Robustness of cluster synchronous patterns in small-world networks with inter-cluster co-competition balance Jianbao Zhang, Zhongjun Ma, and Guanrong Chen Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 023111 (2014); doi: 10.1063/1.4873524 View online: http://dx.doi.org/10.1063/1.4873524 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of spike-time-dependent plasticity on the stochastic resonance of small-world neuronal networks Chaos 24, 033125 (2014); 10.1063/1.4893773 Role of social environment and social clustering in spread of opinions in coevolving networks Chaos 23, 043123 (2013); 10.1063/1.4833995 Effects of time delay on the stochastic resonance in small-world neuronal networks Chaos 23, 013128 (2013); 10.1063/1.4790829 Small-world topology of functional connectivity in randomly connected dynamical systems Chaos 22, 033107 (2012); 10.1063/1.4732541 Multistability, local pattern formation, and global collective firing in a small-world network of nonleaky integrateand-fire neurons Chaos 19, 015109 (2009); 10.1063/1.3087432
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CHAOS 24, 023111 (2014)
Robustness of cluster synchronous patterns in small-world networks with inter-cluster co-competition balance Jianbao Zhang,1 Zhongjun Ma,2,a) and Guanrong Chen3 1
School of Science, Hangzhou Dianzi University, Hangzhou 310018, China School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China 3 Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, China 2
(Received 26 November 2013; accepted 15 April 2014; published online 29 April 2014) All edges in the classical Watts and Strogatz’s small-world network model are unweighted and cooperative (positive). By introducing competitive (negative) inter-cluster edges and assigning edge weights to mimic more realistic networks, this paper develops a modified model which possesses co-competitive weighted couplings and cluster structures while maintaining the common small-world network properties of small average shortest path lengths and large clustering coefficients. Based on theoretical analysis, it is proved that the new model with inter-cluster co-competition balance has an important dynamical property of robust cluster synchronous pattern formation. More precisely, clusters will neither merge nor split regardless of adding or deleting nodes and edges, under the condition of inter-cluster co-competition balance. Numerical simulations demonstrate the robustness of the model against the increase of the coupling strength C 2014 AIP Publishing LLC. and several topological variations. V [http://dx.doi.org/10.1063/1.4873524] Many studies have shown that network dynamics depend heavily on the topological characters on a network. It is also well known that a small-world network displays a stronger synchronizability due to its small average shortest path length, and the type of its synchronization is often cluster synchronization due to its large average clustering coefficient. Based on these two features, therefore, it is an interesting and important topic to discuss how to reach or maintain a desired cluster synchronous pattern in a small-world network. In this paper, we introduce competitive edges into a given small-world network and assign a suitable weight to each edge, so as to obtain a coupling scheme for reaching or maintaining a robust cluster synchronous pattern. The corresponding modified small-world network is of more realistic significance, and the obtained results may provide a better understanding of the robustness of cluster synchronous patterns, in general.
I. INTRODUCTION
Mounting evidence suggests that most of real networks are neither entirely regular nor completely random. In 1998, Watts and Strogatz (WS)1 proposed a small-world network model, which possesses large clustering coefficients and small average shortest path lengths.1–3 For instance, in a friendship network, the property of large clustering coefficients means a large probability that a friend of your friend is also your friend. While the property of small average shortest path lengths is tied to the earlier discovery of “six degree of separation,”4 which implies a large probability that a)
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every two randomly selected persons can be connected by less than six acquaintances on average. In recent years, the small-world properties have been verified by many realworld networks in different fields, including such as protein interaction graphs,5 metabolic networks,6 co-authorships,7 power grids,8 biological neural networks,9 road maps, food webs, the World Wide Web, and so on.10,11 Small-world networks have been proved to be highly effective in relaying information and synchronizing the nodes.1 Generally speaking, such high effectiveness is mainly due to the small average shortest path lengths.12,13 Recently, many other determinant measurements were proposed for the synchronizability of small-world networks, using such as the rewiring probability,14,15 the Cheeger constant,16,17 the edge-adding number and the edge-adding distance,18 and a temporal order characterized by the autocorrelation of a firing rate function.19 Some of these measurements have been applied to forecast and control of collective dynamics in networked complex systems.14,17 However, all edges in the classical WS small-world network model are unweighted and cooperative, and competitive couplings are widespread in many real networks such as trading and commercial networks. In such a network composed of large companies, full synchronization is characterized by the consistence in commercial behavior, i.e., commercial monopoly. In order to break such commercial monopoly, cooperation and competition should be balanced in order to establish a fair and orderly commercial mechanism. In other words, cluster synchronization is beneficial, but full synchronization is forbidden in trading and commercial networks. To the best of our knowledge, very few rigorous theoretical results have been obtained on cluster synchronization in small-world networks. The goal of this paper is to carry out some rigorous theoretical investigations on this subject.
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C 2014 AIP Publishing LLC V
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One of the important issues in the study of cluster synchronization is the cluster synchronous pattern formation.20–22 More precisely, in a complex network, the nodes are split into several or even many clusters, while all the nodes in a same cluster behave in a synchronous fashion, but nodes in different clusters follow distinct timeevolutions; thus, a cluster synchronous pattern emerges.22 Many researches have been carried out to investigate the existence,23–25 the stability26–31 of partially synchronous manifold of coupled dynamical systems, and the algorithms for identifying clusters in a complex network.32–34 If external control is applied, a network can be driven to generate an arbitrarily selected cluster synchronous pattern.35,36 As far as small-world networks are concerned, cluster synchronous patterns may be widespread due to the large clustering coefficients of such networks. Regarding cluster synchronization of complex networks, some questions arise naturally: what is the relation between the synchronous pattern and the network topology? How to achieve a desired synchronous pattern by suitably designing the network topology? How to ensure the robustness of the synchronous pattern? It is well-known that synchronous patterns will be influenced by the network topology and the coupling strength. For instance, the influence of connection topology on synchronization has been investigated in fiberoptic networks of chaotic optoelectronic oscillators.37 When the network topology is varied, e.g., by rewiring edges and adding nodes, the state of the cluster synchronization might be lost, or several clusters might merge into one. This type of transformation of collective behavior patterns is called cluster synchronous bifurcation,38 which implies that the cluster synchronous patterns are not robust against such variations. In this paper, it will be shown that the inter-cluster balance between cooperation and competition (referred to as the inter-cluster co-competition balance) forms a mechanism for robust cluster synchronous patterns. The rest of this paper is organized as follows. Section II provides some preliminaries for the investigation. Then, a small-world network model with inter-cluster co-competition balance is proposed in Sec. III. Theoretical results on cluster synchronization in such small-world networks are given in Sec. IV. Section V presents numerical examples to show the robustness of the cluster synchronous patterns against several types of network topological variations. Finally, the paper is concluded by a brief discussion in Sec. VI. II. PRELIMINARIES
In this section, we present some notations, definitions, and lemmas, which will be useful throughout this paper.
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kðCÞ 2
N [
z : jz cii j
jcij j :
j6¼i
i¼1
If C is symmetric, then it follows from Lemma 1 that X X min cii jcij j kðCÞ max cii þ jcij j :
1iN
1iN
j6¼i
j6¼i
Lemma 2. (Lidskii Theorem)39 Let C1, C2 be N N symmetric matrices. Then, for any choice of indices 1 i1 < < ik < N, k X
kij ðC1 þ C2 Þ
j¼1
k X
kij ðC1 Þ þ
j¼1
k X
kj ðC2 Þ:
j¼1
Next, some concepts and properties of weight matrices are introduced. Definition 1. For a weight matrix C ¼ ðcij ÞNN , if the sum PN of all elements in each row is equal to zero, that is, j¼1 cij ¼ 0; i 2 N, then C is called a diffusively coupled matrix. In addition, if cij ¼ cji > 0 for some i 6¼ j, then the coupling between node i and node j is called a cooperative coupling; if cij ¼ cji < 0 for some i 6¼ j, then the coupling between node i and node j is called a competitive coupling. Lemma 3. Let r > 0 be a constant and C ¼ ðcij ÞNN be a diffusively coupled matrix with cij ¼ cji r for all j 6¼ i. Then, the matrix C has a zero eigenvalue and its other N – 1 nonzero eigenvalues are less than or equal to –rN. Proof. Decompose the matrix C as C ¼ rA þ B, where A ¼ ðaij ÞNN with aij ¼ aji ¼ 1 for j 6¼ i and aii ¼ 1 N. It is well-known that the matrix A has an eigenvalue 0 with multiplicity one, and the other eigenvalues are equal to –N. On the other hand, since both matrices B and C are diffusively coupled matrices with nonnegative couplings, it is easy to verify that k1 ðBÞ ¼ k1 ðCÞ ¼ 0 and kj ðBÞ 0; kj ðCÞ 0; j 2 N. It thus follows from Lemma 2 that k2 ðCÞ k2 ðrAÞ þ k1 ðBÞ ¼ rN þ k1 ðBÞ ¼ rN: ⵧ
The proof is completed. B. Cluster synchronization
Consider a general network consisting of N dynamical nodes, which is divided into n clusters of various sizes. Without loss of generality, let N0 ¼ 0; Nn ¼ N; np ¼ Np Np1 ; n ¼ f1; ; ng, and let G ¼ fG1 ; ; Gn g be a partition of the node index set N ¼ f1; ; Ng, where G1 ¼ f1; ; N1 g; ; Gn ¼ fNn1 þ 1; ; Nn g. Clearly, cluster Gp contains np nodes, p 2 n. Suppose that the dynamical network is described by
A. Properties of the weight matrix
At first, we emphasize that all the matrices in this paper are real and introduce some useful lemmas on eigenvalues of real matrices. For simplicity of notation, let kj(M) denote the jth biggest eigenvalue of a symmetrical matrix M. Lemma 1. (Gersgorin Disk Theorem)39 Every eigenvalue of matrix C ¼ ðcij ÞNN satisfies
X
x_ i ðtÞ ¼ f ðxi ðtÞ; tÞ þ e
n X X
cij Pxj ðtÞ;
(1)
p¼1 j2Gp > is the state vector of node i, f : where xi ¼ ðx1i ; ; xm i Þ m m R ½0; þ1Þ ! R is a continuous function, e > 0 is the coupling strength, C ¼ (cij)NN is the coupling weight matrix
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with cij ¼ cji, and P ¼ diag (p1, ,pm) is a nonzero matrix with pk 0; i; j 2 N; k 2 f1; ; mg. Here, the symmetric matrix C represents the topological structure of the network, and the diagonal matrix P represents the inner coupling components of each node. For more clarity, we rewrite the coupling weight matrix C to be in a block form 0 1 C11 C12 C1n B C B C21 C22 C2n C C C¼B (2) B C: @ A Cn1 Cn2 Cnn
III. A SMALL-WORLD NETWORK WITH CO-COMPETITION BALANCE
Typical small-world network characteristics are large clustering coefficients and small average shortest path lengths. Moreover, intra-cluster connections are dense but inter-cluster connections are sparse.41–43 Motivated by the above preliminaries, two assumptions are made, one is on the inter-cluster couplings while the other is on the intra-cluster couplings. These are discussed, respectively, below. A. An assumption on the inter-cluster couplings
It has been shown that cluster synchronization of network (1) is equivalent to the global attractiveness of the corresponding invariant manifold,40 which is defined as follows. Definition 2. The set n o > > mN j xi ¼ xj ; i; j 2 Gp ; p 2 n S ¼ ðx> 1 ; ; xN Þ 2 R is called the cluster synchronous manifold corresponding to the partition G of network (1). It has be shown that the cluster synchronous manifold S of network (1) is invariant if and only if every sub-matrix Cpq has equal row-sums, for all p; q 2 n.40 The rest of this section will introduce the definition of invariant synchronization manifolds being globally attractive. Definition 3. The manifold S of network (1) is said to be globally attractive, if the index set N is split into clusters Gp, p 2 n, such that the nodes in the same cluster synchronize with each other, namely, lim jjxi ðtÞ xj ðtÞjj ¼ 0
t!þ1
for arbitrary initial values and for all i; j 2 Gp ; p 2 n. Here, as a generalization of the global network synchronization, it is not specifically required the states in different clusters be eventually separated from each other, namely, jjxi ðtÞ xj ðtÞjj > 0 as t !1 for xi 2 Gp and xj 2 Gq with p/q. However, conditions will be derived later for this nontrivial cluster synchronization to be true. Let QUAD(D,Q) be a class of continuous functions f ðu; tÞ : Rm ½0; þ1Þ ! Rm satisfying ðu vÞ> Qf½f ðu; tÞ f ðv; tÞ Dðu vÞg ðu vÞ> ðu vÞ; where u; v 2 Rm ; Q ¼ diagðq1 ; ; qm Þ is a positive-definite diagonal matrix, D ¼ diagðd1 ; ; dm Þ is a diagonal matrix, and > 0 is a constant. Under the condition that f ðu; tÞ 2 QUADðD; QÞ and the manifold S of network (1) is invariant, then the cluster synchronous manifold S of network (1) is globally attractive if
epj kSmax ðCÞ þ dj 0; j ¼ 1; ; m;
(3)
where kSmax ðCÞ ¼ maxfkðCÞ : NC ðkðCÞÞ6S }, NC ðkðCÞÞ is the eigenspace corresponding to the eigenvalue k(C).40
To ensure the invariance of the manifold S, an assumption is imposed on the inter-cluster couplings. P Assumption 1. Every sub-matrix Cpq in (2) satisfies that j2Gq cij ¼ 0 for i 2 Gp ; p; q 2 n, with p 6¼ q. It is a prerequisite that the coupling matrix has a blockmatrix form with every block having an equal row-sum, which ensures invariance of the cluster synchronization manifold.40,44,45 However, one or several proper sub-manifolds may still be globally attractive under this condition. That is, some clusters may merge into one single bigger cluster. To avoid that, we further assume that every block of the coupling matrix in the form of a block matrix with a zero-row-sum. The assumption implies that, for any p; q 2 n with p 6¼ q, the couplings between any node in the p-th cluster and all connected node in the q-th cluster satisfy the property that the sum of cooperative weights is equal to the sum of competitive weights. Therefore, a network topology satisfying Assumption 1 is said to be balanced on the inter-cluster cocompetition (corresponding to the partition G). The inter-cluster co-competition exists in many realworld networks, such as commercial networks in which some individuals in a community may be cooperative and also competitive with some individuals in other communities. In the following, it is assumed that the networks are balanced on inter-cluster co-competition. Note that this assumption is not stringent since it is merely the balance between the two weight sums, rather than the balance between the number of cooperative edges and the number of competitive edges. Theoretically, if every Cpp is a diffusively coupling matrix, then it is straightforward to show that the weight matrix C satisfying Assumption 1. Therefore, the cluster synchronous manifold S of network (1) is invariant. B. An assumption on the intra-cluster couplings
To ensure obtaining a cluster structure, consider the special case where the nodes in each cluster are fully connected. Assumption 2. Assume that every sub-matrix Cpp in (2) is a diffusive coupling matrix and that there is a constant P a satisfying a > maxp2n fbp g, where bp ¼ maxi2Gp f j62Gp jcij jg, such that cij a =np for all i 2 Gp ; j 2 Gp ; i 6¼ j; p 2 n: Assumption 2 may seem to be an overly stringent restriction, but it agrees well with many real-world networks. In a social co-competition network, for example, each individual is influenced by its own neighbors in the same
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community through interactions of a mean-field type. That is, all the individuals in a same community are cooperatively connected with each other; otherwise they will not stay in the same community. Assumption 2 means that, for each node, the absolute weight sum of inter-cluster couplings should be smaller than the weight sum of intra-cluster couplings. Thus, every node is influenced more greatly by the intra-cluster nodes than by the inter-cluster nodes. This situation is very natural and common in the real world. It should also be emphasized that Assumption 2 is only for the ease of estimating eigenvalues, and it can be replaced by another relaxed condition to be presented in the next section. C. Small-world networks with co-competition balance
The now-well-known WS small-world network model was obtained from a nearest-neighbor network through random rewiring of the edges.1 In the WS small-world network model, all edges are cooperative in the sense that they have positive weight values. Since cooperative and competitive couplings often coexist in a real-world network, it is natural to also introduce competitive edges into the small-world network model by means of equipping negative weight values on such edges. Specifically, start with the classical WS small-world network model, in which all the edges are unweighted, i.e., with unity weight therefore are cooperative. Then, the following four-step procedure is carried out: (i)
(ii)
(iii)
(iv)
partition the node index set N ¼ f1; ; Ng of the WS small-world network model into n cluster index sets G1, …, Gn, where G1 ¼ f1; ; N1 g; :::; Gn ¼ fNn1 þ 1; :::; Ng: for every pair of p; q 2 n with p 6¼ q, if a node in the p-th cluster is connected with some nodes in the q-th cluster, then add some competitive edges between the node in the p-th cluster and the other nodes in the q-th cluster. Otherwise, proceed to the next step. assign positive or negative weights to all inter-cluster (cooperative or competitive) edges, so as to ensure the co-competition balance. modify every cluster by adding some edges, so that the cluster becomes fully and cooperatively connected. Then, assign a weight to each intra-cluster edge, so as to meet Assumption 2.
Obviously, inter-cluster edges in the new network are sparse. Also clearly, the model not only has large clustering coefficients and small average shortest path lengths but also has a cluster structure with co-competition couplings. In addition, the intra-cluster edges are strong, whereas the intercluster edges are weak and randomly distributed. For illustration, a schematic example consisting of 12 nodes is shown in Figure 1. The original network is an unweighted cooperative small-world network, shown in Figure 1(a). Following the above four-step procedure, the 12 nodes are first partitioned into some (here, four) clusters, as G ¼ {1,2,3;4,5,6,7;8,9,10;11,12}. Then, several competitive edges are added into the network, as shown in Figure 1(b).
Chaos 24, 023111 (2014)
Moreover, a competitive edge is added between nodes 4 and 9 for Assumption 1. And a cooperative edge is added between nodes 5 and 7, which fully connects the cluster G2 ¼ {4,5,6,7}. Finally, by assigning suitable coupling weights, the small-world network exhibits the property of inter-cluster co-competition balance, as can be verified in Figure 1. IV. ROBUSTNESS OF CLUSTER SYNCHRONOUS PATTERNS
To study the cluster synchronization in the inter-cluster co-competition balanced small-world network, created above, the eigenvalues of the weight matrix are discussed first. Theorem 1. Under Assumptions 1 and 2: (i)
(ii)
the weight matrix C has n zero eigenvalues, k1 ðCÞ ¼ ¼ kn ðCÞ ¼ 0; and the corresponding eigenvectors are np ¼ ðn1p ; ; nNp Þ> , where nip ¼ 0 for i 62 Gp , and nip ¼ 1 for i 2 Gp , p 2 n; the nonzero eigenvalues of the weight matrix C satisfy kN ðCÞ knþ1 ðCÞ b a < 0;
where b ¼ maxp2n bp . Proof. Rewrite the vectors np as ðnp1 ; ; npn Þ> , where npq 2 R1nq are all zero vectors, except npp ¼ ð1; ; 1Þ; p; q 2 n. Under Assumption 1, each sub-matrix Cp0q0 in the form of ~ (2) has zero row-sums, which leads to Cp0 q0 n> pq ¼ 0, and so 0 0 ~ Cnp ¼ 0, for all p ; q ; p; q 2 n. Further, due to the linear independence of the vectors n1 ; ; nn , the multiplicity of the zero eigenvalue is at least n. Denote by B ¼ DiagðC11 ; C22 ; ; Cnn Þ the intra-cluster weight matrix. Then, C – B is the inter-cluster weight matrix. On the one hand, Lemma 1 implies that the eigenvalues of each sub-matrix Cpp are knp ðCpp Þ k1 ðCpp Þ ¼ 0; p 2 n. As a result of Assumption 2 and Lemma 3, one has ( k1 ðCpp Þ ¼ 0; kn ðCpp Þ k2 ðCpp Þ a; where p 2 n. That is, ( k1 ðBÞ ¼ ¼ kn ðBÞ ¼ 0; kN ðBÞ knþ1 ðBÞ a: On the other hand, it is easy to verify that all the diagonal elements of C – B are zero, hence Lemma 1 implies that jki ðC BÞj b; i 2 N. In conclusion, it follows from Lemma 2 that the biggest nonzero eigenvalue satisfies knþ1 ðCÞ knþ1 ðBÞ þ k1 ðC BÞ a þ b: The proof is completed. ⵧ Now, we are in a position to provide a criterion on the robustness of the cluster synchronous pattern formation. Theorem 2. Suppose that x_ i ðtÞ ¼ f ðxi ðtÞ; tÞ is a chaotic oscillator, the function f ðu; tÞ 2 QUADðD; QÞ, and the
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FIG. 1. Creating a small-world network with inter-cluster co-competition balance. The red (blue) lines represent competitive (cooperative) edges, and the dashed line is the edge added to satisfy Assumption 2.
weight matrix C satisfies Assumptions 1 and 2, and moreover epj ðb aÞ þ dj 0; j ¼ 1; ; m:
(4)
Then, the cluster synchronous manifold S of network (1) is globally attractive; any proper sub-manifold of S of network (1) is not globally attractive.
(b)
Proof. (i) It follows from Theorem 1 that the eigenspace of the weight P matrix C corresponding to the zero eigenvalue is NC ð0Þ ¼ f nq¼1 kq nq ; kq 2 Rg S. Thus, kSmax ðCÞ knþ1 ðCÞ b a, which leads to
(d)
(i) (ii)
(c)
epj kSmax ðCÞ þ dj epj ðb aÞ þ dj 0; j ¼ 1; :::; m:
(e)
Therefore, the cluster synchronous manifold S of network (1) is globally attractive. (ii) This is proved by contradiction. Suppose otherwise, 0 and let S be a proper sub-manifold of S of network (1) that is globally attractive. Then, there would exist at least two different clusters merging into a single one for all initial values. Take the initial value x1 ð0Þ ¼ ¼ xN1 ð0Þ ¼ /1 ; xN1 þ1 ð0Þ ¼ ¼ xN2 ð0Þ ¼ /2 ; ; xNn1 þ1 ð0Þ ¼ ¼ xNn ð0Þ ¼ /n , where /p 2 Rm ; p 2 n; are constant vectors different from each other. Then, for any i 2 Gq ; q 2 n, one has x_ i ðt; /q Þ ¼ f ðxi ðt; /q Þ; tÞ þ e
n X X
cij Pxj ðt; /p Þ
p¼1 j2Gp
¼ f ðxi ðt; /q Þ; tÞ:
For a more general case, Assumption 2 is replaced with other relaxing condition, yielding the following new result. Theorem 3. Suppose that the eigenvalues of every submatrix Cpp in (2) satisfy that k2 ðCpp Þ þ b < 0; P where p 2 n; b ¼ maxp2n bp ; bp ¼ maxi2Gp f j62Gp jcij jg. Then, under Assumption 1, the eigenvalues of the weight matrix C satisfy that ( k1 ðCÞ ¼ ¼ kn ðCÞ ¼ 0; kN ðCÞ knþ1 ðCÞ maxp2n k2 ðCpp Þ þ b < 0:
(5)
As can be seen, all the nodes are uncoupled, but the nodes in the same cluster behave in the same fashion. But it is impossible for this to be always true in a general setting, which leads to a contradiction to the beginning hypothesis. ⵧ Remark 1. Theorem 2 implies that every cluster does not split into several different clusters, and different clusters do not merge into one single cluster. That is to say, the cluster synchronous pattern formation is robust against any topological variation under the conditions of Theorem 2. As examples, some common topological variations under the conditions of Theorem 2 are listed below. (a)
and j 2 Gq ; p 6¼ q, then the weights of the original cooperative (or competitive) edges between node i and other nodes in Gq, besides j and other nodes in Gp, should be adjusted to maintain a co-competition balance. Moreover, bp a and bq a should also be satisfied. deleting inter-cluster edges. This is similar to the above case. rewiring inter-cluster edges. All the inter-cluster edges can be rewired between any two nodes in different clusters under Assumptions 1 and 2. removing nodes from a cluster. If n0 nodes are removed from cluster Gp, the original intra-cluster coupling weights should be increased and then inter-cluster coupling weights related to the n0 nodes should be adjusted. adding nodes to a cluster. This is similar to the above case.
adding inter-cluster edges. If a weighted cooperative (or competitive) edge is added between nodes i 2 Gp
Remark 2. According to Theorem 3, after replacing Assumption 2 and condition (4) with the following condition: epj max k2 ðCpp Þ þ b þ dj 0; j ¼ 1; ; m; p2n
Theorem 2 still holds. A small number of competitive edges can also be added to connect some nodes in the same cluster. Therefore, the constraints on the intra-cluster couplings have been greatly relaxed. V. NUMERICAL SIMULATIONS
Consider the network (1) composing of 12 neural networks of the form
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x_ i ¼ Dxi þ Tgðxi Þ þ e
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12 X
cij Cxj ; i ¼ 1; :::; 12;
(6)
j¼1
where xi 2 R3 , D and C are both identity matrices, gðxi Þ ¼ ðgðx1i Þ; gðx2i Þ; gðx3i ÞÞ> ; gðsÞ ¼ ðjs þ 1j js 1jÞ=2, and 0 1 1:25 3:2 3:2 T ¼ @ 3:2 1:1 4:4 A: 3:2 4:4 1:0 It has been shown that f ðu; tÞ 2 QUADðD; QÞ by taking dmin ¼ 5.685.46 Denote by Ak 2 Rkk the fully coupled matrix, in which the off-diagonal elements are all equal to 1, and the diagonal elements are all equal to 1– k. Let the original network topology be the one given in Figure 1(c), where the intra-cluster weight sub-matrices are C11 ¼ 3A3 ;
C22 ¼ 2A4 ;
C33 ¼ 3A3 ;
C44 ¼ 4A2 ;
the inter-cluster weights are all equal to zero, except c25 ¼ c26 ¼ c35 ¼ c36 ¼ c4;10 ¼ c49 ¼ c69 ¼ c6;10 ¼ c9;11 ¼ c9;12 ¼ c10;12 ¼ c10;11 ¼ 1; c52 ¼ c62 ¼ c53 ¼ c63 ¼ c10;4 ¼ c94 ¼ c96 ¼ c10;6 ¼ c11;9 ¼ c12;9 ¼ c12;10 ¼ c11;10 ¼ 1; and c34 ¼ c43 ¼ 2; c24 ¼ c42 ¼ 2: According to the partition, G ¼ f1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12g define the intra-cluster errors X kxi ðtÞ xNp ðtÞk; ep ¼ i2Gp
e0p
¼
X
FIG. 2. Trajectories of the 12 dynamical nodes. The nodes are split into four clusters, where all the nodes in the same cluster evolve along the same trajectory.
B. Simulation 2
According to Remark 1, the following five types of topological variations are tested on the network, as shown in Figure 1(c). (a)
(b)
(c) (d)
kxi ðt0 Þ xNp ðt0 Þk;
i2Gp
and the inter-cluster errors ep1 ¼k xNp ðtÞ xN1 ðtÞk; e0p1 ¼k xNp ðt0 Þ xN1 ðt0 Þk; where p 2 n ¼ f1; 2; 3; 4g.
(e)
adding inter-cluster edges. Add an edge c68 ¼ c86 ¼ c48 ¼ c84 ¼ 0:5, so the edge c69 ¼ c96 ¼ c49 ¼ c94 should be reduced to 0.5. deleting inter-cluster edges. Delete the edge c26 ¼ c62 ¼ c36 ¼ c63 ¼ 1, so the edge c25 ¼ c52 ¼ c35 ¼ c53 should be increased to 2. rewiring inter-cluster edges. Rewire the edge c26 ¼ c62 ¼ c36 ¼ c63 ¼ 1 as c27 ¼ c72 ¼ c37 ¼ c73 ¼ 1. removing nodes from a cluster. Remove the node 7 from the cluster G2, so C22 ¼ 2A3. To satisfy Assumptions 1 and 2, redefine the coupling weights as c49 ¼ c94 ¼ c4;10 ¼ c10;4 ¼ c69 ¼ c96 ¼ c6;10 ¼ c10;6 ¼ 0:5. adding nodes to a cluster. Add a node 13 to the cluster G4, so the intra-cluster weight sub-matrix can be reduced to 3A3.
For every type of the above topological variations, Figure 4 shows that the intra-cluster errors tend to zero, but the inter-cluster errors do not tend to zero. Thus, the cluster
A. Simulation 1
Take e ¼ dmin =2 following Theorem 2. The trajectories of the 12 dynamical nodes are shown in Figure 2. As can be seen, the nodes in the same clusters behave in the same fashion, and the nodes in different clusters behave in different fashions. For clarity, the time evolutions of the inter-cluster errors and the intra-cluster errors are shown in Figure 4(o). The two figures both indicated that a cluster synchronous pattern has been formed. To verify the robustness of the pattern formation against the increasing of the coupling strength, Figure 3 is plotted by increasing the coupling strength from 0 to 50 at t0 ¼ 100. As anticipated, different clusters do not merge into one single cluster even though the coupling strength becomes very large.
FIG. 3. Robustness of the cluster synchronous pattern formation against the increase of the coupling strength. The inter-cluster errors never tend to zero.
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FIG. 4. Robustness of the cluster synchronous pattern formation against five types of topological variations.
synchronous pattern formation is indeed robust against the above topological variations. VI. CONCLUSIONS
In this paper, the classical unweighted and cooperative small-world network model has been modified to be a model with co-competitive weighted couplings and cluster structures while maintaining the original common small-world network properties such as small average shortest path lengths and large clustering coefficients. Moreover, in the new model, intra-cluster connections are dense but inter-cluster connections are sparse. It has been theoretically proved and also numerically verified that the cluster synchronous pattern formation in the new model is robust against several typical network topological variations. It has furthermore been proved that the cluster
synchronous manifold satisfying inter-cluster co-competition balance is the minimal globally attractive manifold for network cluster synchronization. In other words, the inter-cluster co-competition balance is an important mechanism to form robust cluster synchronous patterns. The modified model can be applied to describe and characterize many real-world networks with both cooperation and competition. ACKNOWLEDGMENTS
This project was supported by Zhejiang NSF (No. LQ12A01003), NNSF of China (Nos. 11162004, 11171084, and 61203155) and the Science Foundation of Guangxi Province (No. 2013GXNSFAA019006).
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We are very grateful to the anonymous reviewers’ comments, which were very helpful for us in revising the paper. 1
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CHAOS 24, 023111 (2014)
Robustness of cluster synchronous patterns in small-world networks with inter-cluster co-competition balance Jianbao Zhang,1 Zhongjun Ma,2,a) and Guanrong Chen3 1
School of Science, Hangzhou Dianzi University, Hangzhou 310018, China School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China 3 Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, China 2
(Received 26 November 2013; accepted 15 April 2014; published online 29 April 2014) All edges in the classical Watts and Strogatz’s small-world network model are unweighted and cooperative (positive). By introducing competitive (negative) inter-cluster edges and assigning edge weights to mimic more realistic networks, this paper develops a modified model which possesses co-competitive weighted couplings and cluster structures while maintaining the common small-world network properties of small average shortest path lengths and large clustering coefficients. Based on theoretical analysis, it is proved that the new model with inter-cluster co-competition balance has an important dynamical property of robust cluster synchronous pattern formation. More precisely, clusters will neither merge nor split regardless of adding or deleting nodes and edges, under the condition of inter-cluster co-competition balance. Numerical simulations demonstrate the robustness of the model against the increase of the coupling strength C 2014 AIP Publishing LLC. and several topological variations. V [http://dx.doi.org/10.1063/1.4873524] Many studies have shown that network dynamics depend heavily on the topological characters on a network. It is also well known that a small-world network displays a stronger synchronizability due to its small average shortest path length, and the type of its synchronization is often cluster synchronization due to its large average clustering coefficient. Based on these two features, therefore, it is an interesting and important topic to discuss how to reach or maintain a desired cluster synchronous pattern in a small-world network. In this paper, we introduce competitive edges into a given small-world network and assign a suitable weight to each edge, so as to obtain a coupling scheme for reaching or maintaining a robust cluster synchronous pattern. The corresponding modified small-world network is of more realistic significance, and the obtained results may provide a better understanding of the robustness of cluster synchronous patterns, in general.
I. INTRODUCTION
Mounting evidence suggests that most of real networks are neither entirely regular nor completely random. In 1998, Watts and Strogatz (WS)1 proposed a small-world network model, which possesses large clustering coefficients and small average shortest path lengths.1–3 For instance, in a friendship network, the property of large clustering coefficients means a large probability that a friend of your friend is also your friend. While the property of small average shortest path lengths is tied to the earlier discovery of “six degree of separation,”4 which implies a large probability that a)
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every two randomly selected persons can be connected by less than six acquaintances on average. In recent years, the small-world properties have been verified by many realworld networks in different fields, including such as protein interaction graphs,5 metabolic networks,6 co-authorships,7 power grids,8 biological neural networks,9 road maps, food webs, the World Wide Web, and so on.10,11 Small-world networks have been proved to be highly effective in relaying information and synchronizing the nodes.1 Generally speaking, such high effectiveness is mainly due to the small average shortest path lengths.12,13 Recently, many other determinant measurements were proposed for the synchronizability of small-world networks, using such as the rewiring probability,14,15 the Cheeger constant,16,17 the edge-adding number and the edge-adding distance,18 and a temporal order characterized by the autocorrelation of a firing rate function.19 Some of these measurements have been applied to forecast and control of collective dynamics in networked complex systems.14,17 However, all edges in the classical WS small-world network model are unweighted and cooperative, and competitive couplings are widespread in many real networks such as trading and commercial networks. In such a network composed of large companies, full synchronization is characterized by the consistence in commercial behavior, i.e., commercial monopoly. In order to break such commercial monopoly, cooperation and competition should be balanced in order to establish a fair and orderly commercial mechanism. In other words, cluster synchronization is beneficial, but full synchronization is forbidden in trading and commercial networks. To the best of our knowledge, very few rigorous theoretical results have been obtained on cluster synchronization in small-world networks. The goal of this paper is to carry out some rigorous theoretical investigations on this subject.
24, 023111-1
C 2014 AIP Publishing LLC V
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One of the important issues in the study of cluster synchronization is the cluster synchronous pattern formation.20–22 More precisely, in a complex network, the nodes are split into several or even many clusters, while all the nodes in a same cluster behave in a synchronous fashion, but nodes in different clusters follow distinct timeevolutions; thus, a cluster synchronous pattern emerges.22 Many researches have been carried out to investigate the existence,23–25 the stability26–31 of partially synchronous manifold of coupled dynamical systems, and the algorithms for identifying clusters in a complex network.32–34 If external control is applied, a network can be driven to generate an arbitrarily selected cluster synchronous pattern.35,36 As far as small-world networks are concerned, cluster synchronous patterns may be widespread due to the large clustering coefficients of such networks. Regarding cluster synchronization of complex networks, some questions arise naturally: what is the relation between the synchronous pattern and the network topology? How to achieve a desired synchronous pattern by suitably designing the network topology? How to ensure the robustness of the synchronous pattern? It is well-known that synchronous patterns will be influenced by the network topology and the coupling strength. For instance, the influence of connection topology on synchronization has been investigated in fiberoptic networks of chaotic optoelectronic oscillators.37 When the network topology is varied, e.g., by rewiring edges and adding nodes, the state of the cluster synchronization might be lost, or several clusters might merge into one. This type of transformation of collective behavior patterns is called cluster synchronous bifurcation,38 which implies that the cluster synchronous patterns are not robust against such variations. In this paper, it will be shown that the inter-cluster balance between cooperation and competition (referred to as the inter-cluster co-competition balance) forms a mechanism for robust cluster synchronous patterns. The rest of this paper is organized as follows. Section II provides some preliminaries for the investigation. Then, a small-world network model with inter-cluster co-competition balance is proposed in Sec. III. Theoretical results on cluster synchronization in such small-world networks are given in Sec. IV. Section V presents numerical examples to show the robustness of the cluster synchronous patterns against several types of network topological variations. Finally, the paper is concluded by a brief discussion in Sec. VI. II. PRELIMINARIES
In this section, we present some notations, definitions, and lemmas, which will be useful throughout this paper.
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kðCÞ 2
N [
z : jz cii j
jcij j :
j6¼i
i¼1
If C is symmetric, then it follows from Lemma 1 that X X min cii jcij j kðCÞ max cii þ jcij j :
1iN
1iN
j6¼i
j6¼i
Lemma 2. (Lidskii Theorem)39 Let C1, C2 be N N symmetric matrices. Then, for any choice of indices 1 i1 < < ik < N, k X
kij ðC1 þ C2 Þ
j¼1
k X
kij ðC1 Þ þ
j¼1
k X
kj ðC2 Þ:
j¼1
Next, some concepts and properties of weight matrices are introduced. Definition 1. For a weight matrix C ¼ ðcij ÞNN , if the sum PN of all elements in each row is equal to zero, that is, j¼1 cij ¼ 0; i 2 N, then C is called a diffusively coupled matrix. In addition, if cij ¼ cji > 0 for some i 6¼ j, then the coupling between node i and node j is called a cooperative coupling; if cij ¼ cji < 0 for some i 6¼ j, then the coupling between node i and node j is called a competitive coupling. Lemma 3. Let r > 0 be a constant and C ¼ ðcij ÞNN be a diffusively coupled matrix with cij ¼ cji r for all j 6¼ i. Then, the matrix C has a zero eigenvalue and its other N – 1 nonzero eigenvalues are less than or equal to –rN. Proof. Decompose the matrix C as C ¼ rA þ B, where A ¼ ðaij ÞNN with aij ¼ aji ¼ 1 for j 6¼ i and aii ¼ 1 N. It is well-known that the matrix A has an eigenvalue 0 with multiplicity one, and the other eigenvalues are equal to –N. On the other hand, since both matrices B and C are diffusively coupled matrices with nonnegative couplings, it is easy to verify that k1 ðBÞ ¼ k1 ðCÞ ¼ 0 and kj ðBÞ 0; kj ðCÞ 0; j 2 N. It thus follows from Lemma 2 that k2 ðCÞ k2 ðrAÞ þ k1 ðBÞ ¼ rN þ k1 ðBÞ ¼ rN: ⵧ
The proof is completed. B. Cluster synchronization
Consider a general network consisting of N dynamical nodes, which is divided into n clusters of various sizes. Without loss of generality, let N0 ¼ 0; Nn ¼ N; np ¼ Np Np1 ; n ¼ f1; ; ng, and let G ¼ fG1 ; ; Gn g be a partition of the node index set N ¼ f1; ; Ng, where G1 ¼ f1; ; N1 g; ; Gn ¼ fNn1 þ 1; ; Nn g. Clearly, cluster Gp contains np nodes, p 2 n. Suppose that the dynamical network is described by
A. Properties of the weight matrix
At first, we emphasize that all the matrices in this paper are real and introduce some useful lemmas on eigenvalues of real matrices. For simplicity of notation, let kj(M) denote the jth biggest eigenvalue of a symmetrical matrix M. Lemma 1. (Gersgorin Disk Theorem)39 Every eigenvalue of matrix C ¼ ðcij ÞNN satisfies
X
x_ i ðtÞ ¼ f ðxi ðtÞ; tÞ þ e
n X X
cij Pxj ðtÞ;
(1)
p¼1 j2Gp > is the state vector of node i, f : where xi ¼ ðx1i ; ; xm i Þ m m R ½0; þ1Þ ! R is a continuous function, e > 0 is the coupling strength, C ¼ (cij)NN is the coupling weight matrix
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with cij ¼ cji, and P ¼ diag (p1, ,pm) is a nonzero matrix with pk 0; i; j 2 N; k 2 f1; ; mg. Here, the symmetric matrix C represents the topological structure of the network, and the diagonal matrix P represents the inner coupling components of each node. For more clarity, we rewrite the coupling weight matrix C to be in a block form 0 1 C11 C12 C1n B C B C21 C22 C2n C C C¼B (2) B C: @ A Cn1 Cn2 Cnn
III. A SMALL-WORLD NETWORK WITH CO-COMPETITION BALANCE
Typical small-world network characteristics are large clustering coefficients and small average shortest path lengths. Moreover, intra-cluster connections are dense but inter-cluster connections are sparse.41–43 Motivated by the above preliminaries, two assumptions are made, one is on the inter-cluster couplings while the other is on the intra-cluster couplings. These are discussed, respectively, below. A. An assumption on the inter-cluster couplings
It has been shown that cluster synchronization of network (1) is equivalent to the global attractiveness of the corresponding invariant manifold,40 which is defined as follows. Definition 2. The set n o > > mN j xi ¼ xj ; i; j 2 Gp ; p 2 n S ¼ ðx> 1 ; ; xN Þ 2 R is called the cluster synchronous manifold corresponding to the partition G of network (1). It has be shown that the cluster synchronous manifold S of network (1) is invariant if and only if every sub-matrix Cpq has equal row-sums, for all p; q 2 n.40 The rest of this section will introduce the definition of invariant synchronization manifolds being globally attractive. Definition 3. The manifold S of network (1) is said to be globally attractive, if the index set N is split into clusters Gp, p 2 n, such that the nodes in the same cluster synchronize with each other, namely, lim jjxi ðtÞ xj ðtÞjj ¼ 0
t!þ1
for arbitrary initial values and for all i; j 2 Gp ; p 2 n. Here, as a generalization of the global network synchronization, it is not specifically required the states in different clusters be eventually separated from each other, namely, jjxi ðtÞ xj ðtÞjj > 0 as t !1 for xi 2 Gp and xj 2 Gq with p/q. However, conditions will be derived later for this nontrivial cluster synchronization to be true. Let QUAD(D,Q) be a class of continuous functions f ðu; tÞ : Rm ½0; þ1Þ ! Rm satisfying ðu vÞ> Qf½f ðu; tÞ f ðv; tÞ Dðu vÞg ðu vÞ> ðu vÞ; where u; v 2 Rm ; Q ¼ diagðq1 ; ; qm Þ is a positive-definite diagonal matrix, D ¼ diagðd1 ; ; dm Þ is a diagonal matrix, and > 0 is a constant. Under the condition that f ðu; tÞ 2 QUADðD; QÞ and the manifold S of network (1) is invariant, then the cluster synchronous manifold S of network (1) is globally attractive if
epj kSmax ðCÞ þ dj 0; j ¼ 1; ; m;
(3)
where kSmax ðCÞ ¼ maxfkðCÞ : NC ðkðCÞÞ6S }, NC ðkðCÞÞ is the eigenspace corresponding to the eigenvalue k(C).40
To ensure the invariance of the manifold S, an assumption is imposed on the inter-cluster couplings. P Assumption 1. Every sub-matrix Cpq in (2) satisfies that j2Gq cij ¼ 0 for i 2 Gp ; p; q 2 n, with p 6¼ q. It is a prerequisite that the coupling matrix has a blockmatrix form with every block having an equal row-sum, which ensures invariance of the cluster synchronization manifold.40,44,45 However, one or several proper sub-manifolds may still be globally attractive under this condition. That is, some clusters may merge into one single bigger cluster. To avoid that, we further assume that every block of the coupling matrix in the form of a block matrix with a zero-row-sum. The assumption implies that, for any p; q 2 n with p 6¼ q, the couplings between any node in the p-th cluster and all connected node in the q-th cluster satisfy the property that the sum of cooperative weights is equal to the sum of competitive weights. Therefore, a network topology satisfying Assumption 1 is said to be balanced on the inter-cluster cocompetition (corresponding to the partition G). The inter-cluster co-competition exists in many realworld networks, such as commercial networks in which some individuals in a community may be cooperative and also competitive with some individuals in other communities. In the following, it is assumed that the networks are balanced on inter-cluster co-competition. Note that this assumption is not stringent since it is merely the balance between the two weight sums, rather than the balance between the number of cooperative edges and the number of competitive edges. Theoretically, if every Cpp is a diffusively coupling matrix, then it is straightforward to show that the weight matrix C satisfying Assumption 1. Therefore, the cluster synchronous manifold S of network (1) is invariant. B. An assumption on the intra-cluster couplings
To ensure obtaining a cluster structure, consider the special case where the nodes in each cluster are fully connected. Assumption 2. Assume that every sub-matrix Cpp in (2) is a diffusive coupling matrix and that there is a constant P a satisfying a > maxp2n fbp g, where bp ¼ maxi2Gp f j62Gp jcij jg, such that cij a =np for all i 2 Gp ; j 2 Gp ; i 6¼ j; p 2 n: Assumption 2 may seem to be an overly stringent restriction, but it agrees well with many real-world networks. In a social co-competition network, for example, each individual is influenced by its own neighbors in the same
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community through interactions of a mean-field type. That is, all the individuals in a same community are cooperatively connected with each other; otherwise they will not stay in the same community. Assumption 2 means that, for each node, the absolute weight sum of inter-cluster couplings should be smaller than the weight sum of intra-cluster couplings. Thus, every node is influenced more greatly by the intra-cluster nodes than by the inter-cluster nodes. This situation is very natural and common in the real world. It should also be emphasized that Assumption 2 is only for the ease of estimating eigenvalues, and it can be replaced by another relaxed condition to be presented in the next section. C. Small-world networks with co-competition balance
The now-well-known WS small-world network model was obtained from a nearest-neighbor network through random rewiring of the edges.1 In the WS small-world network model, all edges are cooperative in the sense that they have positive weight values. Since cooperative and competitive couplings often coexist in a real-world network, it is natural to also introduce competitive edges into the small-world network model by means of equipping negative weight values on such edges. Specifically, start with the classical WS small-world network model, in which all the edges are unweighted, i.e., with unity weight therefore are cooperative. Then, the following four-step procedure is carried out: (i)
(ii)
(iii)
(iv)
partition the node index set N ¼ f1; ; Ng of the WS small-world network model into n cluster index sets G1, …, Gn, where G1 ¼ f1; ; N1 g; :::; Gn ¼ fNn1 þ 1; :::; Ng: for every pair of p; q 2 n with p 6¼ q, if a node in the p-th cluster is connected with some nodes in the q-th cluster, then add some competitive edges between the node in the p-th cluster and the other nodes in the q-th cluster. Otherwise, proceed to the next step. assign positive or negative weights to all inter-cluster (cooperative or competitive) edges, so as to ensure the co-competition balance. modify every cluster by adding some edges, so that the cluster becomes fully and cooperatively connected. Then, assign a weight to each intra-cluster edge, so as to meet Assumption 2.
Obviously, inter-cluster edges in the new network are sparse. Also clearly, the model not only has large clustering coefficients and small average shortest path lengths but also has a cluster structure with co-competition couplings. In addition, the intra-cluster edges are strong, whereas the intercluster edges are weak and randomly distributed. For illustration, a schematic example consisting of 12 nodes is shown in Figure 1. The original network is an unweighted cooperative small-world network, shown in Figure 1(a). Following the above four-step procedure, the 12 nodes are first partitioned into some (here, four) clusters, as G ¼ {1,2,3;4,5,6,7;8,9,10;11,12}. Then, several competitive edges are added into the network, as shown in Figure 1(b).
Chaos 24, 023111 (2014)
Moreover, a competitive edge is added between nodes 4 and 9 for Assumption 1. And a cooperative edge is added between nodes 5 and 7, which fully connects the cluster G2 ¼ {4,5,6,7}. Finally, by assigning suitable coupling weights, the small-world network exhibits the property of inter-cluster co-competition balance, as can be verified in Figure 1. IV. ROBUSTNESS OF CLUSTER SYNCHRONOUS PATTERNS
To study the cluster synchronization in the inter-cluster co-competition balanced small-world network, created above, the eigenvalues of the weight matrix are discussed first. Theorem 1. Under Assumptions 1 and 2: (i)
(ii)
the weight matrix C has n zero eigenvalues, k1 ðCÞ ¼ ¼ kn ðCÞ ¼ 0; and the corresponding eigenvectors are np ¼ ðn1p ; ; nNp Þ> , where nip ¼ 0 for i 62 Gp , and nip ¼ 1 for i 2 Gp , p 2 n; the nonzero eigenvalues of the weight matrix C satisfy kN ðCÞ knþ1 ðCÞ b a < 0;
where b ¼ maxp2n bp . Proof. Rewrite the vectors np as ðnp1 ; ; npn Þ> , where npq 2 R1nq are all zero vectors, except npp ¼ ð1; ; 1Þ; p; q 2 n. Under Assumption 1, each sub-matrix Cp0q0 in the form of ~ (2) has zero row-sums, which leads to Cp0 q0 n> pq ¼ 0, and so 0 0 ~ Cnp ¼ 0, for all p ; q ; p; q 2 n. Further, due to the linear independence of the vectors n1 ; ; nn , the multiplicity of the zero eigenvalue is at least n. Denote by B ¼ DiagðC11 ; C22 ; ; Cnn Þ the intra-cluster weight matrix. Then, C – B is the inter-cluster weight matrix. On the one hand, Lemma 1 implies that the eigenvalues of each sub-matrix Cpp are knp ðCpp Þ k1 ðCpp Þ ¼ 0; p 2 n. As a result of Assumption 2 and Lemma 3, one has ( k1 ðCpp Þ ¼ 0; kn ðCpp Þ k2 ðCpp Þ a; where p 2 n. That is, ( k1 ðBÞ ¼ ¼ kn ðBÞ ¼ 0; kN ðBÞ knþ1 ðBÞ a: On the other hand, it is easy to verify that all the diagonal elements of C – B are zero, hence Lemma 1 implies that jki ðC BÞj b; i 2 N. In conclusion, it follows from Lemma 2 that the biggest nonzero eigenvalue satisfies knþ1 ðCÞ knþ1 ðBÞ þ k1 ðC BÞ a þ b: The proof is completed. ⵧ Now, we are in a position to provide a criterion on the robustness of the cluster synchronous pattern formation. Theorem 2. Suppose that x_ i ðtÞ ¼ f ðxi ðtÞ; tÞ is a chaotic oscillator, the function f ðu; tÞ 2 QUADðD; QÞ, and the
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Zhang, Ma, and Chen
Chaos 24, 023111 (2014)
FIG. 1. Creating a small-world network with inter-cluster co-competition balance. The red (blue) lines represent competitive (cooperative) edges, and the dashed line is the edge added to satisfy Assumption 2.
weight matrix C satisfies Assumptions 1 and 2, and moreover epj ðb aÞ þ dj 0; j ¼ 1; ; m:
(4)
Then, the cluster synchronous manifold S of network (1) is globally attractive; any proper sub-manifold of S of network (1) is not globally attractive.
(b)
Proof. (i) It follows from Theorem 1 that the eigenspace of the weight P matrix C corresponding to the zero eigenvalue is NC ð0Þ ¼ f nq¼1 kq nq ; kq 2 Rg S. Thus, kSmax ðCÞ knþ1 ðCÞ b a, which leads to
(d)
(i) (ii)
(c)
epj kSmax ðCÞ þ dj epj ðb aÞ þ dj 0; j ¼ 1; :::; m:
(e)
Therefore, the cluster synchronous manifold S of network (1) is globally attractive. (ii) This is proved by contradiction. Suppose otherwise, 0 and let S be a proper sub-manifold of S of network (1) that is globally attractive. Then, there would exist at least two different clusters merging into a single one for all initial values. Take the initial value x1 ð0Þ ¼ ¼ xN1 ð0Þ ¼ /1 ; xN1 þ1 ð0Þ ¼ ¼ xN2 ð0Þ ¼ /2 ; ; xNn1 þ1 ð0Þ ¼ ¼ xNn ð0Þ ¼ /n , where /p 2 Rm ; p 2 n; are constant vectors different from each other. Then, for any i 2 Gq ; q 2 n, one has x_ i ðt; /q Þ ¼ f ðxi ðt; /q Þ; tÞ þ e
n X X
cij Pxj ðt; /p Þ
p¼1 j2Gp
¼ f ðxi ðt; /q Þ; tÞ:
For a more general case, Assumption 2 is replaced with other relaxing condition, yielding the following new result. Theorem 3. Suppose that the eigenvalues of every submatrix Cpp in (2) satisfy that k2 ðCpp Þ þ b < 0; P where p 2 n; b ¼ maxp2n bp ; bp ¼ maxi2Gp f j62Gp jcij jg. Then, under Assumption 1, the eigenvalues of the weight matrix C satisfy that ( k1 ðCÞ ¼ ¼ kn ðCÞ ¼ 0; kN ðCÞ knþ1 ðCÞ maxp2n k2 ðCpp Þ þ b < 0:
(5)
As can be seen, all the nodes are uncoupled, but the nodes in the same cluster behave in the same fashion. But it is impossible for this to be always true in a general setting, which leads to a contradiction to the beginning hypothesis. ⵧ Remark 1. Theorem 2 implies that every cluster does not split into several different clusters, and different clusters do not merge into one single cluster. That is to say, the cluster synchronous pattern formation is robust against any topological variation under the conditions of Theorem 2. As examples, some common topological variations under the conditions of Theorem 2 are listed below. (a)
and j 2 Gq ; p 6¼ q, then the weights of the original cooperative (or competitive) edges between node i and other nodes in Gq, besides j and other nodes in Gp, should be adjusted to maintain a co-competition balance. Moreover, bp a and bq a should also be satisfied. deleting inter-cluster edges. This is similar to the above case. rewiring inter-cluster edges. All the inter-cluster edges can be rewired between any two nodes in different clusters under Assumptions 1 and 2. removing nodes from a cluster. If n0 nodes are removed from cluster Gp, the original intra-cluster coupling weights should be increased and then inter-cluster coupling weights related to the n0 nodes should be adjusted. adding nodes to a cluster. This is similar to the above case.
adding inter-cluster edges. If a weighted cooperative (or competitive) edge is added between nodes i 2 Gp
Remark 2. According to Theorem 3, after replacing Assumption 2 and condition (4) with the following condition: epj max k2 ðCpp Þ þ b þ dj 0; j ¼ 1; ; m; p2n
Theorem 2 still holds. A small number of competitive edges can also be added to connect some nodes in the same cluster. Therefore, the constraints on the intra-cluster couplings have been greatly relaxed. V. NUMERICAL SIMULATIONS
Consider the network (1) composing of 12 neural networks of the form
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Zhang, Ma, and Chen
x_ i ¼ Dxi þ Tgðxi Þ þ e
Chaos 24, 023111 (2014)
12 X
cij Cxj ; i ¼ 1; :::; 12;
(6)
j¼1
where xi 2 R3 , D and C are both identity matrices, gðxi Þ ¼ ðgðx1i Þ; gðx2i Þ; gðx3i ÞÞ> ; gðsÞ ¼ ðjs þ 1j js 1jÞ=2, and 0 1 1:25 3:2 3:2 T ¼ @ 3:2 1:1 4:4 A: 3:2 4:4 1:0 It has been shown that f ðu; tÞ 2 QUADðD; QÞ by taking dmin ¼ 5.685.46 Denote by Ak 2 Rkk the fully coupled matrix, in which the off-diagonal elements are all equal to 1, and the diagonal elements are all equal to 1– k. Let the original network topology be the one given in Figure 1(c), where the intra-cluster weight sub-matrices are C11 ¼ 3A3 ;
C22 ¼ 2A4 ;
C33 ¼ 3A3 ;
C44 ¼ 4A2 ;
the inter-cluster weights are all equal to zero, except c25 ¼ c26 ¼ c35 ¼ c36 ¼ c4;10 ¼ c49 ¼ c69 ¼ c6;10 ¼ c9;11 ¼ c9;12 ¼ c10;12 ¼ c10;11 ¼ 1; c52 ¼ c62 ¼ c53 ¼ c63 ¼ c10;4 ¼ c94 ¼ c96 ¼ c10;6 ¼ c11;9 ¼ c12;9 ¼ c12;10 ¼ c11;10 ¼ 1; and c34 ¼ c43 ¼ 2; c24 ¼ c42 ¼ 2: According to the partition, G ¼ f1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12g define the intra-cluster errors X kxi ðtÞ xNp ðtÞk; ep ¼ i2Gp
e0p
¼
X
FIG. 2. Trajectories of the 12 dynamical nodes. The nodes are split into four clusters, where all the nodes in the same cluster evolve along the same trajectory.
B. Simulation 2
According to Remark 1, the following five types of topological variations are tested on the network, as shown in Figure 1(c). (a)
(b)
(c) (d)
kxi ðt0 Þ xNp ðt0 Þk;
i2Gp
and the inter-cluster errors ep1 ¼k xNp ðtÞ xN1 ðtÞk; e0p1 ¼k xNp ðt0 Þ xN1 ðt0 Þk; where p 2 n ¼ f1; 2; 3; 4g.
(e)
adding inter-cluster edges. Add an edge c68 ¼ c86 ¼ c48 ¼ c84 ¼ 0:5, so the edge c69 ¼ c96 ¼ c49 ¼ c94 should be reduced to 0.5. deleting inter-cluster edges. Delete the edge c26 ¼ c62 ¼ c36 ¼ c63 ¼ 1, so the edge c25 ¼ c52 ¼ c35 ¼ c53 should be increased to 2. rewiring inter-cluster edges. Rewire the edge c26 ¼ c62 ¼ c36 ¼ c63 ¼ 1 as c27 ¼ c72 ¼ c37 ¼ c73 ¼ 1. removing nodes from a cluster. Remove the node 7 from the cluster G2, so C22 ¼ 2A3. To satisfy Assumptions 1 and 2, redefine the coupling weights as c49 ¼ c94 ¼ c4;10 ¼ c10;4 ¼ c69 ¼ c96 ¼ c6;10 ¼ c10;6 ¼ 0:5. adding nodes to a cluster. Add a node 13 to the cluster G4, so the intra-cluster weight sub-matrix can be reduced to 3A3.
For every type of the above topological variations, Figure 4 shows that the intra-cluster errors tend to zero, but the inter-cluster errors do not tend to zero. Thus, the cluster
A. Simulation 1
Take e ¼ dmin =2 following Theorem 2. The trajectories of the 12 dynamical nodes are shown in Figure 2. As can be seen, the nodes in the same clusters behave in the same fashion, and the nodes in different clusters behave in different fashions. For clarity, the time evolutions of the inter-cluster errors and the intra-cluster errors are shown in Figure 4(o). The two figures both indicated that a cluster synchronous pattern has been formed. To verify the robustness of the pattern formation against the increasing of the coupling strength, Figure 3 is plotted by increasing the coupling strength from 0 to 50 at t0 ¼ 100. As anticipated, different clusters do not merge into one single cluster even though the coupling strength becomes very large.
FIG. 3. Robustness of the cluster synchronous pattern formation against the increase of the coupling strength. The inter-cluster errors never tend to zero.
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023111-7
Zhang, Ma, and Chen
Chaos 24, 023111 (2014)
FIG. 4. Robustness of the cluster synchronous pattern formation against five types of topological variations.
synchronous pattern formation is indeed robust against the above topological variations. VI. CONCLUSIONS
In this paper, the classical unweighted and cooperative small-world network model has been modified to be a model with co-competitive weighted couplings and cluster structures while maintaining the original common small-world network properties such as small average shortest path lengths and large clustering coefficients. Moreover, in the new model, intra-cluster connections are dense but inter-cluster connections are sparse. It has been theoretically proved and also numerically verified that the cluster synchronous pattern formation in the new model is robust against several typical network topological variations. It has furthermore been proved that the cluster
synchronous manifold satisfying inter-cluster co-competition balance is the minimal globally attractive manifold for network cluster synchronization. In other words, the inter-cluster co-competition balance is an important mechanism to form robust cluster synchronous patterns. The modified model can be applied to describe and characterize many real-world networks with both cooperation and competition. ACKNOWLEDGMENTS
This project was supported by Zhejiang NSF (No. LQ12A01003), NNSF of China (Nos. 11162004, 11171084, and 61203155) and the Science Foundation of Guangxi Province (No. 2013GXNSFAA019006).
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023111-8
Zhang, Ma, and Chen
We are very grateful to the anonymous reviewers’ comments, which were very helpful for us in revising the paper. 1
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