=\lceil\frac{\tau}{\inf\{k_{m , x}-k_{m-1,x}\}}\rceil$$\end{document}. given      1 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_{\sigma ( k_{m , x})}\le \mu p_{\sigma ( k_{m+1,x})}$$\end{document } , for any \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in [ k_{m , x},k_{m+1,x}-1]$$\end{document } in corresponding sub - state (km , x ) , 
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				 \begin{document}$$p_{\sigma(k_{m , x})}-\left[\frac{\lambda_{\rm max}(\uppi_{\sigma(k_{m , x } ) } ) } { \lambda_{\rm min}\left(p_{\sigma(k_{m , x})}^{-1}\right)}+ \mu q_{\sigma(k_{m , x } ) } \frac{\lambda_{\rm max}(\upomega_{\sigma(k_{m , x})})}{\lambda_{\rm min } \left(p_{\sigma(k_{m - m_{\tau},x})}^{-1}\right)}\right]\geq0,$$\end{document}where 
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				 \begin{document}$$\begin{aligned } \uppi_{\sigma(k_{m , x})}&=a^{t}p_{\sigma(k_{m , x})}a+l^{t}b^{t}p_{\sigma(k_{m , x})}bl \\ & \quad + l^{t}b^{t}p_{\sigma(k_{m , x})}^{t}q_{1,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}bl \\ & \quad + a^{t}q_{1,\sigma(k_{m , x})}a + a^{t}q_{2,\sigma(k_{m , x})}^{-1}a \\ & \quad -nc_{\sigma(k_{m , x})}a^{t}q_{3,\sigma(k_{m , x})}^{-1}ac_{\sigma(k_{m , x } ) } \\ & \quad -l^{t}nc_{\sigma(k_{m , x})}b^{t}q_{5,\sigma(k_{m , x})}bc_{\sigma(k_{m , x})}l \\ & \quad + l^{t}b^{t}q_{4,\sigma(k_{m , x})}bl \\ \upomega_{\sigma(k_{m , x})}&=l^{t}d^{t}p_{\sigma(k_{m , x } ) } dl - nc_{\sigma(k_{m , x})}^{2}\upgamma^{t}p_{\sigma(k_{m , x})}\upgamma \\ & \quad + l^{t}d^{t}p_{\sigma(k_{m , x})}^{t}q_{2,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}dl \\ &
\quad -\upgamma^{t}p_{\sigma(k_{m , x})}^{t}q_{3,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}\upgamma \\ & \quad + l^{t}d^{t}p_{\sigma(k_{m , x})}^{t}q_{4,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}dl \\ & \quad -\upgamma^{t}p_{\sigma(k_{m , x})}^{t}q_{5,\sigma(k_{m , x})}p_{\sigma(k_{m , x})}\upgamma \\ & \quad -l^{t}nc_{\sigma(k_{m , x})}d^{t}q_{6,\sigma(k_{m , x})}^{-1}dc_{\sigma(k_{m , x})}l \\ &
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				 \begin{document}$$\mu\lambda_{\rm max}^{2}(1+j_u(k_{m , x}))<\text{e}^{\varepsilon_{\sigma(k_{m , x } ) } ( k_{m+1,x}-k_{m , x})}$$\end{document}. \documentclass[12pt]{minimal } 
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				 \begin{document}$$q_{\sigma(k_{m , x})}\geq \text{e}^{\varepsilon_{\sigma(k_{m , x})}(k_{m+1,x}-k_{m , x}+1)+ \sum\nolimits_{i=0}^{m_{\tau}-1}\varepsilon_{k_{m - i , x}}(k_{m+1-i , x}-k_{m - i , x})}$$\end{document } , where \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_{\tau}=\lceil\frac{\tau}{\inf\{k_{m , x}-k_{m-1,x}\}}\rceil$$\end{document}. consider the following lyapunov function : 
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				 \begin{document}$$v(k)=x^{t}(k)(w\otimes p_{\sigma(k)})x(k),$$\end{document}for any \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in[k_{m , x},k_{m+1,x}-1],\;\quad m=1,2,\ldots$$\end{document } where 
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				 \begin{document}$$w=\left[\begin{array}{cccc } n-1 & -1 & \ldots & -1 \\ -1 & n-1 & \ldots & -1 \\ \ldots & \ldots & \ldots & \ldots \\ -1 & -1 & \ldots & n-1\end{array}\right].$$\end{document } let \documentclass[12pt]{minimal } 
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				 \begin{document}$$\varphi(\|\phi_{i}(\theta)-\phi_{j}(\theta)\|^{2 } ) = \lambda_{\rm max}(p_{\sigma(\theta)})\|\phi_{i}(\theta)-\phi_{j}(\theta)\|^{2}$$\end{document } , thus for the case \documentclass[12pt]{minimal } 
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				 \begin{document}$${k\in \mathbb{z}_{\tau}}$$\end{document } , 
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				 \begin{document}$$\begin{aligned } v(\theta)&=x^{t}(\theta)(w\otimes p_{\sigma(0)})x(\theta ) \\ & = \sum\limits_{i=1}^{n-1}\sum\limits_{j = i+1}^{n}(x_i(\theta)-x_j(\theta))^{t}p_{\sigma(0 ) } ( x_i(\theta)-x_j(\theta ) ) \\ & \leq \varphi(\|\phi_i(\theta)-\phi_j(\theta)\|_{\tau}^{2 } ) \\ & = \varphi(\|\phi(\theta)\|_{\tau}^{2 } ) .
\end{aligned}$$\end{document } choose m    1 , such that 
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				 \begin{document}$$\begin{aligned } \varphi(\|\phi(\theta)\|_{\tau}^{2})&\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2 } ) \text{e}^{-\lambda(k_{1,x}-k_{0,x})}\text{e}^{-\varepsilon_{\sigma(k_{0,x})}(k_{1,x}-k_{0,x } ) } \\ & < m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } \\ & < q_{\sigma(k_{0,x})}\varphi(\|\phi(\theta)\|_{\tau}^{2 } ) .
\end{aligned}$$\end{document } by claiming that 
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				 \begin{document}$$v(k)\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{m , x}-k_{0,x } ) } , k\in [ k_{m-1,x},k_{m , x}-1 ] , m\in n.$$\end{document } we first need to show that 
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				 \begin{document}$$v(k)\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x})},k\in [ k_{0,x},k_{1,x}-1].$$\end{document } obviously , for \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in [ k_{0,x}-\tau , k_{0,x}-1]$$\end{document } , 
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				 \begin{document}$$\begin{aligned } v(k)&\leq \varphi(\|\phi(\theta)\|_{\tau}^{2 } ) \\ & \leq m\varphi(\|\phi(\theta)\|_{\tau}^{2 } ) \text{e}^{-\lambda(k_{1,x}-k_{0,x})}\text{e}^{-\varepsilon_{\sigma(k_{0,x})}(k_{1,x}-k_{0,x})}. \end{aligned}$$\end{document } if ( 19 ) is not true , then there exists certain instant of time \documentclass[12pt]{minimal } 
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				 \begin{document}$$\bar{k}\in [ k_{0,x},k_{1,x}-1]$$\end{document } satisfying 
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				 \begin{document}$$\bar{k}=\text{min } \left \{k\in[k_{0,x},k_{1,x}-1]:v(k ) > m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x})}\right\}$$\end{document}such that 
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				 \begin{document}$$\begin{aligned } v(\bar{k})&>m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } \\ & > m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } \text{e}^{-\varepsilon_{\sigma(k_{0,x})}(k_{1,x}-k_{0,x } ) } \\ & > \varphi(\|\phi(\theta)\|_{\tau}^{2 } ) .
\\ \end{aligned}$$\end{document}and 
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				 \begin{document}$$v(k)\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } , k\in [ k_{0,x}-\tau,\bar{k}-1]$$\end{document } considering that 
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				 \begin{document}$$k^{\ast}= \text{max } \left \{k\in[k_{0,x},\bar{k}]:v(k ) \leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } \text{e}^{-\varepsilon_{\sigma(k_{0,x})}(k_{1,x}-k_{0,x})}\right\},$$\end{document } from which we have 
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				 \begin{document}$$v(k ) > m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } \text{e}^{-\varepsilon_{\sigma(k_{0,x})}(k_{1,x}-k_{0,x } ) } , k\in [ k^{\ast}+1,\bar{k}-1].$$\end{document } therefore , 
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				 \begin{document}$$v(k^{\ast})\leq v(k)\leq v(\bar{k}),k\in [ k^{\ast},\bar{k}].$$\end{document } and for any \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in [ k^{\ast},\bar{k}-1]$$\end{document } , 
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				 \begin{document}$$\begin{aligned } v(k+s)&\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } \\ & \leq \text{e}^{\varepsilon_{\sigma(k_{0,x})}(k_{1,x}-k_{0,x})}v(k^{\ast } ) \\ & \leq q_{\sigma(k_{0,x})}v(k ) .
\end{aligned}$$\end{document } consider the increment of v(k ) along the solution of discrete complex networks ( 12 ) in the interval \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in [ k^{\ast},\bar{k}-1]$$\end{document } , one observes 
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				 \begin{document}$$\begin{aligned }    \delta v(k ) &   = v(k + 1 ) - v(k ) \\      & \ ; = [ ( i_{n }   \otimes a)x(k ) + ( i_{n }   \otimes b)f(x(k ) ) \\    
\quad   + ( i_{n }   \otimes d)f(x(k - \tau ) ) + i(k ) \\    
\quad   + ( c_{{\sigma ( k ) } }   \otimes \gamma ) x(k - \tau){]}^{t } ( w \otimes p_{{\sigma ( k + 1 ) } } ) [ ( i_{n }   \otimes a)x(k ) \\    
\quad   + ( i_{n }   \otimes b)f(x(k ) ) + ( i_{n }   \otimes d)f(x(k - \tau ) ) \\    
\quad   + i(k ) + ( c_{{\sigma ( k ) } }   \otimes \gamma ) x(k - \tau ) ] - x^{t } ( k)p_{{\sigma ( k ) } } x(k ) \\      & \ ; = [ ( i_{n }   \otimes a)x(k ) + ( i_{n }   \otimes b)f(x(k ) ) \\    
\quad   + ( i_{n }   \otimes d)f(x(k - \tau ) ) + i(k ) \\     \quad   + ( c_{{\sigma ( k ) } }   \otimes \gamma ) x(k - \tau ) { ] } ^{t } ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) [ ( i_{n }   \otimes a)x(k ) \\    
\quad   + ( i_{n }   \otimes b)f(x(k ) ) + ( i_{n }   \otimes d)f(x(k - \tau ) ) \\    
\quad   + i(k ) + ( c_{{\sigma ( k ) } }   \otimes \gamma ) x(k - \tau ) ] - x^{t } ( k)p_{{\sigma ( k_{{0,x } } ) } } x(k ) \\      & \ ; = x^{t } ( k)(i_{n }   \otimes a^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( i_{n }   \otimes a)x(k ) \\    
\quad   + f^{t } ( x(k))(i_{n }   \otimes b^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( i_{n }   \otimes b)f(x(k ) ) \\    
\quad   + f^{t } ( x(k - \tau ) ) ( i_{n }   \otimes d^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( i_{n }   \otimes d)f(x(k - \tau ) ) \\    
\quad   + i^{t } ( k)(w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) i(k ) \\    
\quad   + x^{t } ( k - \tau ) ( c_{{\sigma ( k_{{0,x } } ) } } ^{t }   \otimes \gamma ^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( c_{{\sigma ( k_{{0,x } } ) } }   \otimes \gamma ) x(k - \tau ) \\    
\quad   + 2x^{t } ( k)(i_{n }   \otimes a^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( i_{n }   \otimes b)f(x(k ) ) \\    
\quad   + 2x^{t } ( k)(i_{n }   \otimes a^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( i_{n }   \otimes d)f(x(k - \tau ) ) \\    
\quad   + 2x^{t } ( k)(i_{n }   \otimes a^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) i(k ) \\    
\quad   + 2x^{t } ( k)(i_{n }   \otimes a^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( c_{{\sigma ( k_{{0,x } } ) } }   \otimes \gamma ) x(k - \tau ) \\    
\quad   + 2f^{t } ( x(k))(i_{n }   \otimes b^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( i_{n }   \otimes d)f(x(k - \tau ) ) \\    
\quad   + 2f^{t } ( x(k))(i_{n }   \otimes b^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) i(k ) \\    
\quad   + 2f^{t } ( x(k))(i_{n }   \otimes b^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( c_{{\sigma ( k_{{0,x } } ) } }   \otimes \gamma ) x(k - \tau ) \\    
\quad   + 2f^{t } ( x(k))(i_{n }   \otimes b^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( i_{n }   \otimes d)f(x(k - \tau ) ) \\    
\quad   + 2f^{t } ( x(k))(i_{n }   \otimes b^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) i(k ) \\    
\quad   + 2f^{t } ( x(k))(i_{n }   \otimes b^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( c_{{\sigma ( k_{{0,x } } ) } }   \otimes \gamma ) x(k - \tau ) \\    
\quad   + 2f^{t } ( x(k - \tau ) ) ( i_{n }   \otimes d^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) i(k ) \\    
\quad   + 2f^{t } ( x(k - \tau ) ) ( i_{n }   \otimes d^{t } ) ( w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( c_{{\sigma ( k_{{0,x } } ) } }   \otimes \gamma ) x(k - \tau ) \\    
\quad   + 2i^{t } ( k)(w \otimes p_{{\sigma ( k_{{0,x } } ) } } ) ( c_{{\sigma ( k_{{0,x } } ) } }   \otimes \gamma ) x(k - \tau ) \\    
\quad   - x^{t } ( k)p_{{\sigma ( k_{{0,x } } ) } } x(k ) \\  
\end{aligned}$$\end{document } note that 
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				 \begin{document}$$(w\otimes p_{\sigma(k)})i(k)=0 \quad and \quad i^{t}(k)(w\otimes p_{\sigma(k)})=0.$$\end{document } for the simplicity of calculation , we define 
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				 \begin{document}$$c_{\sigma ( k_{0,x})}w c_{\sigma ( k_{0,x})}=n c_{\sigma(k_{0,x})}^{2}.$$\end{document}31\documentclass[12pt]{minimal } 
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				 \begin{document}$$wc_{\sigma ( k_{0,x})}=nc_{\sigma ( k_{0,x})}.$$\end{document}32\documentclass[12pt]{minimal } 
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				 \begin{document}$$x_{ij}(k)=x_i(k)-x_j(k).$$\end{document}33\documentclass[12pt]{minimal } 
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				 \begin{document}$$x_{ij}(k-\tau)=x_i(k-\tau)-x_j(k-\tau).$$\end{document}34\documentclass[12pt]{minimal } 
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				 \begin{document}$$f_{ij}(x(k))=f_i(x(k))-f_j(x(k)).$$\end{document}35\documentclass[12pt]{minimal } 
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				 \begin{document}$$f_{ij}(x(k-\tau))=f_i(x(k-\tau))-f_j(x(k-\tau)).$$\end{document } according to lemma 1 , we can obtain that 
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				 \begin{document}$$\begin{aligned }    \delta v(k ) &   = v(k + 1 ) - v(k ) \\      & \ ; = \sum\limits_{{i = 1}}^{{n - 1 } } { \sum\limits_{{j
= i + 1}}^{n } { [ x_{{ij}}^{t } ( k)a^{t } p_{{\sigma ( k_{{0,x } } ) } } ax_{{ij } } ( k ) } }   \\    
\quad   + x_{{ij}}^{t } ( k)l^{t } b^{t } p_{{\sigma ( k_{{0,x } } ) } } blx_{{ij } } ( k ) \\    
\quad   + x_{{ij}}^{t } ( k - \tau ) l^{t } d^{t } p_{{\sigma ( k_{{0,x } } ) } } dlx_{{ij } } ( k - \tau ) \\    
\quad   + x_{{ij}}^{t } ( k - \tau ) nc_{{\sigma ( k_{{0,x } } ) } } ^{2 } \gamma ^{t } p_{{\sigma ( k_{{0,x } } ) } } \gamma x_{{ij } } ( k - \tau ) \\    
\quad   + x_{{ij}}^{t } ( k)a^{t } q_{{1,\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } ax_{{ij } } ( k ) \\    
\quad   + x_{{ij}}^{t } ( k)l^{t } b^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{1,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } blx_{{ij } } ( k ) \\    
\quad   + x_{{ij}}^{t } ( k)a^{t } q_{{2,\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } ax_{{ij } } ( k ) \\    
\quad   + x_{{ij}}^{t } ( k - \tau ) l^{t } d^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{2,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } dlx_{{ij } } ( k - \tau ) \\    
\quad   - x_{{ij}}^{t } ( k)nc_{{\sigma ( k_{{0,x } } ) } } a^{t } q_{{3,\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } ac_{{\sigma ( k_{{0,x } } ) } } x_{{ij } } ( k ) \\    
\quad   - x_{{ij}}^{t } ( k - \tau ) \gamma ^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{3,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } \gamma x_{{ij } } ( k - \tau ) \\    
\quad   + x_{{ij}}^{t } ( k)l^{t } b^{t } q_{{4,\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } blx_{{ij } } ( k ) \\    
\quad   + x_{{ij}}^{t } ( k - \tau ) l^{t } d^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{4,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } dlx_{{ij } } ( k - \tau ) \\    
\quad   - x_{{ij}}^{t } ( k)l^{t } nc_{{\sigma ( k_{{0,x } } ) } } b^{t } q_{{5,\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } bc_{{\sigma ( k_{{0,x } } ) } } lx_{{ij } } ( k ) \\    
\quad   - x_{{ij}}^{t } ( k - \tau ) \gamma ^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{5,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } \gamma x_{{ij } } ( k - \tau ) \\    
\quad   - x_{{ij}}^{t } ( k - \tau ) l^{t } nc_{{\sigma ( k_{{0,x } } ) } } d^{t } q_{{6,\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } dc_{{\sigma ( k_{{0,x } } ) } } lx_{{ij } } ( k - \tau ) \\    
\quad   - x_{{ij}}^{t } ( k - \tau ) \gamma ^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{6,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } \gamma x_{{ij } } ( k - \tau ) \\    
\quad   - x_{{ij}}^{t } ( k)p_{{\sigma ( k_{{0,x } } ) } } x_{{ij } } ( k ) ] \\      & \ ; = \sum\limits_{{i = 1}}^{{n - 1 } } { \sum\limits_{{j = i + 1}}^{n } { \ { x_{{ij}}^{t } ( } } k)[a^{t } p_{{\sigma ( k_{{0,x } } ) } } a + a^{t } q_{{1,\sigma ( k_{{0,x } } ) } } a \\     \quad   + l^{t } b^{t } p_{{\sigma ( k_{{0,x } } ) } } bl + a^{t } q_{{2,\sigma ( k_{{m , x } } ) } } ^ { { - 1 } } a \\     \quad   + l^{t } b^{t } p_{{\sigma ( k_{{m , x } } ) } } ^{t } q_{{1,\sigma ( k_{{m , x } } ) } } p_{{\sigma ( k_{{m , x } } ) } } bl \\     \quad   - nc_{{ij,\sigma ( k_{{0,x } } ) } } a^{t } q_{{3,\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } ac_{{ij,\sigma ( k_{{0,x } } ) } }   \\     \quad   + l^{t } b^{t } q_{{4,\sigma ( k_{{0,x } } ) } } bl \\    
\quad   - l^{t } nc_{{\sigma ( k_{{0,x } } ) } } b^{t } q_{{5,\sigma ( k_{{0,x } } ) } } bc_{{\sigma ( k_{{0,x } } ) } } l - p_{{\sigma ( k_{{0,x } } ) } } ] x_{{ij } } ( k ) \\    
\quad   + x_{{ij}}^{t } ( k - \tau ) [ l^{t } d^{t } p_{{\sigma ( k_{{0,x } } ) } } dl - nc_{{\sigma ( k_{{0,x } } ) } } ^{{(2 ) } } \gamma ^{t } p_{{\sigma ( k_{{0,x } } ) } } \gamma   \\    
\quad   + l^{t } d^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{2,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } dl \\    
\quad   - \gamma ^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{3,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } \gamma   \\     \quad   + l^{t } d^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{4,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } dl \\    
\quad   - \gamma ^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{5,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } \gamma   \\     \quad   - l^{t }
nc_{{\sigma ( k_{{0,x } } ) } } d^{t } q_{{6,\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } dc_{{\sigma ( k_{{0,x } } ) } } l \\     \quad   - \gamma ^{t } p_{{\sigma ( k_{{0,x } } ) } } ^{t } q_{{6,\sigma ( k_{{0,x } } ) } } p_{{\sigma ( k_{{0,x } } ) } } \gamma ] x_{{ij } } ( k - \tau ) \ }   \\      & \ ; = \left [ { \frac{{\lambda _ { { \max } } ( \pi _ { { \sigma ( k_{{0,x } } ) } } ) } } { { \lambda _ { { \min } } ( p_{{\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } ) } } - 1 } \right]v(k ) + \frac{{\lambda _ { { \max } } ( \omega _ { { \sigma ( k_{{0,x } } ) } } ) } } { { \lambda _ { { \min } } ( p_{{\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } ) } } v(k - \tau ) \\      & \ ; = \left [ { \frac{{\lambda _ { { \max } } ( \pi _ { { \sigma ( k_{{0,x } } ) } } ) } } { { \lambda _ { { \min } } ( p_{{\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } ) } } + \mu q_{{\sigma ( k_{{0,x } } ) } } \frac{{\lambda _ { { \max } } ( \omega _ { { \sigma ( k_{{0,x } } ) } } ) } } { { \lambda _ { { \min } } ( p_{{\sigma ( k_{{0,x } } ) } } ^ { { - 1 } } ) } } - 1 } \right]v(k ) \\   \end{aligned}$$\end{document } therefore , we obtain 
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				 \begin{document}$$\begin{aligned } v(\bar{k})&\leq\left[\frac{\lambda_{\rm max}(\uppi_{\sigma(k_{0,x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{0,x})}^{-1 } ) } + \mu q_{\sigma(k_{0,x})}\frac{\lambda_{\rm max}(\upomega_{\sigma(k_{0,x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{0,x})}^{-1})}\right]v(\bar{k}-1 ) \\ & \quad \leq\left[\frac{\lambda_{\rm max}(\uppi_{\sigma(k_{0,x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{0,x})}^{-1 } ) } + \mu q_{\sigma(k_{0,x})}\frac{\lambda_{\rm max}(\upomega_{\sigma(k_{0,x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{0,x})}^{-1})}\right]^{(\bar{k}-k^{\ast})}v(k^{\ast } ) \\ & \quad \leq\left[\frac{\lambda_{\rm max}(\uppi_{\sigma(k_{0,x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{0,x})}^{-1 } ) } + \mu q_{\sigma(k_{0,x})}\frac{\lambda_{\rm max}(\upomega_{\sigma(k_{0,x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{0,x})}^{-1})}\right]^{(k_{1,x}-k_{0,x } ) } \\ & \quad \cdot m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } \text{e}^{-\varepsilon_{\sigma(k_{0,x})}(k_{1,x}-k_{0,x } ) } \\ & \quad \leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{1,x}-k_{0,x } ) } \\ & \quad < v(\bar{k } ) .
next comes that we suppose the claim ( 18 ) holds for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\hbox{m}= 1,2,\ldots , \bar{m}$$\end{document } , such that 
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				 \begin{document}$$v(k)\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{m , x}-k_{0,x } ) } , k\in [ k_{m-1,x},k_{m , x}-1].$$\end{document } correspondingly , we should prove the eq .
( 18 ) holds for \documentclass[12pt]{minimal } 
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				 \begin{document}$$\hbox{m}=\bar{m}+1$$\end{document } , 
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				 \begin{document}$$v(k)\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x } ) } , k\in [ k_{\bar{m},x},k_{\bar{m}+1,x}-1].$$\end{document } note that from ( 13 ) and condition ( ii ) , we have 
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				 \begin{document}$$\begin{aligned } v(k_{\bar{m},x})&=\sum\limits_{i=1}^{n-1}\sum\limits_{j = i+1}^{n}x_{ij}^{t}(k_{\bar{m},x}-1 ) ( i+j_u({k_{\bar{m},x}}))^{t } \\ & \cdot p_{\sigma(k_{\bar{m},x})}(i+j_u({k_{\bar{m},x}}))x_{ij}(k_{\bar{m},x}-1 ) \\ & \leq \mu \lambda_{\rm max}^{2}(i+j_u({k_{\bar{m},x } } ) ) \\ & \cdot \sum\limits_{i=1}^{n-1}\sum\limits_{j = i+1}^{n}x_{ij}^{t } ( k_{\bar{m},x}-1)p_{\sigma(k_{\bar{m},x})}x_{ij}(k_{\bar{m},x}-1 ) \\ & = \mu \lambda_{\rm max}^{2}(i+j_u({k_{\bar{m},x}}))v(k_{\bar{m},x}-1 ) \\ & \leq \mu \lambda_{\rm max}^{2}(i+j_u({k_{\bar{m},x } } ) ) m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m},x}-k_{0,x } ) } \\ & < \mu \lambda_{\rm max}^{2}(i+j_u({k_{\bar{m},x } } ) ) m\varphi(\|\phi(\theta)\|_{\tau}^{2 } ) \\ & \cdot \text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x})}\text{e}^{\varepsilon_{\sigma(k_{\bar{m},x } ) } ( k_{\bar{m}+1,x}-k_{\bar{m},x } ) } \\ & \leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x } ) } \text{e}^{-\varepsilon_{\sigma(k_{\bar{m},x})}(k_{\bar{m}+1,x}-k_{\bar{m},x } ) } \\ \end{aligned}$$\end{document } if ( 39 ) is not true , then there exists a natural number \documentclass[12pt]{minimal } 
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				 \begin{document}$$\bar{k}$$\end{document } satisfying 
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				 \begin{document}$$\bar{k}=\text{min}\left\{k\in[k_{\bar{m},x},k_{\bar{m}+1,x}-1]:v(k ) > m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x})}\right\}.$$\end{document } consequently , we have 
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				 \begin{document}$$\bar{k}>k_{\bar{m},x } \quad \text { and } \quad v(\bar{k})>v(k_{\bar{m},x}).$$\end{document } define 
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				 \begin{document}$$\begin{aligned } k^{\ast}&=\text { max}\left\{k\in [ k_{\bar{m},x},\bar{k } ] : v(k)\leq \right.\\ & \left.m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x } ) } \text{e}^{-\varepsilon_{\sigma(k_{\bar{m},x})}(k_{\bar{m}+1,x}-k_{\bar{m},x})}\right\}. \end{aligned}$$\end{document } thus , there exists \documentclass[12pt]{minimal } 
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				 \begin{document}$$k^{\ast}$$\end{document } such that \documentclass[12pt]{minimal } 
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				 \begin{document}$$k^{\ast}<\bar{k}$$\end{document}. for any \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in[k^{\ast}+1,\bar{k}]$$\end{document } , we have 
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				 \begin{document}$$v(k)>m\varphi(\|\phi(\theta)\|_{\tau}^{2 } ) \cdot \text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x})}\text{e}^{-\varepsilon_{\sigma(k_{\bar{m},x } ) } ( k_{\bar{m}+1,x}-k_{\bar{m},x})},$$\end{document}and for any \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in[k^{\ast},\bar{k}]$$\end{document } , 
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				 \begin{document}$$v(k^{\ast})\leq v(k ) \leq v(\bar{k}).$$\end{document } then , for any \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in[k_{\bar{m},x},k_{\bar{m}+1,x}-1]$$\end{document } , we can obtain 
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				 \begin{document}$$v(k+1)\leq \left[\frac{\lambda_{\rm max}(\uppi_{\sigma(k_{\bar{m},x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{\bar{m},x})}^{-1})}+\mu q_{\sigma(k_{\bar{m},x } ) } \frac{\lambda_{\rm max}(\upomega_{\sigma(k_{\bar{m},x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{\bar{m}-m_{\tau},x})}^{-1})}\right]v(k).$$\end{document } according to the condition ( iii ) , one observes that for any \documentclass[12pt]{minimal } 
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				 \begin{document}$${s\in \mathbb{z}_{\tau},\;k+s\in [ k_{\bar{m}-m_{\tau},x},\bar{k}]}$$\end{document}. meanwhile for any \documentclass[12pt]{minimal } 
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				 \begin{document}$$k\in[k^{\ast},\bar{k}-1]$$\end{document } , 
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				 \begin{document}$$\begin{aligned } v(k+s)&\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}-m_{\tau}+1,x}-k_{0,x } ) } \\ & \leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x } ) } \\ & \cdot \text{e}^{\sum\nolimits_{i=0}^{m_{\tau}-1}\varepsilon_{\sigma(k_{\bar{m}-i , x } ) } ( k_{\bar{m}+1-i , x}-k_{\bar{m}-i , x } ) } \\ & \leq \text{e}^{-\varepsilon_{\sigma(k_{\bar{m},x})}(k_{\bar{m}+1,x}-k_{\bar{m},x } ) } m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x } ) } \\ & \cdot \text{e}^{\varepsilon_{\sigma(k_{\bar{m},x})}(k_{\bar{m}+1,x}-k_{\bar{m},x } ) } \text{e}^{\sum\nolimits_{i=0}^{m_{\tau}-1}\varepsilon_{\sigma(k_{\bar{m}-i , x } ) } ( k_{\bar{m}+1-i , x}-k_{\bar{m}-i , x } ) } \\ & \leq \text{e}^{\varepsilon_{\sigma(k_{\bar{m},x})}(k_{\bar{m}+1,x}-k_{\bar{m},x } ) + \sum\limits_{i=0}^{m_{\tau}-1}\varepsilon_{\sigma(k_{\bar{m}-i , x } ) } ( k_{\bar{m}+1-i , x}-k_{\bar{m}-i , x } ) } \\ & \cdot v(k^{\ast}+1 ) \\ & < \text{e}^{\varepsilon_{\sigma(k_{\bar{m},x})}(k_{\bar{m}+1,x}-k_{\bar{m},x}+1 ) + \sum\limits_{i=0}^{m_{\tau}-1 } \varepsilon_{\sigma(k_{\bar{m}-i , x})}(k_{\bar{m}+1-i , x}-k_{\bar{m}-i , x } ) } \\ & \cdot v(k^{\ast } ) \\ & \leq q_{\sigma(k_{\bar{m},x})}v(k ) . \\ \end{aligned}$$\end{document } consider that \documentclass[12pt]{minimal } 
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				 \begin{document}$$k=\bar{k}$$\end{document } , we have 
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				 \begin{document}$$\begin{aligned } v(\bar{k})&\leq \left[\frac{\lambda_{\rm max}(\uppi_{\sigma(k_{\bar{m},x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{\bar{m},x})}^{-1 } ) } \right.\\ & \left.+\mu q_{\sigma(k_{\bar{m},x})}\frac{\lambda_{\rm max}(\upomega_{\sigma(k_{\bar{m},x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{\bar{m}-m_{\tau},x})}^{-1})}\right]^{(\bar{k}-k^{\ast } ) } v(k^{\ast } ) \\ & \leq \left[\frac{\lambda_{\rm max}(\uppi_{\sigma(k_{\bar{m},x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{\bar{m},x})}^{-1 } ) } \right.\\ & \left.+\mu q_{\sigma(k_{\bar{m},x})}\frac{\lambda_{\rm max}(\upomega_{\sigma(k_{\bar{m},x } ) } ) } { \lambda_{\rm min}(p_{\sigma(k_{\bar{m}-m_{\tau},x})}^{-1})}\right ] ^{(k_{\bar{m}+1,x}-k_{\bar{m},x } ) } \\ & \cdot m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x } ) } \text{e}^{-\varepsilon_{\sigma(k_{\bar{m},x})}(k_{\bar{m}+1,x}-k_{\bar{m},x } ) } \\ & < m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k_{\bar{m}+1,x}-k_{0,x } ) } \\ & < v(\bar{k } ) , \\
( 39 ) holds for \documentclass[12pt]{minimal } 
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				 \begin{document}$$k=\bar{k}+1$$\end{document}. and by virtue of mathematical induction , the claim ( 18 ) is true for each \documentclass[12pt]{minimal } 
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				 \begin{document}$${k\in \mathbb{n}}$$\end{document}. in view of ( 18 ) and definition 1 , it can be obtained that 
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				 \begin{document}$$v(k)\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k - k_{0,x } ) } , k\in[k_{m-1,x},k_{m , x}-1 ] , m\in{\mathbb{n}}.$$\end{document } for any \documentclass[12pt]{minimal } 
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				 \begin{document}$${k\in\mathbb{n}}$$\end{document } , 
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				 \begin{document}$$\begin{aligned } & \text{min}\left\{\lambda_{\rm min}(p_{\sigma(k)})\right\ } \sum\limits_{i=1}^{n-1}\sum\limits_{j = i+1}^{n}\|x_i(k)-x_j(k)\|^{2 } \\ & \;\leq \sum\limits_{i=1}^{n-1}\sum\limits_{j = i+1}^{n}(x_i(k)-x_j(k))^{t}p_{\sigma(k ) } ( x_i(k)-x_j(k ) ) \\ & \;=v(k ) \\ & \;\leq m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k - k_{0,x})}. \\ \end{aligned}$$\end{document } therefor , for any \documentclass[12pt]{minimal } 
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				 \begin{document}$${k\in \mathbb{n}}$$\end{document } , 
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				 \begin{document}$$\begin{aligned } & \sum\limits_{i=1}^{n-1}\sum\limits_{j = i+1}^{n}\|x_i(k)-x_j(k)\|^{2 } \\ & \;\leq \text{min}\left\{\lambda_{\rm min}^{-1}(p_{\sigma(k)})\right\ } m\varphi(\|\phi(\theta)\|_{\tau}^{2})\text{e}^{-\lambda(k - k_{0,x } ) } , \\
\end{aligned}$$\end{document}which implies 
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				 \begin{document}$$\|x_i(k)-x_j(k)\|\leq m_0\text{e}^{-\lambda(k - k_{0,x})},\quad 1\leq
i \leq j\leq n.$$\end{document } therefore , the discrete complex networks ( 1 ) are globally exponentially synchronized under impulsive control .
we consider a multiple lyapunov function for each sub - network with arbitrarily fast switching signal in our theorem , which results in a less conservation criterion . in the switched lyapunov function , p(k )
gives an upper bound on the estimation of divergence rate for each running sub - network . by condition ( ii ) of theorem 1 ,
the impulsive control gain is designed to compensate divergence from system itself and deteriorating effect from arbitrarily fast switching .
if some certain sub - networks could be self - synchronizing , the impulsive control gain only needs to compensate deteriorating effect .
let us consider a 2-dimensional discrete chaotic neural network is given as the isolated node of a small - world network with 30 nodes , 
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				 \begin{document}$$x(k+1)=ax(k)+bf(x(k))+df(x(k-\tau(k)))+i(k),$$\end{document}where \documentclass[12pt]{minimal } 
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				 \begin{document}$$x(k)=(x_1(k),x_2(k))^{t},\;f(x(k))=(\text{tan}h(x_1(k)),\;\text{tan}h(x_2(k)))^{t},\;i(k)=(0,0)^t , \;a=\left[\begin{array}{cc } -1 & 0 \\ 0 & -1\end{array}\right],\;b=\left[\begin{array}{cc } 2 & -0.11 \\ -5 & 3.2\end{array}\right],\;d=\left[\begin{array}{cc } -1.6 & -0.1 \\ -0.18 & -2.4\end{array}\right],$$\end{document } and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tau(k)=\frac{\text{e}^{k}}{1+\text{e}^{(k)}}.$$\end{document } obviously , lipschitz constants can be 1 here . consider a small - world model involved with three different sub - systems as follow 
 30 nodes are arranged in a ring , while each node i is adjacent to its neighbor node ; each pair of nodes are coupled to the whole network by probability p  =  0.02 .
1a.30 nodes are arranged in a ring , while each node i is adjacent to its neighbor node ; each pair of nodes are coupled to the whole network by probability p  =  0.01 .
1b.30 nodes are arranged in a ring , while each node i is adjacent to its 2 neighbor node ; each pair of nodes are coupled to the whole network by probability p  =  0.04 .
a 
 n  =  30 ,  k  =  2 ,  p  =  0.02 ; b 
 n  =  30 ,  k  =  2 ,  p  =  0.01 ; c 
 n  =  30 ,  k  =  4 ,  p  =  0.04 ; d chaotic trajectory of each single node 30 nodes are arranged in a ring , while each node i is adjacent to its neighbor node ; each pair of nodes are coupled to the whole network by probability p  =  0.02 .
30 nodes are arranged in a ring , while each node i is adjacent to its neighbor node ; each pair of nodes are coupled to the whole network by probability p  =  0.01 .
30 nodes are arranged in a ring , while each node i is adjacent to its 2 neighbor node ; each pair of nodes are coupled to the whole network by probability p  =  0.04 .
1c . 
 a 
 n  =  30 ,  k  =  2 ,  p  =  0.02 ; b 
 n  =  30 ,  k  =  2 ,  p  =  0.01 ; c 
 n  =  30 ,  k  =  4 ,  p  =  0.04 ; d chaotic trajectory of each single node the trajectory of each single node of this small - world model has been portrayed in fig .
1d with random initial values in the interval [ 0.3 , 3 ] and [ 3 ,  0.3 ] , respectively .
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				 \begin{document}$$\upgamma = diag\{0.15,0.3\}$$\end{document } , we can obtain the state response of each sub - network .
note that there is no switched rule in each sub - network here , and the initial values are randomly chosen in the interval [ 0.3 , 3 ] and [ 3 ,  0.3 ] , respectively . given a switching signal (t ) in fig . 
3a , we have the state response of the switched complex networks , see fig . 
, for each sub - network , we have 
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				 \begin{document}$$p_{1}=\left[\begin{array}{cc } 3.0001 & 0 \\ 0 & 3.0001\end{array}\right],\;q_{1,1}=\left[\begin{array}{cc } 5.0690 & 0 \\ 0 & 5.0690\end{array}\right],\;q_{2,1}=\left[\begin{array}{cc } 9.3812 & 0 \\ 0 & 9.3812\end{array}\right],\;q_{3,1}=\left[\begin{array}{cc } 33.7971 & 0 \\ 0 & 33.7971\end{array}\right],\;q_{4,1}=\left[\begin{array}{cc } 11.4594 & 0 \\ 0 & 11.4594\end{array}\right],\;q_{5,1}=\left[\begin{array}{cc } 50.9300 & 0 \\ 0 & 50.9300\end{array}\right],\;q_{6,1}=\left[\begin{array}{cc } 20.1879 & 0 \\ 0 & 20.1879\end{array}\right]$$\end{document}. and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\varepsilon_1=0.6213,\;q_1<22.42$$\end{document}. thus , \documentclass[12pt]{minimal } 
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				 \begin{document}$$j_{1}=\left[\begin{array}{cc } -0.6667 & 0
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				 \begin{document}$$p_{2}=\left[\begin{array}{cc } 2.9998 & 0 \\ 0 & 2.9998\end{array}\right],\;q_{1,2}=\left[\begin{array}{cc } 5.1333 & 0 \\ 0 & 5.1333\end{array}\right],\;q_{2,2}=\left[\begin{array}{cc } 9.9398 & 0 \\ 0 & 9.9398\end{array}\right],\;q_{3,2}=\left[\begin{array}{cc } 29.7933 & 0 \\ 0 & 29.7933\end{array}\right],\;q_{4,2}=\left[\begin{array}{cc } 11.2764 & 0 \\ 0 & 11.2764\end{array}\right],\;q_{5,2}=\left[\begin{array}{cc } 52.0308 & 0 \\ 0 & 52.0308\end{array}\right],\;q_{6,2}=\left[\begin{array}{cc } 19.9970 & 0 \\ 0 & 19.9970\end{array}\right]$$\end{document}. and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\varepsilon_2=0.8870,\;q_2<4.22$$\end{document}. thus , \documentclass[12pt]{minimal } 
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				 \begin{document}$$j_{2}=\left[\begin{array}{cc } -0.4079 & 0 \\ 0 & -0.4079\end{array}\right]$$\end{document}.\documentclass[12pt]{minimal } 
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				 \begin{document}$$p_{3}=\left[\begin{array}{cc } 13.4267 & 0 \\ 0 & 13.4267\end{array}\right],\;q_{1,3}=\left[\begin{array}{cc } 10.2609 & 0 \\ 0 & 10.2609\end{array}\right],\;q_{2,3}=\left[\begin{array}{cc }
3.1001 & 0 \\ 0 & 3.1001\end{array}\right],\;q_{3,3}=\left[\begin{array}{cc } 16.6667 & 0 \\ 0 & 16.6667\end{array}\right],\;q_{4,3}=\left[\begin{array}{cc } 5.2060 & 0 \\ 0 & 5.2060\end{array}\right],\;q_{5,3}=\left[\begin{array}{cc } 30.8123 & 0 \\ 0 & 30.8123\end{array}\right],\;q_{6,3}=\left[\begin{array}{cc } 3.0112 & 0 \\ 0 & 3.0112\end{array}\right]$$\end{document}. and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\varepsilon_3=0.1062,\;q_3<12.62$$\end{document}. thus , \documentclass[12pt]{minimal } 
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				 \begin{document}$$j_{3}=\left[\begin{array}{cc } -1.1576 & 0 \\ 0 & -1.1576\end{array}\right]$$\end{document}. \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_{1}=\left[\begin{array}{cc } 3.0001 & 0 \\ 0 & 3.0001\end{array}\right],\;q_{1,1}=\left[\begin{array}{cc } 5.0690 & 0 \\ 0 & 5.0690\end{array}\right],\;q_{2,1}=\left[\begin{array}{cc }
9.3812 & 0 \\ 0 & 9.3812\end{array}\right],\;q_{3,1}=\left[\begin{array}{cc } 33.7971 & 0 \\ 0 & 33.7971\end{array}\right],\;q_{4,1}=\left[\begin{array}{cc } 11.4594 & 0 \\ 0 & 11.4594\end{array}\right],\;q_{5,1}=\left[\begin{array}{cc } 50.9300 & 0 \\ 0 & 50.9300\end{array}\right],\;q_{6,1}=\left[\begin{array}{cc } 20.1879 & 0 \\ 0 & 20.1879\end{array}\right]$$\end{document}. and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\varepsilon_1=0.6213,\;q_1<22.42$$\end{document}. thus , \documentclass[12pt]{minimal } 
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				 \begin{document}$$j_{1}=\left[\begin{array}{cc } -0.6667 & 0 \\ 0 & -0.667\end{array}\right]$$\end{document}. \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_{2}=\left[\begin{array}{cc } 2.9998 & 0 \\ 0 & 2.9998\end{array}\right],\;q_{1,2}=\left[\begin{array}{cc } 5.1333 & 0 \\ 0 & 5.1333\end{array}\right],\;q_{2,2}=\left[\begin{array}{cc } 9.9398 & 0 \\ 0 & 9.9398\end{array}\right],\;q_{3,2}=\left[\begin{array}{cc } 29.7933 & 0 \\ 0 & 29.7933\end{array}\right],\;q_{4,2}=\left[\begin{array}{cc } 11.2764 & 0 \\ 0 & 11.2764\end{array}\right],\;q_{5,2}=\left[\begin{array}{cc } 52.0308 & 0 \\ 0 & 52.0308\end{array}\right],\;q_{6,2}=\left[\begin{array}{cc } 19.9970 & 0 \\ 0 & 19.9970\end{array}\right]$$\end{document}. and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\varepsilon_2=0.8870,\;q_2<4.22$$\end{document}. thus , \documentclass[12pt]{minimal } 
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				 \begin{document}$$j_{2}=\left[\begin{array}{cc } -0.4079 & 0 \\ 0 & -0.4079\end{array}\right]$$\end{document}. \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_{3}=\left[\begin{array}{cc } 13.4267 & 0 \\ 0 & 13.4267\end{array}\right],\;q_{1,3}=\left[\begin{array}{cc } 10.2609 & 0 \\ 0 & 10.2609\end{array}\right],\;q_{2,3}=\left[\begin{array}{cc } 3.1001 & 0 \\ 0 & 3.1001\end{array}\right],\;q_{3,3}=\left[\begin{array}{cc } 16.6667 & 0 \\ 0 & 16.6667\end{array}\right],\;q_{4,3}=\left[\begin{array}{cc } 5.2060 & 0 \\ 0 & 5.2060\end{array}\right],\;q_{5,3}=\left[\begin{array}{cc } 30.8123 & 0 \\ 0 & 30.8123\end{array}\right],\;q_{6,3}=\left[\begin{array}{cc } 3.0112 & 0 \\ 0 & 3.0112\end{array}\right]$$\end{document}. and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\varepsilon_3=0.1062,\;q_3<12.62$$\end{document}. thus , \documentclass[12pt]{minimal } 
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				 \begin{document}$$j_{3}=\left[\begin{array}{cc } -1.1576 & 0 \\ 0 & -1.1576\end{array}\right]$$\end{document}. it is shown from fig . 2a 
c that all of nodes in each sub - network could not reach into a synchronous state without a control .
indeed , the switched signal plays a role of deterioration accelerator to diverge the synchronous state , shown in fig . 3b .
once the feasible impulsive controller is placed on discrete complex networks with topology switching , such complex networks would be synchronized , see fig . 3c.fig .  2 
 a the state responses of sub - networks 1 ; b the state responses of sub - networks 2 ; c the state responses of sub - networks 3 
 a the state responses of sub - networks 1 ; b the state responses of sub - networks 2 ; c the state responses of sub - networks 3 
 a the switching signal (t ) ; b the state responses of the switched system ; c the synchronized state under impulsive control
in this paper , we have investigated impulsive synchronization control of a discrete delayed complex network with switching topology by using lyapunov - ruzimiki method .
a time - varying delay - dependent criteria for exponential synchronization is presented guarantee the switched discrete complex networks tending to be a synchronous manifold .
it is worthwhile to see time - varying delay can take any value , even larger than any dwell time of a sub - network .
furthermore , a numerical example with three sub - networks is presented by using the impulsive control technique .
 in this paper , global exponential synchronization of a class of discrete delayed complex networks with switching topology has been investigated by using lyapunov - ruzimiki method . 
 the impulsive scheme is designed to work at the time instant of switching occurrence . a time - varying delay - dependent criterion for impulsive synchronization 
 is given to ensure the delayed discrete complex networks switching topology tending to a synchronous state . 
 furthermore , a numerical simulation is given to illustrate the effectiveness of main results 
Introduction
Preliminary
Main results
Example and numerical simulations
Conclusion