# Adaptive Exponential Synchronization of Coupled Complex Networks on General Graphs

Authors: Song Liu; Xianfeng Zhou; Wei Jiang; Yizheng Fan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the synchronization in complex dynamical networks, where the coupling configuration corresponds to a weighted graph. An adaptive synchronization method on general coupling configuration graphs is given. The networks may synchronize at an arbitrarily given exponential rate by enhancing the updated law of the variable coupling strength and achieve synchronization more quickly by adding edges to original graphs. Finally, numerical simulations are provided to illustrate the effectiveness of our theoretical results.


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## Body

## 1. Introduction

In general, complex networks consist of a large number of nodes and links among them, in which a node is a fundamental cell with specific activity, hence, complex networks and graphs closely contact each other. The dynamics on complex networks is one on graphs, though the graphs may have different characteristic, for example, classical random graph model [1], small-world model [2–4], scale-free model [5], or others that are closely related to natural structure.Synchronization of coupled complex dynamical networks is a universal phenomenon in various fields of science and society. All kinds of synchronization, including adaptive synchronization, global synchronization, antisynchronization, phase synchronization, projection synchronization, and generalized synchronization have been studied [6–14], in particular on the synchronization of an array of linearly coupled identical systems [15–18]. Researches [15, 19] imply that the structural properties of a network must have inevitable effect on the ability and speed of synchronization, but such work still does not see more. In addition, as pointed in [18], the coupling strength may be self-adaptive due to the spontaneousness of updated law, not be calculated numerically in many other works.In the paper [18], an adaptive synchronization method is introduced, and the networks can synchronize by enhancing the coupling strength automatically under a simple updated law. However, their work is limited to tree-like networks and cannot be applied to general networks. In fact, a tree is a graph without cycles; in this paper, we try to extend the work in [18] to general networks or graphs. We find that their method is in fact effective to general graphs by a rigorous proof, and the networks can also achieve synchronization at an arbitrarily given exponential rate by increasing coupling strength. Based on the knowledge of spectral graph theory, networks can synchronize more quickly by increasing the algebraic connectivity of the graphs, which can be realized by adding edges to original graphs. Finally, numerical simulations are provided to illustrate the effectiveness of theoretical results.


## 2. Preliminaries

In this section, we now introduce some notations and preliminaries.Consider the complex dynamical network consisting ofN linearly and diffusively coupled identical nodes with full diagonal coupling, and each node is an n-dimensional dynamical oscillator which may be chaotic. The state equations of the network are
(1)x˙i=f(xi)+c∑j=1N‍aijΓxj,i=1,2,…,N,
where xi=(xi1,xi2,…,xin)T∈ℝn is a state vector of node i, f(xi)=(f1(xi),f2(xi),…,fn(xi))T:ℝn→ℝn is a given nonlinear vector valued function describing the dynamics of the nodes, c>0 represents coupling strength, and the inner coupling link matrix is a diagonal matrix Γ=diag{r1,r2,…,rn} with ri>0. The coupling configuration matrix A=(aij)N×N is a zero row sums matrix with nonnegative off-diagonal entries, representing the topological structure of the network.There is a weighted graph corresponding to the coupling configuration matrixA, called the coupling configuration graph, defined as a graph G on vertices 1,2,…,N which contains an edge ij(i≠j) with weight aij if and only if aij>0. Giving an arbitrary orientation of the edges of G, so that each edge has a head and a tail, and a labeling of edges as e1,e2,…,eE, where E denotes the number of edges of G. We obtain an edge-vertex incidence matrix of G, denoted as M:=M(A)=(mij)E×N, which is defined as mij=aij  (resp., mij=-aij) if the the edge ei has the vertex j as a head (resp., a tail), and mij=0 otherwise. The Laplacian matrix of G is defined as L=MTM=D-B, where D is a diagonal matrix, and its ith diagonal entry of which is exactly the degree of the vertex i, that is, di=∑j∈N(i)‍aij, where N(i) is the set of neighbors of i in the graph G (or the vertices joining i by edges), and B=(bij) is a weighted adjacency matrix of G such that bij=aij if ij is an edge of G and bij=0 otherwise. One can find that L=-A and is symmetric and positive semidefinite, so its eigenvalues can be arranged as
(2)0=λ0≤λ1≤λ2≤⋯≤λN-1,
where λ0=0 as L has zero row sums, and λ1>0 if and only if G is connected and is called the algebraic connectivity of G by Fiedler [20] in the case of G is simple (i.e., all edges have weight 1). If G is connected, the corresponding eigenvector of the eigenvalue 0 is an all one vector (up to a scalar multiple), denoted by 1. One can refer to Chung [21] and Merris [22] for the details of Laplacian matrices of graphs.If we denotex=(x1T,x2T,…,xNT)T,F(x)=(fT(x1),fT(x2),…,fT(xN))T and substitute -L for A, (1) is transformed as
(3)x˙=F(x)-cL⊗Γx.
Taking the transformation y=xeat(a>0), (3) is written as
(4)y˙=eatF(e-aty)-(cL⊗Γ-aINn)y,
where INn=IN⊗In, and Im denotes an identity matrix of order m.In this paper, we adopt thel2-norm for vectors and the induced spectral norm for matrices. We always suppose that the function F in (3) is Lipschitz continuous, or equivalently f in (1) is Lipschitz continuous, that is, there exists a constant l>0 such that for any x,y∈Rn,
(5)∥f(x)-f(y)∥≤l·∥x-y∥.Denote byM=M⊗In, one can obtain the following lemma.Lemma 1 (see [15]).
Suppose that the coupling configuration graph corresponding toA=-L is connected. Then, the dynamical networks (4) achieve synchronization if and only if limt→∞∥My∥=0.


## 3. Main Results

If the network (4) achieves synchronization, surely the network (3) or (1) achieves synchronization with exponential rate a. So, we directly discuss (4), the main result of this paper is stated as follows.Theorem 2.
Suppose thatF is Lipschitz continuous, and the coupling configuration graph corresponding to A=-L is connected. Then, the network (4) achieves synchronization, and in particular the network (1) achieves exponential synchronization with rate a, when the coupling strength c is adapted duly according to the following updated law:
(6)c˙=kyT(t)(L2⊗Γ)y(t),
where k>0 is an arbitrary constant.Proof.
Construct a Lyapunov function(7)V(t)=12∥My(t)∥2+12k(c-h)2,
where h is a sufficiently large constant.
Noting thatMTM=L⊗In, and substituting (6) for c˙, we get the derivative of V(t) along the trajectories of (4) as follows:
(8)V˙=(My)T(My˙)+1k(c-h)c˙=(My)TM[eatF(e-aty)-(cL⊗Γ-aINn)y]+(c-h)yT(L2⊗Γ)y=eat(My)TMF(e-aty)-hyT(L2⊗Γ)y+ayT(L⊗In)y-cyTMTM(L⊗Γ)y+cyT(L2⊗Γ)y.
Noting that
(9)MTM(L⊗Γ)=(L⊗In)(L⊗Γ)=L2⊗Γ,
we have
(10)V˙=eat(My)TMF(e-aty)-hyT(L2⊗Γ)y+ayT(L⊗In)y.
LetG be the coupling configuration graph corresponding to A=-L. Let G→ be the directed graph after an arbitrary orientation of the edges of G, and let ℰ(G→) be the edge set of G→. An oriented edge in ℰ(G→) is denoted by e=i→j, where j is the head of e and i is the tail of e.
Observe that(11)(My)TMF(e-aty)=∑e=i→j∈ℰ(G→)‍mij(yj-yi)T·mij[f(e-atyj)-f(e-atyi)]≤e-at·l·∑e=i→j∈ℰ(G→)‍mij2∥yj-yi∥2=le-at(My)T(My)=le-atyT(L⊗In)y.
Hence,
(12)V˙≤(l+a)yT(L⊗In)y-hyT(L2⊗Γ)y≤(l+a)yT(L⊗In)y-hryT(L2⊗In)y,
where r=min1≤i≤nri>0.
Note thatL is a real and symmetric matrix, so we may let z0:=(1/n)1,z1,…,zN-1 as unit orthogonal eigenvectors corresponding to eigenvalues λ0=0,λ1,…,λN-1 of the matrix L, respectively. Let z¯0,z¯1,…,z¯n-1 be an arbitrary orthogonal base of ℝn. Thus, zi⊗z¯j, i=0,1,…,N-1, j=0,1,…,n-1, form an orthogonal base of ℝNn, which are also the unit orthogonal eigenvectors of L⊗In and L2⊗In. As the associated graph of L or A is connected, λ1>0, we now write
(13)y(t)=∑i,j‍αij(t)zi⊗z¯j.
Then, we obtain
(14)V˙≤(l+a)∑i≥0,j‍λiαij(t)2-hr∑i≥0,j‍λi2αij(t)2=(l+a)∑i≥1,j‍λiαij(t)2-hr∑i≥1,j‍λi2αij(t)2≤(l+a)∑i≥1,j‍λiαij(t)2-hr∑i≥1,j‍λiλ1αij(t)2=(l+a-hrλ1)∑i≥1,j‍λiαij(t)2=(l+a-hrλ1)yT(L⊗In)y=(l+a-hrλ1)(My)T(My).
Hence, we can choose a sufficiently large h such that β:=l+a-hrλ1<0. Thus,
(15)V˙≤β(My(t))T(My(t)).
It is obvious thatV˙=0 if and only if My=0. In addition, if My=0, then y=1⊗y1, and V˙=0 by (10). Therefore, the set E={y∣My=0} is the largest invariant set contained in V˙=0 for the system (4). Then, we obtain limt→∞∥My∥=0 based on the LaSalle invariant principle of differential equations. Hence, by Lemma 1, the complex dynamical network (4) is synchronized under the updated law (6) of coupling strength c. In particular, the complex dynamical network (1) is exponentially synchronized with rate a.Remark 3.
In [18], the updated law of coupling strength c is given as follows:
(16)c˙=Kx(t)TMTMMTMΓx(t)=Kx(t)TL2⊗Γx(t),
where Γ=IN⊗Γ; the synchronization speed of complex dynamical network (1) or (3) can be controlled by the constant K in (16). Here, we take k=Ke2at; the synchronization is realized with exponential rate a. So, we make a further illustration of the above statement in [18] from (6) and Theorem 2.Remark 4.
From (15) in the proof of Theorem 2, if β is taken smaller, V(t) will decrease quickly, and then the synchronization of the network will be attained quickly. This will be done if the Laplacian matrix of the coupling configuration graph has a larger eigenvalue λ1. Based on the knowledge of spectral graph theory (or see [23]), λ1 is not decreased by adding edges to original graph. Therefore, we can add edges to the coupling configuration graph such that the complex network achieves synchronization more quickly.Remark 5.
As the coupling configuration graph is connected and Lemma1 holds, the dynamical network (4) achieves synchronization. If the graph is disconnected, the network only can synchronize in each connected component of the graph but not in the whole graph.


## 4. Numerical Simulations

In this section, we present several numerical examples to show the theoretical results. In particular, we consider complex dynamical networks with the Lorenz system as each node(17)[x˙i1x˙i2x˙i3]=[α(xi2-xi1)γxi1-xi1xi3-xi2xi1xi2-βxi3],i=1,2,3,4,5,
which is chaotic when α=10, β=8/3, γ=28. Three small networks with five nodes are shown in Figure 1, where the weight of every edge in each network is 1. Choose the inner coupling link matrix to be Γ=diag{1,2,3}.Three small networks.
(a)
(b)
(c)The corresponding coupling configuration matrices are listed in the same order as follows:(18)[-110001-210001-210001-210001-1],[-110001-310101-210001-210101-2],[-110001-411101-210011-310101-2].According to Theorem2, the updated law of coupling strength c is chosen as c˙=KxT(t)(L2⊗Γ)x(t) (i.e., k=K, a=0 in (6)). Here K=0.0001, the initial coupling strength c=0, and the initial value of x is taken as [-8,2,35,1,-9,-3,-4,5,1,2,-4,1,-7,8,4]T. The synchronization errors ∥Mx∥ of three networks are shown in Figure 2, respectively, where the figure listed below is a local magnification of the above figure, and Figure 2 also shows that the method in [18] is effective to general networks. Numerical simulations show that synchronization can be reached more quickly by increasing the algebraic connectivity of the graph, which can be realized by adding edges to original graph.Synchronization errors∥Mx∥ of three networks in Figure 1.
(a)
(b)Figure3 shows that adaptive exponential synchronization is achieved for complex dynamical network in Figure 1(a), where k=0.0001, a=0, and a=3 in (6). The same results also hold for the networks in Figures 1(b) and 1(c).Figure 3
Exponential Synchronization errors∥Mx∥ of the network in Figure 1(a).


## 5. Conclusions

In this paper, an adaptive synchronization method on general networks or graphs is obtained. According to our method, networks may synchronize at an arbitrarily given exponential rate by increasing coupling strength and synchronize more quickly by increasing the algebraic connectivity of the corresponding graphs, which can be achieved by adding edges to original graphs. The obtained results also extend the work in [18] as a tree is a graph without cycles. Finally, numerical simulations are provided to illustrate the effectiveness of our theoretical results.



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