Synchronization, Consensus of Complex Networks and their relationships
aa r X i v : . [ n li n . AO ] A p r Synchronization, Consensus of Complex Networksand their relationships
Tianping Chen,
Senior Member, IEEE
Abstract —In this paper, we focus on the topic Synchronizationand consensus of Complex Networks and their relationships. Itis revealed that two topics are closely relating to each other andall results given in [1] can be obtained by the results in [2].
Index Terms —Consensus, Synchronization, SynchronizationManifold.
I. I
NTRODUCTION
In recent decades, the synchronization problem of multia-gent systems has received compelling attention from variousscientific communities due to its broad applications. Manynatural and synthetic systems, such as neural systems, socialsystems, WWW, food webs, electrical power grids, can all bedescribed by complex networks. In such a network, every noderepresents an individual element of the system, while edgesrepresent relations between nodes. For decades, complex net-works have been focused on by scientists from various fields,for instance, sociology, biology, mathematics and physics.In the pioneer work [4] (also see [ ? ]), the authors proposeda master stability function near a trajectory, by which localsynchronization was investigated. In [ ? ], a distance betweennode state and synchronization manifold was introduced andglobal synchronization was discussed.In [2], a general framework is presented to analyze synchro-nization stability of Linearly Coupled Ordinary DifferentialEquations (LCODEs). The uncoupled dynamical behavior ateach node is general, which can be chaotic or others; thecoupling configuration is also general, without assuming thecoupling matrix be symmetric or irreducible. It was revealedthat the left and right eigenvectors corresponding to eigenvaluezero of the coupling matrix play key roles in the stability anal-ysis of the synchronization manifold. Different from previouspapers, a non-orthogonal projection on the synchronizationmanifold was first introduced. With this projection, a newapproach to investigate the stability of the synchronizationmanifold of coupled oscillators was proposed. Novel masterstability function near the projection was proposed.It is clear that linearly Coupled linear system as well asconsensus are special cases of the linearly Coupled OrdinaryDifferential Equations (LCODEs), which are also hot topics indecades. For example, the synchronization of observer basedlinear systems (see [5], [6], [1] and others). This work is supported by the National Natural Sciences Foundation ofChina under Grant Nos. 61273211.T. Chen was with the School of Computer Sciences/Mathematics, FudanUniversity, Shanghai 200433, China ([email protected]).
II. U
NIFIED MODEL AND GENERAL APPROACH
In this section, we present some definitions, denotations andlemmas required throughout the paper.In [2], following model was discussed dx i ( t ) dt = f ( x i ( t ) , t ) + c N X j =1 l ij Γ x j ( t ) , i = 1 , · · · , N (1)where x i ( t ) ∈ R n is the state variable of the i − th node, t ∈ [0 , + ∞ ) is a continuous time, f : R × [0 , + ∞ ) → R n is continuous map, L = ( l ij ) ∈ R N × N is the couplingmatrix with zero-sum rows and l ij ≥ , for i = j , which isdetermined by the topological structure of the LCODEs, and Γ ∈ R n × n is an inner coupling matrix. Some time, picking Γ = diag { γ , γ , · · · , γ n } with γ i ≥ , for i = 1 , · · · , n . dx i ( t ) dt = Ax i ( t ) + c N X j =1 l ij Γ x j ( t ) , i = 1 , · · · , N (2)where A ∈ R n × n .When n = 1 , A = 0 , we get the following consensus model dx i ( t ) dt = N X j =1 l ij x j ( t ) , i = 1 , · · · , N (3)In case that the state variables x i ( t ) are not observed. Then,instead of coupling x i ( t ) (because they are not available), theauthors coupled the measured output ˙ ζ i ( t ) = N X j =1 l ij y i ( t ) and following observer based synchronization model dx i ( t ) dt = Ax i ( t ) + c N X j =1 l ij F Cx j ( t ) , i = 1 , · · · , N (4)is proposed, where y ( t ) = Cx ( t ) is observer measurement C ∈ R q × n , and C ∈ R n × q , was discussed in [1] and [5], [6].In [5], [6], the model was written as dx i ( t ) dt = Ax i ( t ) + BL N X j =1 c L ij x j ( t ) , i = 1 , · · · , N, (5)where L ∈ R p × n , and B ∈ R N × p . It is a special case whenthe relative states between neighboring agents are available.It is clear that all these model are special cases of the mostgeneral and universal model (1). Therefore, the results givenin [2] can apply to these special cases. First, we give some basic concepts and necessary back-ground knowledge.Following Lemma can be found in [2] (see Lemma 1 in[2]).
Lemma 1. If L is a coupling matrix with Rank(L)=N-1,then the following items are valid: If λ is an eigenvalue of L and λ = 0 , then Re ( λ ) < ; L has an eigenvalue with multiplicity 1 and the righteigenvector [1 , , . . . , ⊤ ; Suppose ξ = [ ξ , ξ , · · · , ξ m ] ⊤ ∈ R m (without loss ofgenerality, assume m P i =1 ξ i = 1 ) is the left eigenvector of A corresponding to eigenvalue . Then, ξ i ≥ holdsfor all i = 1 , · · · , m ; more precisely, L is irreducible if and only if ξ i > holds for all i =1 , · · · , m ; L is reducible if and only if for some i , ξ i = 0 . Insuch case, by suitable rearrangement, assume that ξ ⊤ =[ ξ ⊤ + , ξ ⊤ ] , where ξ + = [ ξ , ξ , · · · , ξ p ] ⊤ ∈ R p , with all ξ i > , i = 1 , · · · , p , and ξ = [ ξ p +1 , ξ p +2 , · · · , ξ N ] ⊤ ∈ R N − p with all ξ j = 0 , p + 1 ≤ j ≤ N . Then, L canbe rewritten as (cid:20) L L L L (cid:21) where L ∈ R p,p isirreducible and L = 0 . By definition, any reducible coupling matrix can be rewrit-ten as (see [3]) L = (cid:20) L L L (cid:21) or more generally, (see [2]) L = L · · · L L · · · ... . . . ... ... L p L p · · · L pp where and for each q = 2 , · · · , p , L qq ∈ R m q ,m q , isirreducible. Remark 1. (see [2]) In fact, if L = [ l ij ] ∈ R N × N is acoupling matrix, then − L is a singular M-matrix. Thus, • If λ is an eigenvalue of L then Re ( λ ) ≤ ; • L has a spanning tree with root L , if and only if for any q = 2 , · · · , p , there is some ¯ q < q , such that L ¯ qq = 0 ; • By M-matrix theory, for any q = 2 , · · · , p , L qq is a non-singular matrix, if and only if there is some ¯ q < q , suchthat L ¯ qq = 0 . Equivalently, L qq is a singular matrix (acoupling matrix), if and only if for all ¯ q < q , L ¯ qq = 0 . Inthis case, L is not a root and L has no spanning tree; • Therefore, L has an eigenvalue with multiplicity 1, ifand only if L has a spanning tree with root L .It is also clear that the so-called master-slave system isa special case of this model. The nodes in root are mastersand others are slaves. Based on previous settings, there isno difference between strongly connected networks and thenetworks with spanning trees. Therefore, in the following, weassume the networks are strongly connected. Let [ ξ , · · · , ξ N ] T be the left eigen-vector corresponding tothe eigenvalue for the matrix L = [ l ij ] . For the model (1)with directed coupling, a nonorthogonal projection of x ( t ) onthe synchronization manifold S, ¯ X ( t ) = [¯ x ( t ) , · · · , ¯ x ( t )] T ,where ¯ x ( t ) = P Ni =1 ξ i x i ( t ) , was first introduced in [2]. Itplays a key role in discussing synchronization problem. Forthe orthogonal projection ¯ x ( t ) = N P Ni =1 x i ( t ) see [3]. Basedon the projection, synchronization is reduced to proving thedistance between all nodes x i ( t ) and the synchronization state δx i ( t ) = x i ( t ) − ¯ x ( t ) → . And (1) can be rewritten as dδx i ( t ) dt = Df (¯ x ( t ) , t ) δx i ( t ) + m X j =1 l ij Γ δx j ( t ) (6)Following theorem and corollary were proved in [2] (seeTheorem 1 and Corollary 1 in [2]). Theorem 1.
Let λ , λ , · · · , λ l be the non-zero eigenvaluesof the coupling matrix L . If all variational equations dz ( t ) dt = [ Df (¯ x ( t ) , t ) + cλ k Γ] z ( t ) , k = 2 , , · · · , l (7) are exponentially stable, then the synchronization manifold S is local exponentially stable for the general synchronizationmodel (1). That is δx i ( t ) = x i ( t ) − ¯ x ( t ) → exponentially. Remark 2.
It is clear that Theorem 1 is based on L ∞ norm.Following theorem is based on L norm. Theorem 2.
Let λ k = α k + jβ k , k = 2 , · · · , m , where j isthe imaginary unit, be the eigenvalues of the coupling matrix.If there exist a positive definite matrix P and ǫ > such that (cid:26) P ( D ( t ) + cλ k Γ) (cid:27) s < − ǫE n , k = 2 , , · · · , m (8) where D ( t ) = ( D ij ( t )) denotes the Jacobian matrix Df (¯ x ( t ) , t ) , H s = ( H ∗ + H ) / , H ∗ is Hermite conjugate of H , and E n ∈ r n × n is identity matrix, then the synchronizationmanifold S is locally exponentially stable for the coupledsystem (1).A. Applications to Consensus It is clear that for linear systems, globally stable and locallystable are equivalent. Therefore, applying Theorem 1 to themodels (2), (3) and (4), we have
Corollary 1.
Let λ , λ , · · · , λ l be the non-zero eigenvaluesof the coupling matrix L . If all variational equations dz ( t ) dt = [ A + cλ k Γ] z ( t ) , k = 2 , , · · · , l (9) are exponentially stable, then the synchronization manifold S is globally exponentially stable for the models (2), (3) and (4)with Γ =
F C . Corollary 2.
Let λ , λ , · · · , λ l be the non-zero eigenvalues ofthe coupling matrix L . If there exist a positive definite matrix P and ǫ > such that (cid:26) P ( A + cλ k Γ) (cid:27) s < − ǫE n , k = 2 , , · · · , m (10) are exponentially stable, then the synchronization manifold S is globally exponentially stable for the models (2), (3) and (4)with Γ =
F C . In case ( A, C ) is detectable, one can find F constructively.Because ( A, C ) is detectable, for a fixed T , P = Z T e − A T t C T Ce − At dt > P A + A T P = − Z T ddt [ e − A T t C T Ce − At ] dt = C T C − e − A T t C T Ce − At < Therefore, there exists ǫ > such that P A + A T P − C T C < − e − A T t C T Ce − A T t < − ǫI n and pick F = P − C T . (cid:26) P ( A + cλ k P − C T C ) (cid:27) s = P A + A T P − cRe ( λ k ) C T C (11)If for all k = 2 , , · · · , m , cRe ( λ k ) > . (cid:26) P ( A + cλ k F C ) (cid:27) s < − ǫE n , k = 2 , , · · · , m (12)Therefore, we can give following result. Corollary 3.
Suppose (A,C) is detectable. λ , λ , · · · , λ l bethe non-zero eigenvalues of the coupling matrix L . If for all k = 2 , , · · · , m , cRe ( λ k ) > . Then, the model dx i ( t ) dt = Ax i ( t ) + c N X j =1 l ij P − C T Cx j ( t ) , i = 1 , · · · , N (13) can be synchronized exponentially, i.e. x i ( t ) − ¯ x ( t ) → exponentially. It was also given in [1]. Here, we reveal the relationsbetween [1] and [2].Based on stabilizable and detectable theory for linear sys-tems, in [1], authors discussed following consensus of multi-agent systems and synchronization of complex networks ˙ x i ( t ) = Ax i ( t ) + Bu i ( t ) , y i ( t ) = Cx i ( t ) (14)where x i ( t ) ∈ R n is the stat, u i ( t ) ∈ R p is the controlinput, and y i ( t ) ∈ R q is the measured output. A ∈ R n × n , B ∈ R n × p , C ∈ R q × n . It is assumed that is stabilizable anddetectable.An observer-type consensus protocol ˙ v i ( t ) = ( A + BK ) v i ( t ) + F ( N X j =1 Cl ij v j ( t ) − ζ i ( t )) (15)is proposed, which can also be written as (cid:26) ˙ v i ( t ) = ( A + BK ) v i ( t ) + P Nj =1 F Cl ij ( v j ( t ) − x j ( t ))˙ x i ( t ) = Ax i ( t ) + BKv i ( t ) (16)where K ∈ R p × n , F ∈ R n × q . Let e i ( t ) = v i ( t ) − x i ( t ) , one can transfer (16) to (cid:26) ˙ e i ( t ) = Ae i ( t ) + F C P Nj =1 l ij e j ( t )˙ x i ( t ) = ( A + BK ) x i ( t ) + BKe i ( t ) (17)Denote ¯ x ( t ) = P Ni =1 ξ i x i ( t ) , ¯ e ( t ) = P Ni =1 ξ i e i ( t ) , δx i ( t ) = x i ( t ) − ¯ x ( t ) , δ ¯ e i ( t ) = e i ( t ) − ¯ e ( t ) .Therefore, as a special case of Theorem 1, we have Theorem 3.
Let λ , λ , · · · , λ l be the non-zero eigenvaluesof the coupling matrix L . If dz ( t ) dt = [ A + λ k F C ] z ( t ) , k = 2 , , · · · , l (18) are exponentially stable, then δ ¯ e i ( t ) = e i ( t ) − ¯ e ( t ) , i =1 , · · · , N , converge to zero exponentially.Additionally, if A + BK is Hurwiz, then, δ ¯ x i ( t ) = x i ( t ) − ¯ x ( t ) , i = 1 , · · · , N , converge to zero exponentially.In particular, if ( A, C ) is detectable, we can pick F = P − C T and for all k = 2 , , · · · , m , cRe ( λ k ) > . Then,the model (16) converges. Remark 3.
By Theorem 1, it is clear that under the conditionsthat A + λ k F C, k = 2 , , · · · , l are Hurwiz, then followingalgorithm discussed in [5], [6] ˙ x i ( t ) = Ax i ( t ) + F N X j =1 l ij y j ( t ) (19) where y ( t ) = Cx ( t ) , reaches consensus exponentially.B. Applications to Pinning Control In this section, we apply general results given in to pinningcontrol of multi-agents consensus.Consider dx ( t ) dt = Ax ( t ) + c m P j =1 l j Γ x j ( t ) − cε ( x ( t ) − s ( t )) , dx i ( t ) dt = Ax i ( t ) + c m P j =1 l ij Γ x j ( t ) ,i = 2 , · · · , m (20)In particular, in case ( A, B ) is controllable, dx ( t ) dt = Ax ( t ) + c m P j =1 l j P − BB T x j ( t ) − cε ( x ( t ) − s ( t )) , dx i ( t ) dt = Ax i ( t ) + c m P j =1 l ij P − BB T x j ( t ) ,i = 2 , · · · , m (21)where ˙ s ( t ) = As ( t ) . Proposition 1. If A = ( a ij ) mi,j =1 is an irreducible Mezlermatrix with Rank ( A ) = m − . Then, the real part of alleigenvalues of the matrix ˜ L = l − ε l · · · l m l l · · · l m ... ... . . . ... l m l m · · · l mm are negative. Denote δx i ( t ) = x i ( t ) − s ( t ) , then for i = 1 , · · · , m , wehave δdx i ( t ) dt = Ax i ( t ) + c m X j =1 ˜ l ij Γ δx j ( t ) (22)and dδx i ( t ) dt = Aδx i ( t ) + c m X j =1 ˜ l ij P − BB T δx j ( t ) , (23)Denote the eigenvalues of ˜ L by µ , · · · , µ N and by samearguments, just replacing ¯ x ( t ) by s ( t ) , we have Corollary 4.
Let µ , µ , · · · , µ m be the eigenvalues of thecoupling matrix ˜ L . If all variational equations dz ( t ) dt = [ Df (¯ x ( t ) , t ) + cλ k Γ] z ( t ) , k = 2 , , · · · , l (24) are exponentially stable, then for the model (22) , δx i ( t ) = x i ( t ) − s ( t ) → exponentially. Corollary 5.
Suppose (A,B) is controllable. µ , µ , · · · , µ m be the eigenvalues of the coupling matrix ˜ L . If for all k =1 , , , · · · , m , cRe ( µ k ) > . Then, for the model (23) , x i ( t ) − s ( t ) → exponentially. Remark 4.
Synchronization (consensus) with or without pin-ning control are two different topics but closely related.For Synchronization (consensus) without pinning control, thesynchronization state is ¯ x ( t ) . Instead, For Synchronization(consensus) pinning control, the synchronization state is asolution s ( t ) of the uncoupled system ˙ s ( t ) = f ( s ( t )) . Remark 5.
For Synchronization (consensus) without pinningcontrol, the coupling matrix L is a singular M-matrix. Instead,For Synchronization (consensus) pinning control, the couplingmatrix L is a nonsingular M-matrix.. In [2], it is revealed that even though the synchronizationmanifold can be stable, the individual state may be unstable.It was also explored that the right and left eigenvectors ofthe coupling matrix corresponding to the eigenvalue 0 playkey roles in the geometrical analysis of the synchronizationmanifold. These two eigenvectors are used to decompose thewhole space into a direct sum of the synchronization mani-fold and the transverse space. By means of this geometricalanalysis, a new approach to investigating the stability of thesynchronization manifold was proposed.III. D
ISCUSSIONS • In [4], the synchronization stability of a network ofoscillators by using the master stability function methodwas introduced.In [1], it was said that [2], [4] (References [22] and[27] in [1]) addressed the synchronization stability ofa network of oscillators by using the master stabilityfunction method.The authors also said that the proposed framework is,in essence, consistent with the master stability function method used in the synchronization of complex networksand yet presents a unified viewpoint to both the consensusof multiagent systems and the synchronization of com-plex networks.In fact, for linear systems, global stability and localstability are equivalent. Therefore, the master stabilityfunction method can be used to prove local stability aswell as global stability.It should be pointed out that the master stability functionsare different in the two papers [2] and [4]. In [2], masterstability function applies based on ¯ x ( t ) . Instead, in [4],master stability function applies based on s ( t ) satisfying ˙ s ( t ) = f ( s ( t )) . Here, in [1], the authors follow the lineand approach proposed in [2]. • There are two fundamental questions about the synchro-nization and consensus problems of coupled systems:how to reach consensus and consensus on what, as saidin [1].In fact, this issue has been addressed in [2] (see Theorem1 and Theorem 2 in [2]). In [2], the following universalapproach based on the decomposition has been proposed. – Synchronization Manifold : S = { x =[ x , · · · , x N ] ∈ R n,N : x i = x j , f or all i, j } – Non-orthogonal transverse subspace L = { x =[ x , · · · , x N ] ∈ R n,N : n P i =1 ξ i x i = 0 } – Decomposition of R n,N : R n,N = S ⊕ L . – For each x = [ x , · · · , x N ] ∈ R n,N , define1. ¯ x = n P j =1 ξ i x i and ¯ X = [¯ x, · · · , ¯ x ] ∈ S
2. Let δx i = x i − ¯ x , for all i . Then δx = x − ¯ X =[ δx , · · · , δx m ] ∈ L , – Decomposition: x = ¯ X + δx , where ¯ X ∈ S and δx ∈ L – Stability of Synchronization manifold
S ⇔ δx → . – δx → answers the question ”How to”. ¯ x = n P j =1 ξ i x i answers ”consensus on what”, i.e., whatis the synchronization state.– The conditions given in Theorem 1 (as well asTheorem 2) ensure δx → . It has been revealed that the Left eigenvector and RightEigenvector of the coupling matrix L with Eigenvalue Play key roles on Synchronization – Right eigenvector = [1 , · · · , T ∈ R N denotesthe direction parallel to S ; – Left eigenvector ξ = [ ξ , ξ , · · · , ξ N ] ⊤ ∈ R N de-notes the direction of the transverse subspace L . • In [1], the so called relative-State consensus protocol(also see [5], [6]) dx i ( t ) dt = Ax i ( t ) + BL N X j =1 c L ij x j ( t ) , i = 1 , · · · , N, (25)where L ∈ R p × n , was discussed.As a direct consequence of Theorem 1, we have Fig. 1. Decomposition of x ( t ) Theorem 4.
Let λ , λ , · · · , λ l be the non-zero eigen-values of the coupling matrix L . If dz ( t ) dt = [ A + λ k BL ] z ( t ) , k = 2 , , · · · , l (26) are exponentially stable, then the synchronization mani-fold S is exponentially stable for the general model (25). On the other hand, in case that the relative states betweenneighboring agents are not available, following protocol dx i ( t ) dt = Ax i ( t ) + B ¯ L N X j =1 c L ij y j ( t ) , i = 1 , · · · , N, (27)where ¯ L ∈ R p × q , B ∈ R n × p , can be used.It can also be rewritten as dx i ( t ) dt = Ax i ( t ) + B ¯ LC N X j =1 c L ij x j ( t ) , i = 1 , · · · , N, (28)Therefore, by Theorem 1, we have Theorem 5.
Let λ , λ , · · · , λ l be the non-zero eigen-values of the coupling matrix L . If dz ( t ) dt = [ A + λ k B ¯ LC ] z ( t ) , k = 2 , , · · · , l (29) are exponentially stable, then the synchronization mani-fold S is exponentially stable for the general model (27). • In [1], following Spacecraft Formation Flying model (cid:20) ˙ r i ¨ r i (cid:21) = (cid:20) I A A (cid:21) (cid:20) r i − h i ˙ r (cid:21) + c N X j =1 a ij (cid:20) F F (cid:21) (cid:20) r i − h i − r j + h j ˙ r i − ˙ r j (cid:21) (30)was discussed. And following result (Corollary 3) wasgiven:Assume that graph has a directed spanning tree. Then,protocol (30) solves the formation flying problem if andonly if the matrices (cid:20) I A A (cid:21) + cλ i (cid:20) F F (cid:21) areHurwitz for i = 2 , · · · , N , where λ i , i = 2 , · · · , N ,denote the nonzero eigenvalues of the Laplacian matrixof L . It is clear that Corollary 1, Corollary 2 and Corollary 3in [1] can be obtained directly from Theorem 1. • It is claimed in [1] that ”It is observed by comparingTheorem 2 and Corollary 2 that even if the consensusprotocol takes the dynamic form (3) or the static form(22), the final consensus value reached by the agents willbe the same, which relies only on the communicationtopology, the initial states, and the agent dynamics.”However, for the coupled system (4), we have x i ( t ) − e At N X i =1 ξ i [ x i (0) − v i (0)] → (31)Instead, for the system (16), we have x i ( t ) − e At N X i =1 ξ i x i (0) → (32)They are different.It is claimed in [1] that Similar to [23], δ is referred toas the disagreement vector. In fact, the accurate saying isthat it came from [3] and [2]. In particular, from [2]. Conclusions
In this paper, we focus on the topic Syn-chronization and consensus of Complex Networks and theirrelationships. It is revealed that two topics are closely relatingto each other and all results given in [1] can be obtained by theresults in [2]. Several protocols on this topic are also revisitedand the relationships between them are addressed. It is pointedout that the model introduced in [2] and the approach providedthere is universal. Many existed synchronization and consensusmodels and their stability behavior analysis can be derivedeasily from the theoretical results given in [2]. These modelsinclude consensus and synchronization of linear coupled non-linear (or linear) systems, observed-based linear systems andmany others. R
EFERENCES[1] Zhongkui Li, Zhisheng Duan, Guanrong Chen, and Lin Huang Consen-sus of Multiagent Systems and Synchronization of Complex Networks:A Unified Viewpoint,