Adaptive Synchronization of Complex Dynamical Multilinks Networks with Similar Nodes

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.

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.

Model and Preliminaries
The model of complex multilinks network consisting of  similar nodes with  kinds of properties can be described by where   () = ( 1 (),  2 (), . . .,   ())  is the state vector of node ,   is a matrix, (  ()) is a smooth nonlinear vector function,  > 0 is the coupling strength, Γ 0 and Γ  are the inner-coupling matrices,   = ( () ) × , 0 ≤  ≤  − 1 represents the topological structure of the th subnetwork, and   is the time-delay of the th subnetwork compared with the basic network ( 0 = 0).We define  () =  () = 1 if there is a connection between node  and node  ( ̸ = ) in the th subnetwork, otherwise  () =  () = 0.And we define  () = − ∑  =1, ̸ =   () .Network (1) is in a state of asymptotical synchronization, if as  → ∞ (1 ≤  ≤ ), where ()  is a synchronous solution of the node system ẋ  () =     () + (  ()).We define the error vectors as 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.If   and   are matrices with the similar element values, then the node  and  are similar nodes.In the network (1), we define  as the matrix of basic node and   (1 ≤  ≤  − 1) as matrices of other nodes.Because  and   are matrices with similar element values, we define , and ] (0 < ] < 1) represents the minimal similarity of the nodes in the network.The norm of matrix  is ‖‖  = (∑  ,=1  2  ) 1/2 , and   (1 ≤ ,  ≤ ) are elements of matrix .
Remark 2. From Definition 1, we know ] is an important parameter.When ] approaches to 1, the nodes in the network are similar.If ] satisfies a certain condition, then the similar nodes of the network can achieve synchronization without controllers.On the contrary, when ] 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 ] 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.
where  is a positive constant.

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 where   =  + Δ  ,   is the matrix of node , and  is the matrix of basic node.Because  and   are matrices with similar element values, Δ  =   − .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 ] is bigger than the threshold, where the threshold of ] is and it also satisfies the following inequality: then the system (1) is synchronized without controllers.
where ⊗ represents the Kronecker product.Then by Lemma 4, we have Therefore, if we have then V() ≤ 0. So we get the synchronization criterion as follows: If ] satisfies (15), the nodes are synchronized.Thus we complete the proof.
Remark 7. The matrix of basic node can be chosen at random from   (1 ≤  ≤ ).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 where   ()  is the zero mean bounded noise.Using system (16), we can easily get the following error system: then we get 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 then [  ()] approaches to zero.
The proof process of Theorem 8 is similar to the proof process of Theorem 6, so here it is omitted.

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: And the updating laws are where   (1 ≤  ≤ ) are positive constants.The adaptive controllers (20) are widely used in solving many synchronous problems.
Then the controlled network can be characterized as According to system (22), the following error dynamical system can be derived: 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: where  * is a sufficiently large positive constant to be determined, then the system (1) is synchronized.
where ⊗ represents the Kronecker product.Therefore, if we have then V() ≤ 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 Likewise, we can design the controllers as in (20) and (21).If the following condition holds: then the system (31) is synchronized, where  * 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: and the controllers are designed as in ( 20)- (21).If the following condition holds: then the system (33) is synchronized, where  * is a sufficiently large positive constant to be determined.

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: where 1 ⩽  ⩽ 30,  0 = ( (0) ) 30 × 30 , and  1 = ( (1) ) 30 × 30 are symmetrically diffusive coupling matrixes with  (0) ,  (39) Then we compute the ] = 0.9983, 0.5388 < ] < 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.
Furthermore, in order to verify Theorem 9, we choose the model (16) as the second example, where the Brownian motion satisfies () = 0, () = 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/ noise [26], and the stochastic bounded model like [27] in the complex network with similar nodes will be studied.
Next, another example as the third one describes the controlled network using Lü systems and considers the network consisting of 30 nodes.The node dynamical system is ẋ  = (−36 1 + 36 2 ; 20 2 −  1  3 ; −3 3 +  1  2 ), for  = 1, 2, . . ., 30.And Δ  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  1 = 25,  2 = 30,     According to the adaptive synchronization criteria, we add the adaptive controllers (20)  To be more persuadable, with the same calculation method, Figures 5(a)-5(d) plot the synchronous errors of networks with links owning 3 properties.Figures 5(a)-5(d) have different network models, and  1 = 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.

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.

Figure 1 :
Figure 1: Separate synchronous variables  1 ,  2 ,  3 (1 ⩽  ⩽ 30) of different network models where  1 are shown by the red line,  2 are shown by the blue line, and  3 are shown by the green line. = 2;  = 0.01.(a)  0 and  1 are small-world network models, and the rewiring probability among nodes is 0.3 and 0.6.(b)  0 and  1 are scale-free network models, and their minimum degrees are 2 and 3. (c)  0 and  1are random network models, and their connection probability among nodes is 0.1 and 0.3.(d)  0 is a small-world network model, and the rewiring probability among nodes is 0.3. 1 is a scale-free network model, and the minimum degree is 2. Those similar nodes can achieve synchronization without controllers (color online).

Figure 2 :
Figure 2: The similar nodes can achieve synchronization without controllers when the Brownian motion satisfies () = 0, () = 1.And separate synchronous variables  1 ,  2 ,  3 (1 ⩽  ⩽ 30) of different network models where  1 are shown by the red line,  2 are shown by the blue line, and  3 are shown by the green line. = 2;  = 0.01.The network models of (a)-(d) are the same with Figures 1(a)-1(d) (color online).

Figure 3 :
Figure 3: Separate synchronous variables  1 ,  2 ,  3 (1 ⩽  ⩽ 30) of different network models where  1 are shown by the red line,  2 are shown by the blue line, and  3 are shown by the green line. = 2;  = 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).

the function of rand can produce a random number between 0 and 1 .Figure 4 :
Figure 4: Separate synchronous variables  1 ,  2 ,  3 (1 ⩽  ⩽ 30) of different network models where  1 are shown by the red line,  2 are shown by the blue line, and  3 are shown by the green line. = 2;  = 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).

Figure 5 :
Figure5: Separate synchronous variables  1 ,  2 ,  3 (1 ⩽  ⩽ 30) of different network models where  = 2,  = 0.01, and  1 = 0.01.(a)  0 ,  1 , and  2 are small-world network models, the rewiring probability among nodes is 0.3, 0.6, and 0.7.(b)  0 ,  1 , and  2 are scale-free network models, and their minimum degrees are 1, 2, and 3. (c)  0 ,  1 , and  2 are random network models, and their connection probability among nodes is 0.1, 0.3, and 0.5.(d)  0 is a small-world network model, and the rewiring probability among nodes is 0.3. 1 is a scale-free network model, and the minimum degree is 2.  2 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).

( 41 )
It does not satisfy Theorem 6.So the nodes cannot achieve synchronization without controllers.Simulation results are given in Figures3(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.
1, and  2 are small-world network models, the rewiring probability among nodes is 0.3, 0.6, and 0.7.(b)  0 ,  1 , and  2 are scale-free network models, and their minimum degrees are 1, 2, and 3. (c)  0 ,  1 , and  2 are random network models, and their connection probability among nodes is 0.1, 0.3, and 0.5.(d)0 is a small-world network model, and the rewiring probability among nodes is 0.3. 1 is a scale-free network model, and the minimum degree is 2.  2 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).thenwe can know that  = 64.6142.And other parameters are the same with the first example.We have  max () = 20, and ( max () +  +  max ( 0 )