# Adaptive Synchronization of Complex Dynamical Multilinks Networks with Similar Nodes

Authors: Weiping Wang; Lixiang Li; Haipeng Peng; Jialiang Yuan; Jinghua Xiao; Yixian Yang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper studies the synchronization of complex dynamical networks
with multilinks and similar nodes. The dynamics of all the nodes in
the networks are impossible to be completely identical due to the differences of parameters or the existence of perturbations. Networks with
similar nodes are universal in the real world. In order to depict
the similarity of the similar nodes, we give the definition of the
minimal similarity of the nodes in the network for the first time.
We find the threshold of the minimal similarity of the nodes in the
network. If the minimal similarity of the nodes is bigger than the
threshold, then the similar nodes can achieve synchronization
without controllers. Otherwise, adaptive synchronization method is
adopted to synchronize similar nodes in the network. Some new
synchronization criteria are proposed based on the Lyapunov stability
theory. Finally, numerical simulations are given to illustrate the
feasibility and the effectiveness of the proposed theoretical
results.


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## Body

## 1. Introduction

Complex dynamical networks have attracted increasing attention in recent years, since they have been widely exploited to model many complex systems in the science, engineering, and society [1, 2]. Synchronization of complex network has been found to be a universal phenomenon in nature and it has important potential applications to real-word dynamical systems. As an important and interesting collective behavior, synchronization of complex network has been studied extensively [3–8], such as complete synchronization, projective synchronization [9], impulsive synchronization [10, 11], exponential synchronization [12], adaptive synchronization [13–15], and pinning synchronization [16–21].Most previous research assumes that the dynamics of all nodes are identical. Consequently, the synchronization problem is significantly simplified. However, the assumption that the nodes are completely identical is not realistic in many real-world networks [22], such as in the neural networks, where the internal neurons in the nervous system are impossible to be completely identical due to the differences of the parameters. And the authors of [23, 24] studied synchronization of complex dynamical networks with nonidentical nodes. While, in normal circumstances, the neurons are not completely identical or completely nonidentical. They are similar to each other and they will achieve synchronization to transmit information which shows that the neural system has certain robustness. At this time, we want to know the answers of the following questions, which have a practical meaning for us to analyze and control many realistic networks with similar nodes. How to depict the similarity of the similar nodes? What is the condition that the similar nodes have to satisfy in the network in order to achieve synchronization without controllers? If there is a mutation or a pathological change, then some neurons may have many different characteristics, and they can not synchronize with other neurons. When the similarity of the similar nodes is broken, how to synchronize the nodes in the network?Furthermore, enormous works have been done on the synchronization in complex networks with single-link, and a lot of meaningful conclusions have been obtained. The authors of [19] propose that single-link network is a special case of multilinks network. Therefore, research on multilinks networks are more representative. Multilinks means that there is more than one link between two nodes and each of them has its own property. For instance, there are relationship networks, transportation networks, World Wide Web, and so forth. The transportation network as an example of a network with multilinks, which is made up by combining the corresponding airline network, railway network, and highway network. We can split the multilinks networks into many subnetworks based on the property of the connections. For a transportation network, the transmission speed is different among airline network, railway network, and highway network. In our previous work [19], time-delay was introduced to split complex dynamical networks into subnetworks, upon which a model of complex networks with multilinks has been constructed. However, the important issue of synchronization for complex dynamical networks with similar nodes and multilinks has so far received little attention. The study of the synchronization problem with similar nodes in the complex multilinks network becomes an interesting and challenging topic.In this paper, we give a model of complex multilinks networks with similar nodes. A definition of similar nodes is given and the minimal similarity of similar nodes in the network is analyzed for the first time. We find a threshold of the minimal similarity of similar nodes. If the minimal similarity of similar nodes is bigger than the threshold in the network, then the similar nodes can achieve synchronization. Otherwise, we should add some controllers to the nodes in order to get synchronization. Then some new adaptive synchronization criteria are proposed. Finally, numerical simulations of dynamical networks with similar nodes are presented to demonstrate the feasibility and the effectiveness of the results.


## 2. Model and Preliminaries

The model of complex multilinks network consisting ofN similar nodes with m kinds of properties can be described by
(1)x˙i(t)=Aixi(t)+f(xi(t))+c∑j=1N‍b(0)ijΓ0xj(t)+c∑l=1m-1‍∑j=1N‍b(l)ijΓlxj(t-τl),i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),…,xiN(t))T is the state vector of node i, Ai is a matrix, f(xi(t)) is a smooth nonlinear vector function, c>0 is the coupling strength, Γ0 and Γl are the inner-coupling matrices, Bl=(b(l)ij)N×N, 0≤l≤m-1 represents the topological structure of the lth subnetwork, and τl is the time-delay of the lth subnetwork compared with the basic network (τ0=0). We define b(l)ij=b(l)ji=1 if there is a connection between node i and node j(j≠i) in the lth subnetwork, otherwise b(l)ij=b(l)ji=0. And we define b(l)ii=-∑j=1,j≠iN‍b(l)ij.Network (1) is in a state of asymptotical synchronization, if
(2)x1(t)=x2(t)=⋯=xN(t)→s(t)
as t→∞(1≤i≤N), where s(t)ϵRn is a synchronous solution of the node system x˙i(t)=Aixi(t)+f(xi(t)). We define the error vectors as
(3)ei(t)=xi(t)-si(t).Hereafter, the definitions of similar nodes and the minimal similarity of the similar nodes are given, and a useful assumption and two lemmas are introduced.Definition 1.
IfAi and Aj are matrices with the similar element values, then the node i and j are similar nodes. In the network (1), we define A as the matrix of basic node and Ai(1≤i≤N-1) as matrices of other nodes. Because A and Ai are matrices with similar element values, we define δi=1-(∥A-Ai∥F/∥A∥F), ν=min⁡(δi), (1≤i≤N-1), and ν(0<ν<1) represents the minimal similarity of the nodes in the network. The norm of matrix A is ∥A∥F=(∑i,j=1N‍aij2)1/2, and aij(1≤i,j≤N) are elements of matrix A.Remark 2.
From Definition1, we know ν is an important parameter. When ν approaches to 1, the nodes in the network are similar. If ν satisfies a certain condition, then the similar nodes of the network can achieve synchronization without controllers. On the contrary, when ν is far away from 1, the nodes in the network become not similar, so the nodes cannot achieve synchronization without controllers. That is to say, there exists a threshold, if ν is bigger than the threshold; then the similar nodes in the network can get synchronization without controllers. And the threshold is what we tried to find in the following.Assumption 3.
The smooth nonlinear functionf(·) satisfies the following Lipschitz condition:
(4)∥f(xi(t))-f(s(t))∥≤L∥xi(t)-s(t)∥,
where L is a positive constant.Lemma 4.
For any two vectorsx and y, and a matrix Q>0 with compatible dimensions, one has
(5)2xTy≤xTQx+yTQ-1y.Lemma 5.
IfAϵCN×N, the eigenvalues of A are λi(i=1,2,…,N), then max⁡(λi)≤∥A∥, where ∥A∥ is an arbitrary matrix norm.


## 3. Synchronization Analysis

In this section, suppose there is not a control scheme to synchronize a delayed complex multilinks network with similar nodes. According to system (1), the error dynamical system can be derived as
(6)e˙i(t)=Aiei(t)+f(xi(t))-f(s(t))+c∑j=1N‍b(0)ijΓ0ej(t)+c∑l=1m-1‍∑j=1N‍b(l)ijΓlej(t-τl),
where Ai=A+ΔAi, Ai is the matrix of node i, and A is the matrix of basic node. Because A and Ai are matrices with similar element values, ΔAi=Ai-A. It is easy to see that the synchronization of the complex network (1) is achieved if the zero solution of the error system (6) is globally asymptotically stable, which is ensured by the following theorem. And we find that the minimal similarity of the similar nodes satisfies an inequality for synchronization.Theorem 6.
Consider network (1), if the minimal similarity of the nodes ν is bigger than the threshold, where the threshold of ν is
(7)(λmax⁡(A)+L+cλmax⁡(P0)+c2∑l=1l=m-1‍λmax⁡(PlPlT)+c2(m-1)+∥A∥F∑l=1l=m-1‍)(∥A∥F)-1,
and it also satisfies the following inequality:
(8)λmax⁡(A)+L+cλmax⁡(P0)+(c/2)∑l=1l=m-1λmax⁡(PlPlT)∥A∥F+(c/2)(m-1)+∥A∥F∥A∥F<ν<1,
then the system (1) is synchronized without controllers.Proof.
Construct the following Lyapunov function:(9)V(t)=12∑i=1N‍ei(t)Tei(t)+c2∑l=1m-1‍∫t-τlt‍∑i=1N‍ei(θ)Tei(θ)dθ.
We get
(10)V˙(t)=∑i=1N‍ei(t)Te˙i(t)+c2∑l=1m-1‍∑i=1N‍[ei(t)Tei(t)-ei(t-τl)Tei(t-τl)]=∑i=1N‍ei(t)TAei(t)+∑i=1N‍ei(t)TΔAiei(t)+∑i=1N‍ei(t)[f(xi(t))-f(s(t))]+∑i=1N‍cei(t)T∑j=1N‍b(0)ijΓ0ej(t)+∑i=1N‍cei(t)T∑l=1m-1‍∑j=1N‍b(l)ijΓlej(t-τl)+c2∑i=1N‍∑l=1m-1‍ei(t)Tei(t)-c2∑i=1N‍∑l=1m-1‍ei(t-τl)Tei(t-τl).
Lete(t)=(e1(t),e2(t),…,eN(t)), then we get
(11)V˙(t)≤[e(t)TAe(t)+e(t)TΔAie(t)+Le(t)Te(t)]+ce(t)T(B0⊗Γ0)e(t)+ce(t)T(B1⊗Γ1)e(t-τ1)+ce(t)T(B2⊗Γ2)e(t-τ2)+⋯+ce(t)T(Bm-1⊗Γm-1)e(t-τm-1)+c2(m-1)e(t)Te(t)-c2e(t-τ1)Te(t-τ1)-c2e(t-τ2)Te(t-τ2)-⋯-c2e(t-τm-1)Te(t-τm-1).
Let
(12)P0=B0⊗Γ0,P1=B1⊗Γ1,P2=B1⊗Γ2,⋮Pm-1=Bm-1⊗Γm-1,
where ⊗ represents the Kronecker product. Then by Lemma 4, we have
(13)V˙(t)≤λmax⁡(A)e(t)Te(t)+λmax⁡(ΔAi)e(t)Te(t)+Le(t)Te(t)+cλmax⁡(P0)e(t)Te(t)+c2λmax⁡(P1P1T)e(t)Te(t)+c2λmax⁡(P2P2T)e(t)Te(t)+⋯+c2λmax⁡(Pm-1Pm-1T)e(t)Te(t)+c2(m-1)e(t)Te(t)≤[∑l=1m-1‍λmax⁡(A)+∥ΔAi∥F+L+cλmax⁡(P0)+c2∑l=1m-1‍λmax⁡(PlPlT)+c2(m-1)]e(t)Te(t)≤[∑l=1m-1‍λmax⁡(A)+(1-ν)∥A∥F+L+cλmax⁡(P0)+c2∑l=1m-1‍λmax⁡(PlPlT)+c2(m-1)]e(t)Te(t).
Therefore, if we have
(14)λmax⁡(A)+(1-ν)∥A∥F+L+cλmax⁡(P0)+c2∑l=1m-1‍λmax⁡(PlPlT)+c2(m-1)<0,
then V˙(t)≤0. So we get the synchronization criterion as follows:
(15)λmax⁡(A)+L+cλmax⁡(P0)+(c/2)∑l=1l=m-1λmax⁡(PlPlT)∥A∥F+(c/2)(m-1)+∥A∥F∥A∥F<ν<1.
If ν satisfies (15), the nodes are synchronized. Thus we complete the proof.Remark 7.
The matrix of basic node can be chosen at random fromAi(1≤i≤N). No matter which one we choose, Theorem 6 also holds.
Furthermore, noise plays an important role in the process of synchronization. Here we consider the influence of the noise. If there is an additive noise in the system (1) in the form of
(16)x˙i(t)=Aixi(t)+f(xi(t))+c∑j=1N‍b(0)ijΓ0xj(t)+c∑l=1m-1‍∑j=1N‍b(l)ijΓlxj(t-τl)+ηi(t),i=1,2,…,N,
where ηi(t)ϵRn is the zero mean bounded noise. Using system (16), we can easily get the following error system:
(17)e˙i(t)=(A+ΔAi)ei(t)+f(xi(t))-f(s(t))+c∑j=1N‍b(0)ijΓ0ej(t)+c∑l=1m-1‍∑j=1N‍b(l)ijΓlej(t-τl)+ηi(t),
then we get
(18)E(e˙i(t))=(A+ΔAi)E[ei(t)]+E[f(xi(t))-f(s(t))]+c∑j=1N‍b(0)ijΓ0E[ej(t)]+c∑l=1m-1‍∑j=1N‍b(l)ijΓlE[ej(t-τl)]+E[ηi(t)].
Finally, we get the theorem as follows.Theorem 8.
When there is a noise or perturbation, considering the network (16), if the following condition holds
(19)λmax⁡(A)+L+cλmax⁡(P0)+(c/2)∑l=1l=m-1λmax⁡(PlPlT)∥A∥F+(c/2)(m-1)+∥A∥F∥A∥F<ν<1,
then E[ei(t)] approaches to zero.The proof process of Theorem8 is similar to the proof process of Theorem 6, so here it is omitted.


## 4. Adaptive Synchronization

In this section, a control scheme is developed to synchronize a delayed complex multilinks network with similar nodes, which do not satisfy the synchronization criterion (15). And the following adaptive controllers are used:
(20)ui=-diei(t),1≤i≤N.
And the updating laws are
(21)d˙i=kiei(t)Tei(t),1≤i≤N,
where ki(1≤i≤N) are positive constants. The adaptive controllers (20) are widely used in solving many synchronous problems.Then the controlled network can be characterized as(22)x˙i(t)=Aixi(t)+f(xi(t))+c∑j=1N‍b(0)ijΓ0xj(t)+c∑l=1m-1‍∑j=1N‍b(l)ijΓlxj(t-τl)+ui,i=1,2,…,N.
According to system (22), the following error dynamical system can be derived:
(23)e˙i(t)=(A+ΔAi)ei(t)+f(xi(t))-f(s(t))+c∑j=1N‍b(0)ijΓ0ej(t)+c∑l=1m-1‍∑j=1N‍b(l)ijΓlej(t-τl)-diei(t).It is clear to see that the synchronization of the controlled complex network (22) is achieved if the zero solution of the error system (23) is globally asymptotically stable, which is ensured by the following theorem.Theorem 9.
Consider the network (22) under the actions of the controllers (20) and the updating laws (21). If the following condition holds:
(24)λmax⁡(A)+(1-ν)∥A∥F+L+cλmax⁡(P0)+c2∑l=1l=m-1‍λmax⁡(PlPlT)+c2(m-1)<d*,
where d* is a sufficiently large positive constant to be determined, then the system (1) is synchronized.Proof.
Construct the following Lyapunov function:(25)V(t)=12∑i=1N‍ei(t)Tei(t)+c2∑l=1m-1‍∫t-τlt‍∑i=1N‍ei(θ)Tei(θ)dθ+12∑i=1N‍(di-d*)2ki.
Clearly, V(t) is positive. Then the derivative of V(t) is obtained as
(26)V˙(t)=∑i=1N‍ei(t)Te˙i(t)+c2∑l=1m-1‍∑i=1N[(t-τl)Tei(t)Tei(t)‍-ei(t-τl)Tei(t-τl)]+∑i=1N‍[diei(t)Tei(t)-d*ei(t)Tei(t)]=∑i=1N{∑l=1m-1‍ei(t)TAei(t)+ei(t)TΔAiei(t)+ei(t)‍×[f(xi(t))-f(s(t))]+cei(t)T∑j=1N‍b(0)ijΓ0ej(t)+cei(t)T∑l=1m-1‍∑j=1N‍b(l)ijΓlej(t-τl)-diei(t)Tei(t)+c2∑l=1m-1‍ei(t)Tei(t)-c2∑l=1m-1‍ei(t-τl)Tei(t-τl)+diei(t)Tei(t)-d*ei(t)Tei(t)∑l=1m-1‍}.
Let e(t)=(e1(t),e2(t),…,eN(t)), then we get
(27)V˙(t)≤[e(t)TAe(t)+e(t)TΔAie(t)+Le(t)Te(t)]+ce(t)T(B0⊗Γ0)e(t)+ce(t)T(B1⊗Γ1)e(t-τ1)+ce(t)T×(B2⊗Γ2)e(t-τ2)+⋯+ce(t)T(Bm-1⊗Γm-1)e(t-τm-1)+c2(m-1)e(t)Te(t)-c2e(t-τ1)Te(t-τ1)-c2e(t-τ2)Te(t-τ2)-⋯-c2e(t-τm-1)Te(t-τm-1)-d*e(t)Te(t).
Let
(28)P0=B0⊗Γ0,P1=B1⊗Γ1,P2=B1⊗Γ2,⋮Pm-1=Bm-1⊗Γm-1,
where ⊗ represents the Kronecker product. Then by Lemma 4, we have
(29)V˙(t)≤λmax⁡(A)e(t)Te(t)+λmax⁡(ΔAi)e(t)Te(t)+Le(t)Te(t)+cλmax⁡(P0)e(t)Te(t)+c2λmax⁡(P1P1T)e(t)Te(t)+⋯+c2λmax⁡(Pm-1Pm-1T)e(t)Te(t)+c2(m-1)e(t)Te(t)-d*e(t)Te(t)≤[∑l=1m-1‍λmax⁡(A)+∥ΔAi∥F+L+cλmax⁡(P0)+c2∑l=1m-1‍λmax⁡(PlPlT)+c2(m-1)-d*∑l=1m-1‍]e(t)Te(t)≤[∑l=1m-1‍λmax⁡(A)+(1-ν)∥A∥F+L+cλmax⁡(P0)+c2∑l=1m-1‍λmax⁡(PlPlT)+c2(m-1)-d*∑l=1m-1‍]e(t)Te(t).
Therefore, if we have
(30)λmax⁡(A)+(1-ν)∥A∥F+L+cλmax⁡(P0)+c2∑l=1m-1‍λmax⁡(PlPlT)+c2(m-1)<d*,
then V˙(t)≤0. Here we complete the proof.Remark 10.
If there is not a nonlinear function in system (1), then the network (1) is transferred into
(31)x˙i(t)=Aixi(t)+c∑j=1N‍b(0)ijΓ0xj(t)+c∑l=1m-1‍∑j=1N‍b(l)ijΓlxj(t-τl),i=1,2,…,N.
Likewise, we can design the controllers as in (20) and (21). If the following condition holds:
(32)λmax⁡(A)+(1-ν)∥A∥F+cλmax⁡(P0)+c2∑l=1m-1‍λmax⁡(PlPlT)+c2(m-1)<d*,
then the system (31) is synchronized, where d* is a sufficiently large positive constant to be determined.Remark 11.
The single-link network is a special case of multilinks networks [19]. When there is not a delay, the network (1) is transferred into the following single-link network:
(33)x˙i(t)=Aixi(t)+f(xi(t))+c∑j=1Nb(0)ijΓ0xj(t),i=1,2,…,N,‍
and the controllers are designed as in (20)-(21). If the following condition holds:
(34)λmax⁡(A)+(1-ν)∥A∥F+L+cλmax⁡(P0)<d*,
then the system (33) is synchronized, where d* is a sufficiently large positive constant to be determined.


## 5. Numerical Simulation

In this section, we use some examples to explain the influence of the proposed criteria, and we consider a network consisting of 30 similar nodes. The multilinks network with 2 properties can be described as follows:(35)x˙i(t)=Aixi(t)+f(xi(t))+c∑j=130‍b(0)ijΓ0xj(t)+c∑j=130‍b(1)ijΓ1xj(t-τ),
where 1⩽i⩽30, B0=(b(0)ij)30×30, and B1=(b(1)ij)30×30 are symmetrically diffusive coupling matrixes with b(0)ij, b(1)ij=0 or 1. Γ0=Γ1=diag⁡(1,1,1), c=2, τ=0.01, f(xi(t))=(0.6sin(xi1),0.6sin(xi2),0.6sin(xi3))T. According to Assumption 3, we can know that L=0.6:
(36)Ai=(-10+0.1*rand0.1*rand0.1*rand0.1*rand-10+0.1*rand0.1*rand0.1*rand0.1*rand-10+0.1*rand),
where the function of rand can produce a random number between 0 and 1. According to the definition of similar nodes, we know Ai, 1⩽i⩽30 are matrices of the similar nodes. And
(37)A=(-10000-10000-10)
is the matrix of the basic node. So
(38)ΔAi=(0.1*rand0.1*rand0.1*rand0.1*rand0.1*rand0.1*rand0.1*rand0.1*rand0.1*rand).
According to the precise calculation, λmax⁡(A)=-10, λmax⁡(P0)=0, λmax⁡(P1TP1)=18.5139. Based on the stability analysis, we get (λmax⁡(A)+L+cλmax⁡(P0)+(c/2)∑l=1l=m-1λmax⁡(PlPlT)+(c/2)(m-1)+∥A∥F)(∥A∥F)-1=(-100+0.6+18.5139+1+30000)/30000=0.5388. According to (38), because  0<rand<1, the biggest changes are
(39)ΔAi=(0.10.10.10.10.10.10.10.10.1).Then we compute theν=0.9983, 0.5388<ν<1, so the similar nodes can achieve synchronization which satisfies Theorem 6. From Figures 1(a)–1(d), we know that the similar nodes in the network achieved synchronization under different network models.Separate synchronous variablesei1, ei2, ei3(1⩽i⩽30) of different network models where ei1 are shown by the red line, ei2 are shown by the blue line, and ei3 are shown by the green line. c=2; τ=0.01. (a) B0 and B1 are small-world network models, and the rewiring probability among nodes is 0.3 and 0.6. (b) B0 and B1 are scale-free network models, and their minimum degrees are 2 and 3. (c) B0 and B1 are random network models, and their connection probability among nodes is 0.1 and 0.3. (d) B0 is a small-world network model, and the rewiring probability among nodes is 0.3. B1 is a scale-free network model, and the minimum degree is 2. Those similar nodes can achieve synchronization without controllers (color online).
(a)
(b)
(c)
(d)Furthermore, in order to verify Theorem9, we choose the model (16) as the second example, where the Brownian motion satisfies Eω(t)=0, Dω(t)=1, and the parameters are the same with the first example. Figures 2(a)–2(d) plot the synchronous errors converge to 0 in finite time under different network models with noise, which reflects that similar nodes have a certain robustness. In our future work, we will consider the model (35) with Gaussian noise [25] or 1/f noise [26], and the stochastic bounded model like [27] in the complex network with similar nodes will be studied.The similar nodes can achieve synchronization without controllers when the Brownian motion satisfiesEω(t)=0, Dω(t)=1. And separate synchronous variables ei1, ei2, ei3(1⩽i⩽30) of different network models where ei1 are shown by the red line, ei2 are shown by the blue line, and ei3 are shown by the green line. c=2; τ=0.01. The network models of (a)–(d) are the same with Figures 1(a)–1(d) (color online).
(a)
(b)
(c)
(d)Next, another example as the third one describes the controlled network using Lü systems and considers the network consisting of30 nodes. The node dynamical system is x˙i=(-36xi1+36xi2;20xi2-xi1xi3;-3xi3+xi1xi2), for i=1,2,…,30. And ΔAi are the same with the first example. Since Lü attractor is bounded, we suppose that all nodes are running in the given bounded region. There exists the constants M1=25, M2=30, M3=45 satisfying ∥xij∥2,∥sj∥2⩽Mj for 1⩽i⩽30 and 1⩽j⩽3 [28]. Thus we have
(40)∥f(xi)-f(s)∥≤2M12+M22+M32∥ei∥2≈64.6142∥ei∥2,
then we can know that L=64.6142. And other parameters are the same with the first example. We have λmax⁡(A)=20, and
(41)(∑l=1l=m-1‍λmax⁡(A)+L+cλmax⁡(P0)+c2∑l=1l=m-1λmax⁡(PlPlT)c2(m-1)+∥A∥F‍)(∥A∥F)-1=20+64.6142+18.5139+1+30013001=2.9008.It does not satisfy Theorem6. So the nodes cannot achieve synchronization without controllers. Simulation results are given in Figures 3(a)–3(d) which show the evolution process of 30 state variables in three dimensions. And it verified that the similar nodes cannot achieve synchronization without controllers.Separate synchronous variablesei1, ei2, ei3(1⩽i⩽30) of different network models where ei1 are shown by the red line, ei2 are shown by the blue line, and ei3 are shown by the green line. c=2; τ=0.01. The dynamics of nodes satisfy the Lü system. The nodes of the network cannot achieve synchronization without controllers. The network models of (a)–(d) are the same with Figures 1(a)–1(d) (color online).
(a)
(b)
(c)
(d)According to the adaptive synchronization criteria, we add the adaptive controllers (20) and (21) to these similar nodes of the network. k1=k2=k3=1. The curves of error vectors ei1, ei2, ei3(i=1,2,3) are shown in Figures 4(a)–4(d).Separate synchronous variablesei1, ei2, ei3(1⩽i⩽30) of different network models where ei1 are shown by the red line, ei2 are shown by the blue line, and ei3 are shown by the green line. c=2; τ=0.01. The dynamics of nodes satisfy the Lü system. The nodes are controlled by the adaptive controllers (20) and (21). The network models of (a)–(d) are the same with Figures 1(a)–1(d) (color online).
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(d)To be more persuadable, with the same calculation method, Figures5(a)–5(d) plot the synchronous errors of networks with links owning 3 properties. Figures 5(a)–5(d) have different network models, and τ1=0.01. This demonstrates that our theorem is not only applicable to multilinks network owning two links properties but also to real networks with multiple links. From Figures 1(a)–1(d) to Figures 5(a)–5(d), we attain that our theorems are feasible in different network models under different conditions. This result is more helpful to real networks not just to model networks. From the above simulation results, we can see that these similar nodes can achieve synchronization under the impacts of the adaptive controllers. In the future, we will consider the possible application of this paper to packet delay issue in computer communications.Separate synchronous variablesei1, ei2, ei3(1⩽i⩽30) of different network models where c=2, τ=0.01, and τ1=0.01. (a) B0, B1, and B2 are small-world network models, the rewiring probability among nodes is 0.3, 0.6, and 0.7. (b) B0, B1, and B2 are scale-free network models, and their minimum degrees are 1, 2, and 3. (c) B0, B1, and B2 are random network models, and their connection probability among nodes is 0.1, 0.3, and 0.5. (d) B0 is a small-world network model, and the rewiring probability among nodes is 0.3. B1 is a scale-free network model, and the minimum degree is 2. B2 is a random network model, and the connection probability among nodes is 0.1. The dynamics of nodes satisfy the Lü system. The multilinks network can achieve synchronization under the adaptive controllers (color online).
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## 6. Conclusion

In this paper, we present the definition of similar nodes and analyze their minimal similarity in the network for the first time. We find the threshold of the minimal similarity of the similar nodes if it is bigger than the threshold, then the similar nodes can achieve synchronization without controllers. Otherwise, we have to add some controllers in order to get synchronization. So some new adaptive synchronization criteria are proposed to realize the synchronization of multilinks networks with similar nodes. Finally, numerical simulations are provided to show the effectiveness and the correctness of the proposed criteria. The model and the principles designed in this paper are very useful to analyze and control the dynamical multilinks networks with similar nodes, such as heart cells networks and neural networks.



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*Source: 736585-2013-06-12.xml*