Automatica Global stabilization of complex networks with ... - CiteSeerX

Automatica 46 (2010) 116–121

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Global stabilization of complex networks with digraph topologies via a local pinning algorithmI Wenlian Lu a , Xiang Li b,∗ , Zhihai Rong c a

Centre for Computational Systems Biology and School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China

b

Adaptive Networks and Control Lab, Department of Electronic Engineering, Fudan University, Shanghai 200433, PR China

c

Department of Automation, Donghua University, Shanghai 201620, PR China

article

info

Article history: Received 3 November 2008 Received in revised form 21 February 2009 Accepted 22 September 2009 Available online 20 October 2009 Keywords: Pinning control Directed graph Stabilizability Shortest path Complex network Chaos control

abstract This paper concerns the global stability of controlling a complex network with digraph topology to a homogeneous trajectory of the uncoupled system by the local pinning control algorithm. The derived stability condition indicates that the smallest real part of eigenvalues of the Laplacian sub-matrix corresponding to the unpinned vertices can be used to measure the stabilizability of a digraph with a given pinned vertex set. A pinned vertex set can stabilize a directed network to some unstable trajectories, for instance, to a chaotic trajectory of the uncoupled systems, if and only if the pinned vertex set can access all other vertices in the digraph. Furthermore, in the bigraph case, the analytical estimation of the stabilizability’s lower bound suggests that an optimal pinning strategy should take not only the vertex degree, but also the shortest path between pairs of vertices into considerations. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Since 1998, the new discoveries of small-world (Watts & Strogatz, 1998) and scale-free connectivity patterns (Barabási & Albert, 1999) in many natural and artificial complex networking systems have attracted very wide attention on the complexity of network topology, yielding fruitful results which enrich and deepen the understanding of real-world complex networks, and put an important step forward from the random graph theory built by Erdös and Rényi (1959). Control and stabilization of largescale dynamic systems has been one of the main issues with wide interests over the past decades (Siljak, 1991), and the latest generalization of nonlinear small-gain theorems have been stated to find their application in analysis and control synthesis for largescale network systems (Dashkovskiy, Ruffer, & Wirth, 2007; Jiang & Wang, 2008). More close to the interest of this paper, in Wang and

I This paper was supported partly by the National Key Basic Research and Development Program (No. 2010CB731403), the National Natural Sciences Foundation of China (No. 60874089 and No. 60804044), and partly by Shanghai Rising-Star Program (No. 09QH1400200), and Shanghai Pujiang Program (No. 08PJ14019). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +86 21 65642127; fax: +86 21 65642127. E-mail addresses: [email protected] (W. Lu), [email protected] (X. Li), [email protected] (Z. Rong).

0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.10.006

Chen (2002), Li, Wang, and Chen (2004), Sorrentino, di Bernardo, Garofalo, and Chen (2007), Chen, Liu, and Lu (2007), Xiang and Chen (2007), Wang and Slotine (2006), Porfiri and di Bernardo (2008), Yu, Chen, and Lü (2009), the authors have stabilized a complex bi-directed network to its homogeneous static state with a small fraction of vertices placing local feedback controllers, i.e., the so-called pinning control algorithm (Hu & Qu, 1994; Roy, Murphy, Maier, & Gills, 1992), which was first proposed to control the multi-mode laser systems. It was concluded that the pinning strategies should be more selective according to the detailed complex network topology, and for a class of scale-free networks, the degree-based pinning strategy, i.e., pinning the vertex set with the largest connection degrees, yields better stabilizability than that of randomly placing pinning controllers in the network (Li et al., 2004; Wang & Chen, 2002). More recently, the concept of Lyapunov stability for complex dynamical networks has been proposed and applied with the pinning control algorithm, and the network stability (controllability) is converted to measuring the negative definiteness of one simple matrix characterizing the network topology (Xiang & Chen, 2007), which can be measured by the eigenvalues of such matrices (Porfiri & di Bernardo, 2008; Wang & Slotine, 2006). Also, (Porfiri & di Bernardo, 2008; Wang & Slotine, 2006; Yu et al., 2009) used Lyapunov function to study pinning control in complex networks and gave conditions based on the algebraic properties of the coupling matrix guaranteeing stability. However, up until now, a general stability analysis of pinning control a complex network with directed weighted graph

W. Lu et al. / Automatica 46 (2010) 116–121

topology towards digging the influence of the graph topology is absent. In this paper, we further investigate the pinning control of a complex network with directed graph topology to an arbitrary trajectory of the uncoupled intrinsic system and focus on the relationship between selection of pinned vertices and the underlying graph topology. First, we present several criteria guaranteeing the global stability of a directed network with the pinning control algorithm. Second, we propose a new quantity to measure the stabilizability of a directed graph with a pinned vertex set. Thus, we find the pinned vertex set with minimum numbers to realize controlling stability upon some unstable trajectory including chaos. In the case of bidirected graph (bigraph), we find the lower-bound estimation of the stabilizability depends on two factors: the connection degrees of the pinned vertices, and the shortest paths from the pinned vertex set to the unpinned vertex set. 2. Preliminaries At first, we introduce the notations employed through the paper. Z > defines the transpose of a matrix Z and Z s is the symmetry part of Z . We denote by Z > 0 (≥ 0) that Z is positive (semi-positive) definite and so it is with < 0 and ≤ 0. λ(Z ) is the eigenvalue set of a square matrix Z and λmax (Z ) (λmin (Z )) is the maximum (minimum) eigenvalue of Z (if all real). We denote the 2-norm for both vectors and matrices as k·k2 . Here, Ip is the identity matrix with dimension p. The symbol ⊗ is the Kroneck product. Re(z ) defines the real part of a complex number z. For a set T , T c is its complementary set and then S \ T denotes S ∩ T c . ∅ is the empty set. #E denotes the number of the elements of a finite set E. ! denotes the factorial. Several definitions and a lemma in the graph theory along with the notations are introduced in the following which will be used in the later analysis. The interested readers please refer to some textbooks of graph theory (Brualdi & Ryser, 1991; Godsil & Royle, 2001; Jungnickel, 2005) for more details. Let G = [V , E , W ] denote a weighted directed graph (digraph), where V denotes the vertex set numbered by {1, 2, . . . , m}, E = {e(i, j)} denotes the edge set by that e(i, j) ∈ E if and only if there exists an edge from vertex j to vertex i, where vertex j(i) is said to be the tail (head) of the edge, and W = {wij } where wij > 0 denotes the weight of edge e(i, j). We denote the Laplacian matrix of a weighted digraph G by L = [lij ]m i,j=1 . That is, lij = −wij (i 6= j) if there exists an edge from vertex Pm j to vertex i; otherwise lij = 0, and diagonal elements lii = − j=1,j6=i lij . We denote the edge set from vertex set V2 to V1 by E (V1 , V2 ). We say vertex j accesses vertex i or vertex i can be accessible from vertex j if there exists a path from j to i. And we denote all vertices which can access vertex i via a single edge by N (i). It should be pointed out that digraphs we consider here are supposed to be simple, i.e., without loops and multiple edges. And, we define a digraph as the reversal of the definitions in ordinary graph theory textbooks for the convenient description of dynamical networks. Thus, we define a bigraph G = [V , E ] where V denotes the vertex set, and E = {e(i, j)} denotes the edge set by that e(i, j) ∈ E if there exists an edge between vertex i and vertex j, which implies that e(i, j) ∈ E if and only if e(j, i) ∈ E . We say that a digraph G has a spanning tree if there exists a vertex called root which can access all other vertices. Denote the root set by root(G) composed of the root vertices which can access all other vertices. We say a digraph G is strongly connected, if for any vertex pair (i, j) there exist a directed path from vertex j to vertex i and another directed path from vertex i to vertex j. And, a subgraph Gi of G is said to be a strongly connected component if Gi is strongly connected and no other subgraph containing Gi is strongly connected. For a bigraph, the connection degree of a vertex is the number of the neighbors in its neighborhood. In this paper, for a given vertex subset V1 ⊂ V , we denote the sub-matrix of L composed

117

of the rows and columns with indices in V1 by L(V1 ) and the corresponding subgraph by G(V1 ). d(i, j) denotes the distance form vertex j to vertex i, i.e., the length of the shortest path from vertex j to vertex i, and for two vertex sets V1 and V2 , d(V1 , V2 ) = supj∈V2 infi∈V1 d(i, j) denotes the Hausdorff distance from the vertex set V2 to V1 . m,m Definition 1. Let C = [cij ]m be a nonsingular matrix i,j=1 ∈ R with cij ≤ 0, i, j = 1, . . . , m, i 6= j. C is said to be an M-matrix if all elements of C −1 are nonnegative.

For more details concerned with M-matrix, we refer the interested readers to Refs. (Berman, 2003; Boyd et al., 1994; Wu, 2005). Linearly coupled ordinary differential equations (LCODEs) model a networking coupled system with continuous-time states, which can be generally written as follows: x˙ i (t ) = g (xi (t ), t ) − c

m X

lij Γ xj (t ),

i = 1, . . . , m

(1)

j =1

where xi (t ) = [xi1 (t ), . . . , xin (t )]> ∈ Rn denotes the state variable vector of the ith individual vertex in the coupled system, t ∈ R+ = [0, ∞) denotes the continuous time, g : Rn × R+ → Rn is a continuous map, scalar c denotes the coupling strength, lij ≤ 0 denotes the coupling coefficient from vertex j to vertex i, for all Pm i 6= j, with lii = − j=1,j6=i lij , a semi-positive definite matrix n,n Γ = [γkl ]m denotes the inner connection matrix with k,l=1 ∈ R γkl 6= 0 if two vertices are connected by their kth and lth state component respectively. It is well known that the coupled system (1) can correspond to a digraph G = [V , E , W ], where V is composed of the individuals of the system denoted by {1, . . . , m}, the coupling matrix L = [lij ]m i,j=1 denotes the Laplacian, and by the way we can obtain the corresponding edge set E and weight set W . 3. Pinning control of directed networks: Global stability analysis In this section, we study the pinning control problem of a directed network described by the LCODEs (1) to a homogeneous trajectory satisfying: s˙(t ) = g (s(t ), t )

(2)

where s(t ) can be an equilibrium, limit cycle, or even a chaotic attractor of the uncoupled system. For this purpose, we apply the pinning control strategy with local feedback controllers to a vertex subset D ⊂ V , the pinned vertex set, with a very small fraction, i.e., f = #D /#V  1. Therefore, the pinning controlled network is described as follows: x˙ i (t ) = g (xi (t ), t ) − c

m X

lij Γ xj (t ) − c  Γ

j =1

× [xi (t ) − s(t )],

i∈D

m

x˙ k (t ) = g (xk (t ), t ) − c

X

lkj Γ xj (t ),

k 6∈ D ,

(3)

j =1

where scalar  > 0 is the feedback control gain. For the function g (·, ·), we have the following assumption. Definition 2. Given a square matrix V , a function h : Rn × R+ → Rn is V -uniformly decreasing if there exists δ > 0 such that

(x − y)> V (h(x, t ) − h(y, t )) ≤ −δ(x − y)> (x − y) holds for all x, y ∈ Rn and t ∈ R+ . Considering a single system: u˙ = g (u, t ) −  Γ (u − s) with s˙ = g (s, t ), we can conclude that if g (x, t ) − α Γ is V -uniformly decreasing for some α > 0, then a sufficiently large  can guarantee limt →∞ [u(t ) − s(t )] = 0. However, in the networking system (3), other unpinned vertices in the network may not always be guaranteed to stabilize to s(t ) for any arbitrary large  . To study

118

W. Lu et al. / Automatica 46 (2010) 116–121

these interactional factors on pinning control a complex network, we just leave the pinning control gain  aside by supposing that it can be sufficiently large as mentioned in Li et al. (2004); Wang and Chen (2002). Then, we are in the position to address the main theorem. Theorem 1. Suppose that

= 2(x − S )> (Ξ ⊗ V ){G(x, t ) − G(S , t ) − α(Im ⊗ Γ )(x − S ) −[(cL − α Im ) ⊗ Γ ](x − S ) − c (D ⊗ Γ )(x − S )} ≤ −2δ(x − S )> (Ξ ⊗ In )(x − S ) − 2(x − S )> × {[Ξ (cL + c  D − α Im )] ⊗ (V Γ )}(x − S ) ≤ −2δ(x − S )> (Ξ ⊗ In )(x − S ) L(x, s) . ≤ −2δ kV k2

1. there exist a scalar α and a symmetric positive definite matrix V such that g (x, t ) − α Γ x is V -uniformly decreasing; 2. V Γ is semi-positive definite; 3. the matrix cL(V \ D ) − α Im−p is an M-matrix, where p = #D .

It gives

Then, there exists an 0 > 0 such that for any  > 0 , the pinning controlled directed network (3) is exponentially asymptotically stable at the homogeneous trajectory s(t ).

which completes the proof of this theorem by   estimating the − kVδk t 2 convergent rate as kx(t ) − S (t )k2 = O e . 

Proof. Without loss of generality, we suppose that D = {1, 2, . . . , p} and V \ D = {p + 1, . . . , m}h. By this idecomposition, we rewrite the Laplacian matrix L as L =

L11 L21

L12 L22

, where L11 ∈ Rp,p

and L22 ∈ Rm−p,m−p correspond to the vertex subset D and V \ D , respectively. Owing to the condition (3) that cL22 − α I is an Mmatrix, from the results on M-matrix as shown in Berman (2003), there must exist m − p positive constants ξ1 , ξ2 , . . . , ξm−p such that for the positive definite diagonal matrix Ξ2 = diag h {ξ1 , .i. . , ξm−p },

[Ξ2 (cL22 − α I )]s is positive definite. Let Ξ = h i Ip 0

0 0

Ip 0

0

Ξ2 and D =

. We firstly prove the following claim.

2c  Ip ≥ 2α Ip + (cL12 + cL21 Ξ2 )(c Ξ2 L22 + cL22 Ξ2 − 2α Ξ2 ) >

−1

> × (c Ξ2 L21 + cL> 12 ) − cL11 − cL11

holds for all  > 0 . It gives > cL11 + cL> 11 + 2c  Ip − 2α Ip − (cL12 + cL21 Ξ2 ) −1 × (c Ξ2 L22 + cL> (c Ξ2 L21 + cL> 22 Ξ2 − 2α Ξ2 ) 12 ) > 0

for all  > 0 . Applying the Schur complement (Boyd et al., 1994) > with Q (x) = cL11 + cL> 11 + 2c  Ip , S (x) = cL12 + cL21 Ξ2 , and > R(x) = c Ξ2 L22 + cL22 Ξ2 − 2α Im−p , we have the following linear matrix inequality



2c (L11 )s + 2c  Ip − 2α Ip c Ξ2 L21 + cL> 12

  − kV2δk t 2 L(x(0), S (0))

Theorem 1 tells that under the conditions, a sufficiently large  can guarantee the global stability. From the proof, the lower bound of such an  can be estimated as 1 > −1  ≥ k2α Ip + (cL12 + cL> 21 Ξ2 )(c Ξ2 L22 + cL22 Ξ2 − 2α Ξ2 ) 2c > × (c Ξ2 L21 + cL> (5) 12 ) − cL11 − cL11 k2 for some positive η. However, this bound would be rather coarse

due to the imprecise estimation in the proof. Adaptive algorithm upon  can be utilized to reach a relatively smaller feedback control gain:

˙ = ηc Ξ (x − S )> D ⊗ (V Γ )(x − S ).

Claim 1. There exist an m × m positive definite diagonal matrix Ξ and a constant 0 > 0 such that [Ξ (cL + c  D − α Im )]s is positive definite for all  > 0 . In fact, one can see that there exists some 0 such that >

W (x(t ), S (t )) ≤ e

cL12 + cL> 21 Ξ2 2c (Ξ2 L22 )s − 2α Im−p



> 0,

for all  > 0 . Equivalently, we conclude that [Ξ (cL + c  D − α Im )]s is positive definite for all  > 0 . This proves Claim 1. >

Then, we construct a Lyapunov function. Let x = [x1 , . . . , x ] ∈ Rnm , S = [s> , . . . , s> ]> ∈ Rnm , and G(x, t ) = [g (x1 , t )> , . . . , g (xm , t )> ]> . Thus, the pinned network (3) is rewritten in the following matrix form: m> >

x˙ (t ) = G(x(t ), t ) − c (L ⊗ Γ )x − c (D ⊗ Γ )[x(t ) − S (t )].

(4)

(6)

In fact, if considering a candidate Lyapunov function as W1 = (x − S )> (Ξ ⊗ V )(x − S ) +

1

η

( − 0 )2

with 0 satisfying inequality (5). Similar algebras can lead the result that this adaptive feedback (6) can guarantee the global stability. From Theorem 1, let us consider the infimum of the coupling strength c with which the network can be pinned to any homogeneous trajectory of the uncoupled system with parameter α , which represents a class of function g (·, ·). That is, a smaller infimum indicates that the network can be easier to be stabilized. From the condition 3 of Theorem 1, one has inf{c : cL(V \ D ) − α Im−p is an M-matrix}. To calculate such an infimum of c, we have the following proposition directly from the M-matrix theory (Berman, 2003). Proposition 2. Let B = L(V \ D ). The following statements are equivalent: 1. cB − α Im−p is an M-matrix; 2. c minu∈λ(B) Re(u) > α ; 3. There exists a positive definite diagonal matrix Ξ with dimension m − p such that [Ξ (cB − α Im−p )]s is positive definite. From this proposition, we have

α

inf{c : cL(V \ D ) − α Im−p is an M-matrix} =

u∈λ(L(V \D ))

stab(G, D ) =

W (x, s) = (x − S ) (Ξ ⊗ V )(x − S ).

or equivalently, in the Rayleigh quotient form:

Differentiating W (x, s) along systems (2) and (4), we have d dt

W (x, s)|(2) and (4)

= 2(x − S )> (Ξ ⊗ V )[G(x, t ) − G(S , t ) −c (L ⊗ Γ )x − c (D ⊗ Γ )(x − S )]

Re(u)

.

Hence, the following quantity can be used to measure the stabilizability of a digraph G with the pinned vertex set D :

Define the following candidate Lyapunov function: >

min

stab(G, D ) =

min

u∈λ(L(V \D ))

max

Re(u)

min

W = diag{wi }m >0 z 6=0 i=1

(7)

z > WL(V \ D )z z > Wz

.

(8)

Therefore, the third stability criterion in Theorem 1 is rewritten as c>

α stab(G, D )

.

(9)

W. Lu et al. / Automatica 46 (2010) 116–121

One can see that a larger stab(G, D ) implies a larger region of c to stabilize a network by a pinned vertex set D . Since α > 0 implies that the expected trajectory s(t ) might be unstable for the uncoupled system (2), especially, s(t ) could be chaotic, to stabilize a directed network to such a trajectory, we need stab(G, D ) > 0. The following result gives the selection of pinned vertices leading to stab(G, D ) > 0. Theorem 3. The sufficient and necessary condition for stab(G, D ) > 0 is that the pinned vertex set D can access all other vertices in the digraph G, i.e., for any vertex u ∈ V \ D , there exists a vertex v ∈ D such that there is a directed path from v to u. Proof. Necessity. Let V1 be the maximal vertex set in V \ D which is accessible by D , and V2 = V \(V1 ∪D ). We suppose V2 6= ∅. This implies that no edges entering V2 from V1 ∪ D . Therefore, the submatrix L(V2 ) has the eigenvalue zero because its sum of each row is zero. This causes that one of the eigenvalues of L(V \ D ) is zero, which contradicts with the condition stab(G, D ) > 0. Therefore, V2 = ∅, and the necessity is proved. Sufficiency. With a proper permutation P, we rewrite the matrix B = L(V \ D ) (supposed to be l-reducible) in the following Frobenius form (Godsil & Royle, 2001): B11 0

 

B=P 0 0 0

B12 B22 0 0 0

··· ··· .. .

B1l−1 B2l−1

··· ···

Bl−1l−1 0

.. .

B1l B2l 



.. .

Bl−1l Bll

 > P  

where the diagonal blocks Bkk , k = 1, . . . , l, are irreducible, and the low-triangular blocks are zero. Let Gk be the subgraph corresponding to the sub-matrix Bkk , 1 ≤ k ≤ l. First, we claim that for each 1 ≤ k ≤ l − 1, there exists at least one index j ≥ k such that Bkj 6= 0 and Bll is an M-matrix. Otherwise, suppose that there exists 1 ≤ k ≤ l such that Bkk has all row sums zero, which implies that the vertex set corresponding to Bkk is not accessible from other vertices. This contradicts with the fact that the vertex set D can access all other vertices. Therefore, Bkk is strictly diagonal dominant at one row at least, and we conclude that Bkk is an Mmatrix for all 1 ≤ k ≤ l. By the properties of M-matrices, we finally conclude that stab(G, D ) > 0 since the eigenvalue set of B is composed of the eigenvalue sets of Bkk , 1 ≤ k ≤ l. This completes the proof of the theorem.  This theorem gives a way to choose pinned vertices for stab(G, D ) > 0. Another question arises naturally that what is the minimal number of the pinned vertex set D guaranteeing stab(G, D ) > 0. Namely, min{#D : stab(G, D ) > 0}. With the proof of Theorem 3, we have the following corollary immediately. Corollary 4. Let {C1 , C2 , . . . , Cl } be the subset collection of the vertex vet V satisfying that the corresponding subgraph G(Ci ) is a strongly connected component of a digraph G, for all i = 1, 2, . . . , l. Let p be the number of these strongly connected components satisfying that there are no edges with heads in and tails out for each component. Then, we have p = min{#D , stab(G, D ) > 0} = m − rank(L). Furthermore, without loss of generality, let C1 , . . . , Cp be the collection of strongly connected components of G satisfying that there are no edges with heads in and tails out. The collection of the pinned vertices D satisfying D ∩ Ci 6= ∅ for all i = 1, 2, . . . , p is sufficient Sp and necessary for stab(G, D ) > 0. If D satisfies that D ⊂ i=1 Ci and D ∩ Ci is a singleton set for all i = 1, 2, . . . , p, then D is a pinned vertex set with the minimal number satisfying stab(G, D ) > 0. Jungnickel (2005) presents an algorithm to get the strongly connected components (Algorithms 8.5.4 and 8.5.5). According to

119

Corollary 4, this algorithm can be utilized to obtain the minimal pinned vertex set satisfying stab(G, D ) > 0. These algorithms mainly contain the following steps: firstly, we get all strongly connected components; secondly, we select such strongly connected component which has no edges with heads in and tails out; finally, a vertex set intersecting each of such strongly connected components via a single vertex composes the minimal vertex set for stab(G, D ) > 0. 4. Pinning control of bi-directed networks: The stabilizability bounds A specific bigraph case of the Laplacian of a network is based on the adjacent matrix and widely studied in many previous works. Since a bigraph is a special case of the digraph, the conclusions in the digraph section can be directly applied. Here, we consider a bigraph G = [N , E ]. Let A be the adjacent matrix of the graph. Let D = diag{k1 , . . . , km } be the degree matrix, where ki is the connection degree of vertex i, i = 1, . . . , m. Thus, we define that Laplacian L = (lij )m i,j=1 = D − A. Let V1 be a subset of V . We denote by kmax (V1 ) the maximum degree of the vertex set V1 . The following proposition is the direct consequence from Theorem 1, which revisits and improves the corresponding Theorem 1 in Li et al. (2004). Proposition 5. Assume that the graph of the network is a bigraph and its Laplacian L is symmetry. Suppose that (1) there exist a scalar α and a symmetric and positive definite matrix V such that f (x, t ) − α Γ x is V -uniformly decreasing; (2) V Γ is positive semi-definite; (3) λmin (L(V \ D )) > α . Then, there exists an 0 > 0 such that for any  > 0 , the pinning controlled bi-directed network (3) is exponentially asymptotically stable at the homogeneous trajectory s(t ). According to Theorem 3 as well as Corollary 4, we conclude that stab(G, D ) > 0 if and only if all vertices in G \ D are accessible from the vertex set D . And, the minimal pinned vertex set guaranteeing stab(G, D ) > 0 is such a set with exactly one vertex from each connected component, which coincides with the generalized conclusion in Chen et al. (2007). Furthermore, we analytically estimate the stabilizability bounds of a bigraph. Theorem 6. Let k = kmax (D ), and d = d(V \ D , D ). For any V1 ⊂ V \ D , the following inequalities hold: #E (V1 , V \ (D ∪ V1 )) + #E (V1 , D ) #V1

≥

1 2

  2 k+

1 2

(2k)

−d

−d

.

≥ stab(G, D ) (10)

Proof. The right part of inequality (10) can be directly obtained from Theorem 4 in Wu (2005). We now prove the left part. Noting that L is symmetric, we let L22 = L(V \ D ), and the stabilizability can therefore be written in the following Rayleigh quotient form: stab(G, D ) =

min

z 6=0,z ∈Rm−p

z > L22 z z>z

.

hAnd, without i loss of generality, suppose L22 has the form L22 = W11 W> 12

W12 W22

with W11 corresponding to the vertex subset V1 . Let

z = [z1 , z2 ] , where z1 = [1, . . . , 1]> corresponding to the vertex >

120

W. Lu et al. / Automatica 46 (2010) 116–121

set V1 , and z2 = 0. Let W11 = D1 + V1 , where D1 is a nonnegative diagonal matrix, and V1 is a matrix with each row sum zero. Then, we have z > L22 z z>z

=

z1> W11 z1 z1> z1

=

z1> D1 z1 z1> z1

.

Noting that z1> z1 = #V1 and z1> D1 z1 = #E (V1 , V \ V1 )

Table 1 Stabilizabilities of networks with the two different pinned strategies and their estimations. l0

stab1

stab2

(K −l0 +1)! 2K !

1 2

2 3 4 5

1.0408e−2 1.7268e−3 1.5544e−4 1.9039e−5

3.8493e−3 1.8491e−2 7.4577e−2 0.26795

8.3333e−2 1.6667e−2 4.1667e−3 1.3889e−3

1.3467e−6 1.7506e−5 2.2758e−4 2.9586e−3

(2K + 1)−(K −l0 +1)

pinned vertex set to the others into considerations when designing a more effective pinning strategy.

= #E (V1 , V \ (D ∪ V1 )) + #E (V1 , D ), we have z > L22 z z>z

=

#E (V1 , V \ (D ∪ V1 )) + #E (V1 , D ) #V1

which completes the proof.

,



For this theorem, we have two remarks. First, the right part of inequality (10) indicates that the lower bound of stabilizability is not only affected by the maximal vertex degree in the pinned vertex set, but also depends on the distance between the unpinned vertex set and the pinned vertex set, which suggests that the degree-based selective pinning strategy may not be the optimal one. Therefore, a pinning strategy yields good stabilizability should not only consider the degrees of selected vertices, but also take into account the shortest path traffic between vertices. Second, Please note that the right part of inequality (10) still holds in the digraph case if the shortest directed path distance is considered. In the following, we present an example of an artificial bigraph to verify Theorem 6 and show that the selective pinning strategy based on the degree is not always the optimal one. First, we construct a graph as follows. Consider a tree T with K layers. The lth layer has K !/(K − l + 1)! vertices, and each vertex has K − l + 1 children, 1 ≤ l ≤ K − 1. The K th layer is composed PK of K ! leaves. Then, the whole tree has l=1 K !/l! vertices. There is a ring R having the nearest neighbor connections and 2(K !)/(K − l0 + 1)! vertices. Thus, we construct a bigraph G that the ring R is connected to the tree T together via an arbitrary leaf in T . Pl We consider two pinning strategies with l0=1 K !/(K − l + 1)! vertices pinned for some 2 ≤ l0 ≤ K −1. One is pinning the vertices with the largest degrees, and all pinned vertices are located in the tree T if with multiple choices; the other is a modification from the first one with pinning K !/(K − l0 + 1)! vertices in the ring R by a cuttingly distribution, so that for each pair of neighbored vertices in the ring, only one vertex is pinned, which implies that half of vertices in the ring are pinned and the distance from the unpinned vertex set to the pinned vertex set in this ring is exactly 1. We denote the stabilizability of the bigraph with these two different pinning strategies by stab1 and stab2 , respectively. Comparing both in this artificial example, we have the following result. Proposition 7. If (K −Kl !+1)! > (2K + 1)K −l0 +1 , then stab2 > stab1 0 holds. In fact, by Theorem 6, we have stab1 < stab2 ≥

1 2 1

  2 K+

1 2

(2K )−(K −l0 +1)

(K −l0 +1)! 2K !

and

−(K −l0 +1)

(2K + 1)−(K −l0 +1) . 2 These lead to Proposition 7. Also, we can see that when l0 = K − 1, the condition changes to K ! > 2(2K + 1)2 , which holds for K ≥ 6. Fixing K = 6 and picking l0 from 2 to 5, Table 1 gives the computational values of stab1,2 and their estimations from Theorem 6, which verify our analytical estimations. So, we can see that in this case the degree-based pinning strategy is not always the optimal one. From Theorem 6, one may take the distance from the >

5. Conclusions In conclusion, to pin a complex directed network to a homogeneous trajectory of the uncoupled system, the underlying network topology and selection of pinned vertices play the key role on the network stabilizability. In this paper, we have arrived at several new criteria to guarantee the global stability of pinning control a directed network, and analytically estimated the bounds of the network stabilizability which are affected by not only the degrees of the pinned vertices but also the distance between the pinned vertex set and unpinned vertex set. Therefore, a selective pinning strategy yielding better stabilizability of a complex network should take into account not only the vertex degree, but also the shortest path traffics between the pinned and unpinned vertices, which is unavoidably determined by categories of connectivity patterns in real-world complex networks, and deserves more efforts for further extensive study in near future. Acknowledgements We are grateful to Dr. C. W. Wu and Dr. M. Porfiri for mailing their published or unpublished papers not limited to those listed in the references. References Barabási, A. L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 509–512. Berman, A. (2003). Completely positive matrices. New Jersey: World Scientific Publishing. Boyd, S., et al. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM. Brualdi, R. A., & Ryser, H. J. (1991). Combinatorial matrix theory. Cambridge: Cambridge University Press. Chen, T. P., Liu, X. W., & Lu, W. L. (2007). Pinning complex networks by a single controller. IEEE Transactions on Circuits and Systems -I: Regular Papers, 54(6), 1317–1326. Dashkovskiy, S., Ruffer, B. S., & Wirth, F. R. (2007). An ISS small gain theorem for general networks. Mathematics Control, Signals and Systems, 19, 93–122. Erdös, P., & Rényi, R. (1959). On random graphs. Publications Mathematics, 6, 290–297. Godsil, C., & Royle, G. (2001). Algebraic graph theory. New York: Springer-Verlag. Hu, G., & Qu, Z. L. (1994). Controlling spatiotemporal chaos in coupled map lattice systems. Physical Review Letters, 72, 68–71. Jiang, Z. P., & Wang, Y. (2008). A generalization of the nonlinear small-gain theorem for large-scale complex systems. In Proceeding of the 7th World congress on intelligent control and automation (pp. 1188–1193). Jungnickel, D. (2005). Graphs, networks, and algorithms. Berlin Heidelberg: SpringerVerlag. Li, X., Wang, X. F., & Chen, G. (2004). Pinning a complex dynamical network to its equilibrium. IEEE Transactions on Circuits and Systems-I: Regular Paper, 51(10), 2074–2087. Porfiri, M., & di Bernardo, M. (2008). Criteria for global pinning-controllability of complex networks. Automatica, 44, 3100–3106. Roy, R., Murphy, T. W., Jr, Maier, T. D., & Gills, Z. (1992). Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system. Physical Review Letters, 68, 1259–1262. Siljak, D. D. (1991). Decentralized control of complex systems. New York: Academic Press. Sorrentino, F., di Bernardo, M., Garofalo, F., & Chen, G. (2007). Controllability of complex networks via pinning. Physical Review E, 75, 046103.

W. Lu et al. / Automatica 46 (2010) 116–121

121

Wang, X. F., & Chen, G. (2002). Pinning control of scale-free dynamical networks. Physics A, 310, 521–531. Wang, W., & Slotine, J. J. E. (2006). A theoretical study of different leader roles in networks. IEEE Transactions on Automatic Control, 51(7), 1156–1161. Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442. Wu, C. W. (2005). On bounds of extremal eigenvalues of irreducible and m-reducible matrices. Linear Algebra and Its Applications, 402, 29–45. Xiang, J., & Chen, G. (2007). On the V-stability of complex dynamical networks. Automatica, 43, 1049–1057. Yu, W., Chen, G., & Lü, J. (2009). On pinning synchronization of complex dynamical networks. Automatica, 45, 429–435.

Xiang Li received the B.S. degree and the Ph.D. both in Control Theory and Control Engineering from Nankai University, China, in 1997 and 2002, respectively. He was a postdoc at City University of Hong Kong, Hong Kong from 2002–2004 and an Alexander von Humboldt Fellow at Int. University Bremen, Germany, from 2005–2006. He was an Associate Professor at Shanghai Jiao Tong University from 2004–2007. Since 2008, he is a Professor at Fudan University. He is an IEEE senior member, and received the IEEE Circuits and Systems Society Guillemin–Cauer Award in 2005, and the Natural Science Award of Shanghai (1st class) in 2008. His main research fields include control theories and applications of complex networks and complex systems.

Wenlian Lu received the B.S. degree in Mathematics and the Ph.D. degree in Applied Mathematics from Fudan University, Shanghai, China, in 2000 and 2005, respectively. He was a postdoc at Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany from 2005–2007 and currently an Associate Professor with the School of Mathematical Sciences, Fudan University, Shanghai, China. His research interests include neural networks, nonlinear dynamical systems, and complex systems.

Zhihai Rong was born in Harbin, China, in 1978. He received the B.S. degree in Automation and the M.S. degree in Detection Technology and Automation Equipment from Harbin Institute of Technology, Harbin, China, in 2001 and 2003, respectively, and the Ph.D. degree in control theory and applications from the Shanghai Jiao Tong University, Shanghai, in 2008. Currently, he is a lecturer with the Department of Automation, Donghua University, Shanghai, China. His research interests include model and analysis of networked evolutionary game dynamics.