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Synchronization of stochastic multi-weighted complex networks with Lévy noise based on graph theory
Journal Pre-proof Synchronization of stochastic multi-weighted complex networks with Lévy noise based on graph theory Chunmei Zhang, Yinghui Yang
PII: DOI: Reference:
S0378-4371(19)31951-X https://doi.org/10.1016/j.physa.2019.123496 PHYSA 123496
To appear in:
Physica A
Received date : 25 March 2019 Revised date : 18 September 2019 Please cite this article as: C. Zhang and Y. Yang, Synchronization of stochastic multi-weighted complex networks with Lévy noise based on graph theory, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123496. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Journal Pre-proof
Synchronization of stochastic multi-weighted complex networks with L´evy noise based on graph theory Chunmei Zhang and Yinghui Yang∗
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School of Mathematics, Southwest Jiaotong University, Chengdu 610031, PR China
Abstract
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This paper addresses the synchronization problem of stochastic multi-weighted complex networks with L´evy noise. Based on the drive-response concept and graph theory, global Lyapunov function of the error network between drive-response networks is obtained by the weighted summation of Lyapunov functions of vertex systems. According to the stochastic analysis and state feedback control technique, the rigorous synchronization analysis of drive-response networks in the p-th moment and probability sense is presented. The obtained synchronization criteria are closely related with multi-weights and the intensity of L´evy noise. Finally, some numerical simulations are provided to illustrate the effectiveness of the theoretical results.
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Keywords: Stochastic complex networks; Synchronization; Multi-weights
1. Introduction
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There are many literature results about the dynamics of complex networks or coupled systems on networks. Many of them were about the synchronization of complex networks [1, 2, 3, 4, 5, 6, 7, 8], including pinning synchronization [9, 10], cluster synchronization [11, 12], projective synchronization [13, 14] and so on. Therein, many research results were about the synchronization of complex networks with single weight [15]. However, in some real networks such as public traffic network and social network, there exist multiple kinds of weights among nodes [16, 17, 18]. For instance, in the social network, treating each person as an individual node, people can contact with each other by Phone, Email, MSN et al. Each kind of communication tool has different weights (i.e. efficiency). So, it is better to use coupled systems with multi-weights to model the social network. There are some results in the literature to study the dynamics of coupled systems with multi-weights [19, 20, 21, 22]. Therein, [21, 22] investigated the synchronization of multi-weighted complex dynamical networks. In the real applications, systems are always disturbed by some environmental noise such as white noise. Researchers often use stochastic differential equations driven by Brownion motion to describe the systems perturbed by white noise [23, 24, 25]. It should be noted that complex networks in science and engineering may suffer sudden environmental shocks such as earthquakes and epidemics [26]. However, systems with this sudden shock can not be described by stochastic systems with white noise. Scholars always call this kind of very rare yet extreme shock as L´evy noise or L´evy jump [27, 28, 29] and use stochastic systems driven by compensated Possion random measure to explain this kind of noise [30, 31, 32]. In [33, 34], they investigated the Corresponding author. Tel.: +86 028 66367140. Email address: [email protected], [email protected] (Chunmei Zhang and Yinghui Yang )
Preprint submitted to Physica A: Statistical Mechanics and its Applications
November 9, 2019
Journal Pre-proof
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synchronization of stochastic neural networks with L´evy noise. While in [16, 19, 21, 22], scholars studied the synchronization of determined complex networks with multi-weights. However, few papers were about the dynamics of stochastic complex networks with multi-weights [35]. Motivated by the above discussions, our paper aims to study the synchronization of stochastic multi-weighted complex networks with L´evy noise. Lyapunov method is the classical tool to study the dynamics of systems. But it is an open problem that how to construct the Lyapunov function for any given system. There is no systematic method for this construction, especially for some complex systems. In our paper, both multi-weights and L´evy noise are considered into the complex networks, which makes the construction of Lapunov function much more difficult. Fortunately, Michael Y. Li and his collaborators developed a systematic method to construct the global Lyapunov function for determined complex networks [36]. That is, the Lyapunov function of complex networks can be obtained by the weighted summation of Lyapunov functions of vertex systems. This method is always called as graphtheoretic method since some results in algebraic graph theory are used. Following this pioneering work, many researchers used the graph-theoretic method to investigate other kinds of complex networks, such as stochastic complex networks [37, 38, 39], delayed complex networks [40, 41], impulsive complex networks [42], multiweighted complex networks [35] and so on. Naturally, can we use the graph-theoretic method to study the synchronization of stochastic multi-weighted complex networks with L´evy noise? This is the main task of this paper. The contributions of this paper are threefold. Firstly, the model is general and novel. Both multi-weights and L´evy noise are considered in the model. In detail, by the method of network split and drive-response concept, the drive system is designed as the determined complex networks with multi-weights. While in the response system, L´evy noise and the feedback controller are added in the complex networks with multi-weights. The response system is modelled as stochastic coupled systems driven by Brownian motion and compensated Possion random measure. The error network is then obtained by subtracting the drive network from the response network. Secondly, by using the graph-theoretic method, the global Lyapunov function V(e, t) of the error network is ∑ obtained by the weighted summation of Lyapunov functions of vertex systems, i.e., V(e, t) = lk=1 ck Vk (ek , t), ck is the cofactor of the k-th diagonal element of Laplacian matrix L for (G, (bkh )l×l ). bkh = δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + s · · · + δ s hˇ s akh , k, h = 1, 2, · · · , l. Thirdly, according to stochastic analysis and state feedback control technique, the synchronization criteria of drive-response networks in the p-th moment and probability sense are presented, respectively. A numerical example is also given to illustrate the effectiveness of the theoretical results. 2. Model description and preliminaries
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First, we give some basic notations and preliminaries. Let (Ω, F , F, P) be a complete probability space with a filtration F = {Ft }t≥0 satisfying the usual conditions (i.e. the filtration is right continuous and F0 contains all P-null sets). W(·) is a one dimensional standard Brownian motion defined on the given probability space and E(·) is the mathematical expectation with respect to P. Let | · | be the Euclidean norm for vectors or the trace norm for matrices. AT denotes the transpose of a matrix A. Denote Rn+ = {e ∈ Rn : ei > 0, i = 1, 2, · · · , n}. ˜ Π(dt, dσ) = Π(dt, dσ) − λ(dσ)dt is a compensated Poisson random measure, where Π(dt, dσ) is a Poisson counting measure on [0, +∞] × Y, which is independent of Brownian motion W(t). λ is the intensity measure and Y is a measurable subset of Rn+ such that λ(Y) < ∞. Throughout this paper, we simply write the left limit of x(t) as x(t) for brevity. For many basic concepts of graph theory such as unicyclic graph and spanning subgraph, we omit them to highlight the synchronization results of this paper. For the detail, please see [36]. We just list an important lemma about graph theory. 2
Journal Pre-proof Lemma 1. ([36]) Let l ≥ 2. Assume that (G, B), B = (bkh )l×l , is a weighted digraph. Let Q be the set of all spanning unicyclic graphs Q of (G, B), CQ be the cycle of Q, ω(Q) be the weight of Q and ck be the cofactor of the k-th diagonal element of the Laplacian matrix L for (G, B). Then for arbitrary functions Fkh (ek , eh ), 1 ≤ k, h ≤ l, it holds that l ∑ ∑ ∑ ck bkh Fkh (ek , eh ) = ω(Q) Frs (er , e s ). k,h=1
Q∈Q
(s,r)∈E(CQ )
In addition, if (G, B) is strongly connected, then ck > 0 for k = 1, 2, · · · , l.
x˙k (t) = fk (xk (t), t) + δ1
l ∑
a1kh H1 xh (t)
+ δ2
h=1
l ∑
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In the following, we show the model description. A digraph G with l (l ≥ 2) vertices and s kinds of weights is given. Since multiple weights have different natures, the weights with same nature and l vertices compose a sub-network. Hence, by the theory of network split, the complex network with s kinds of weights can be split into s sub-networks with single weight. That is, the multi-weighted digraph G can be splitted into s digraphs with single weight (G, A(i) ), A(i) = (aikh )l×l , i = 1, 2, · · · , s. The drive system can be described as a coupled system of nonlinear equations on G as follows: a2kh H2 xh (t)
h=1
+ · · · + δs
l ∑ h=1
s akh H s xh (t), k = 1, 2, · · · , l,
(1)
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where xk ∈ Rn is the state variable of vertex k in the drive system, fk : Rn × R1+ → Rn is continuously differentiable for each k and represents the dynamics of vertex k. δi (i = 1, 2, · · · , s) is the coupling strength of the i-th digraph with single weight (G, A(i) ). aikh is the i-th weight from h-th node to k-th node if it exists, and 0 otherwise. The inner coupling Hi = diag(hi11 , hi22 , · · · , hinn ) (i = 1, 2, · · · , s) is a positive diagonal n × n matrix. As mentioned in Introduction, the real systems are always perturbed by environmental noise. In this paper, we mainly consider the effect of L´evy noise on the synchronization of multi-weighted complex networks. Based on the concept of drive-response synchronization and state feedback control technique, the response system can be designed as stochastic coupled systems driven by Brownian motion and compensated Possion random measure as follows: l l l ∑ ∑ ∑ 1 2 s dyk (t) = fk (yk (t), t) + δ1 akh H1 yh (t) + δ2 akh H2 yh (t) + · · · + δ s akh H s yh (t) + uk (t) dt h=1 h=1 h=1 (2) ∫ ˜ + gk (yk (t) − xk (t), t)dW(t) + γk (yk (t) − xk (t), t, σ)Π(dt, dσ), k = 1, 2, · · · , l, Y
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where yk ∈ Rn is the state variable of vertex k in the response system. gk : Rn ×R1+ → Rn and γk ∈ Rn ×R1+ ×Y → Rn are the diffusion coefficient and jumping coefficient on the k-th vertex system, respectively. uk ∈ Rn is the state feedback controller to be designed. Assume that the coefficients of drive-response networks (1) and (2) satisfy some usual conditions (such as local Lipschitz condition and linear growth condition). Therefore, by the existence and uniqueness theorem in [43], there exists a unique solution to (1) and (2) for any given initial data x(t0 ) = x0 and y(t0 ) = y0 in Rnl . Denote them as x(t) and y(t), respectively. Here, x(t) = (x1T (t), x2T (t), · · · , xlT (t))T and y(t) = (yT1 (t), yT2 (t), · · · , yTl (t))T . Define ek (t) = yk (t) − xk (t) and denote e(t) = (eT1 (t), eT2 (t), · · · , eTl (t))T as the synchronization error. Considering the linear state feedback controller uk (t) as uk (t) = νk ( fk (yk (t), t) − fk (xk (t), t)), k = 1, 2, · · · , l, 3
(3)
Journal Pre-proof in which νk , −1 is the gain constant to be designed. With the control law (3), we can get the error network: l l l ∑ ∑ ∑ s dek (t) = (1 + νk )( fk (yk (t), t) − fk (xk (t), t)) + δ1 a1kh H1 eh (t) + δ2 a2kh H2 eh (t) + · · · + δ s akh H s eh (t) dt + gk (ek (t), t)dW(t) +
∫
Y
h=1
h=1
h=1
˜ γk (ek (t), t, σ)Π(dt, dσ), k = 1, 2, · · · , l.
(4)
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Through this paper, assume that gk (0, t) = γk (0, t, σ) = 0, k = 1, 2, · · · , l for any t ≥ t0 and σ ∈ Y. So, e(t) = 0 is the trivial solution to the error network (4). For any initial data e(t0 ) = e0 ∈ Rnl , there exists a unique solution of the error network (4), denote it as e(t; t0 , e0 ). We always write it as e(t) for brevity. Denote C 2,1 (Rn × R1+ ; R1+ ) for the family of all nonnegative function V(e, t) on Rn × R1+ that are continuously twice differentiable in e and once in t. If Vk ∈ C 2,1 (Rn ×R1+ ; R1+ ), define an operator LVk from Rn ×R1+ to R1+ associated with the k-th equation of error network (4) by l l ∑ ∑ ∂Vk (ek , t) ∂Vk (ek , t) LVk (ek , t) = + a1kh H1 eh (t) + δ2 a2kh H2 eh (t) + · · · (1 + νk )( fk (yk (t), t) − fk (xk (t), t)) + δ1 ∂t ∂ek h=1 h=1 2 l ∑ 1 ∂V (ek , t) s +δ s akh H s eh (t) + trace gTk (ek (t), t) k(i) ( j) gk (ek (t), t) 2 ∂ek ∂ek n×n h=1 ] ∫ [ ∂Vk (ek , t) γk (ek (t), t, σ) λ(dσ), + Vk (ek + γk (ek (t), t, σ), t) − Vk (ek , t) − ∂ek Y (5) where
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∂Vk (ek , t) ∂Vk (ek , t) ∂Vk (ek , t) ∂Vk (ek , t) = , , ··· , , ∂ek ∂e(1) ∂e(2) ∂e(n) k k k
∂2 Vk (ek ,t) (1) 2 · · · 2 (∂ek ) ∂Vk (ek , t) .. = ... . (i) ( j) ∂ek ∂ek n×n ∂2 Vk (ek ,t) ··· ∂e(n) ∂e(1) k
k
∂2 Vk (ek ,t) ∂e(1) ∂e(n) k k
∂2 Vk (ek ,t) (n) .. .
(∂ek )2
.
n×n
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Finally, we present some definitions about p-th moment exponential synchronization and stochastic asymptotical synchronization of drive-response networks [23, 38]. Definition 1. The drive-response networks (1) and (2) are said to be p-th moment exponentially synchronized if the p-th moment Lyapunov exponent lim sup t→∞
1 log(E|e(t; t0 , e0 )| p ) < 0, t0 ∈ R1+ , e0 ∈ Rnl . t
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When p = 2, they are said to be exponentially synchronized in mean square. Definition 2. The drive-response networks (1) and (2) are said to be stochastically synchronized if for every ε ∈ (0, 1), c > 0, and t0 ≥ 0, there exists a δ = δ(ε, c, t0 ) > 0 such that P(|e(t; t0 , e0 )| < c, ∀t ≥ t0 ) ≥ 1 − ε,
for any e0 ∈ Rnl satisfying |e0 | < δ. 4
Journal Pre-proof Definition 3. The drive-response networks (1) and (2) are said to be stochastically asymptotically synchronized if it is stochastically synchronized and, furthermore, for every ε ∈ (0, 1) and t0 ≥ 0, there exists a δ = δ(ε, t0 ) > 0 such that P(lim e(t; t0 , e0 ) = 0) ≥ 1 − ε, t→∞
nl
for any e0 ∈ R satisfying |e0 | < δ. 3. Synchronization analysis for drive-response networks (1) and (2)
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In this section, based on the drive-response concept and state feedback control technique, some synchronization results will be provided. Both the exponential synchronization criterion and stochastic synchronization criterion of drive-response networks (1) and (2) will be presented by stochastic analysis and graph theory [35, 36]. We first give some assumptions. (A1). For some p > 0, there exist positive constants αk , βk , ηk and functions Vk , Fkh such that αk |ek | p ≤ Vk (ek , t) ≤ βk |ek | p , p
h=1
bkh Fkh (ek , eh , t), ek , eh ∈ Rn , t ≥ t0 .
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LVk (ek , t) ≤ −ηk |ek | +
l ∑
(A2). Along each directed cycle C of weighted digraph (G, (bkh )l×l ), it holds that ∑ Fkh (ek , eh , t) ≤ 0, ek , eh ∈ Rn , t ≥ t0 . (h,k)∈E(C)
Theorem 1. If conditions (A1)-(A2) hold and (G, (bkh )l×l ) is strongly connected, then drive-response networks (1) and (2) are p-th moment exponentially synchronized under the control law (3). Proof. Define
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V(e, t) =
l ∑
ck Vk (ek , t),
k=1
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in which ck is the cofactor the k-th diagonal element of the Laplacian matrix L for (G, (bkh )l×l ). From condition (A1), it follows easily that there are positive constants α¯ and β¯ such that ¯ p , t ≥ t0 . α|e| ¯ p ≤ V(e, t) ≤ β|e|
From condition (A2) and Lemma 1, we can get that
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LV(e, t) = ≤
l ∑
k=1 l ∑ k=1
=−
ck LVk (ek , t) l ∑ p ck −ηk |ek | + bkh Fkh (ek , eh , t)
l ∑ k=1
ck ηk |ek | p + 5
h=1 l ∑
k,h=1
ck bkh Fkh (ek , eh , t)
Journal Pre-proof =− ≤−
l ∑ k=1
l ∑ k=1
∑
ck ηk |ek | p +
Q∈Q
ω(Q)
∑
Fkh (ek , eh , t)
(h,k)∈E(CQ )
ck ηk |ek | p p
≤ − min {ck ηk }l(1− 2 )∧0 |e| p , −η¯ |e| p , 1≤k≤l
where a ∧ b , min{a, b}. Hence, with the similar discussion as [38], it follows that β¯ − βη¯¯ (t−t0 ) p |e0 | . e α¯
of
E|e(t; t0 , e0 )| p ≤ Clearly,
1 η¯ log(E|e(t; t0 , e0 )| p ) = − < 0. t β¯ t→∞ Therefore, drive-response networks (1) and (2) are p-th moment exponentially synchronized by Definition 1.
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lim sup
Pr e-
Remark 1. It is always known that Lyapunov method is an efficient tool to study the dynamics of systems. In this paper, multi-weights and stochastic perturbations are considered into complex networks. Hence, it is difficult to construct the Lyapunov function for the error network (4) directly. Fortunately, Theorem 1 shows an useful method to solve this difficulty. That is, we only need to construct the Lyapunov function Vk (ek , t) for each vertex system e˙ k (t) = (1 + νk )( fk (yk (t), t) − fk (xk (t), t)) (k = 1, 2, · · · , l) and then use the weighted summation ∑l k=1 ck Vk (ek , t) to get the Lyapunov function V of the error network (4) indirectly. Here, the weight ck is the key point. ck is the cofactor of the k-th diagonal element of Laplacian matrix L for (G, (bkh )l×l ). What is the relationship between the new weighted matrix (bkh )l×l and s kinds of weighted matrices (aikh )l×l , i = 1, 2, · · · , s? We tend to find the answer in the next theorem.
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In Theorem 1, some conditions for vertex Lyapunov functions Vk , k = 1, 2, · · · , l have been given. There is a natural question: can we find some functions to satisfy those conditions in Theorem 1? In the following, we will answer this question. Some assumptions for the coefficients of drive-response networks (1) and (2) are given as follows.
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(B1). There exist positive constants αk and βk such that (y − x)T ( fk (y, t) − fk (x, t)) ≤ −αk |y − x|2 , |gk (x, t)|2 ≤ βk |x|2 , x, y ∈ Rn , k = 1, 2, · · · , l.
(B2). There exists a positive constant ηk such that
(B3). It holds that
|γk (e, t, σ)| ≤ ηk |e|, e ∈ Rn , σ ∈ Y, t ≥ t0 , k = 1, 2, · · · , l.
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(1 + νk )αk >
for some p ≥ 2, where η¯ k =
l ∑
1 bkh + (p − 1)βk + η¯ k λ(Y), k = 1, 2, · · · , l, 2 h=1
1 s , hˇ i = max hij j , i = 1, 2, · · · , s. ((1 + ηk ) p − 1 − pηk ), bkh = δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh 1≤ j≤n p 6
(6)
Journal Pre-proof Theorem 2. Assume that assumptions (B1)−(B3) hold. If (G, (bkh )l×l ) is strongly connected, then drive-response networks (1) and (2) are p-th moment exponentially synchronized under the control law (3). Proof. Define Vk = |ek | p , k = 1, 2, · · · , l. By using the definition of operator (5), it follows that LVk (ek (t), t)
l l ∑ ∑ p−2 T 1 a2kh H2 eh (t) =p|ek | ek (t) (1 + νk )( fk (yk (t), t) − fk (xk (t), t)) + δ1 akh H1 eh (t) + δ2 h=1
h=1
( ) 1 s akh H s eh (t) + trace gTk (ek (t), t)(p|ek (t)| p−2 In + p(p − 2)|ek (t)| p−4 ek (t)eTk (t))gk (ek (t), t) + · · · + δs 2 h=1 ∫ ( ) |ek (t) + γk (ek (t), t, σ)| p − |ek (t)| p − p|ek (t)| p−2 eTk (t)γk (ek (t), t, σ) λ(dσ) + Y
ˇ1
p
≤ − p(1 + νk )αk |ek (t)| + pδ1 h |ek (t)| + pδ2 h |ek (t)|
p−2
a2kh
h=1
1 + p(p − 1)βk |ek | p + 2
∫
(
Y
l ∑
a1kh
h=1
(
|ek (t)|2 |eh (t)|2 + 2 2
)
) ( ) l 2 2 ∑ |ek (t)| |eh (t)|2 |e (t)| |e (t)| k h s p−2 s + akh + + · · · + pδ s hˇ |ek (t)| 2 2 2 2 h=1 2
((1 + ηk ) p − 1 − pηk ) |ek | p λ(dσ)
Pr e-
ˇ2
l ∑
p−2
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l ∑
l
∑( ) p s ≤ − p(1 + νk )αk |ek (t)| p + |ek (t)| p δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh 2 h=1 l
l
+
∑( ∑( ) ) p−2 s s δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh + |eh (t)| p |ek (t)| p 2 h=1 h=1
, − δk |ek (t)| p +
l ∑ h=1
urn
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[ ] 1 + p(p − 1)βk |ek (t)| p + (1 + ηk ) p − 1 − pηk |ek (t)| p λ(Y) 2 l l ∑ ∑ 1 = − p(1 + νk )αk |ek (t)| p + p bkh |ek (t)| p + bkh (|eh (t)| p − |ek (t)| p ) + p(p − 1)βk |ek (t)| p + pη¯ k λ(Y)|ek (t)| p 2 h=1 h=1 l l ∑ ∑ 1 = − p (1 + νk )αk − bkh − (p − 1)βk − η¯ k λ(Y) |ek (t)| p + bkh (|eh (t)| p − |ek (t)| p ) 2 h=1 h=1 bkh (|eh (t)| p − |ek (t)| p ),
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[ ] ∑ where δk = p (1 + νk )αk − lh=1 bkh − (p − 1)βk /2 − η¯ k λ(Y) > 0 and In is the identity matrix of size n. Clearly, along each directed cycle C of the weighted digraph (G, (bkh )l×l ), it holds that ∑ (|eh (t)| p − |ek (t)| p ) = 0. (h,k)∈E(C)
Hence, Theorem 1 tells us that drive-response networks (1) and (2) can be p-th moment exponentially synchronized. 7
Journal Pre-proof Remark 2. In Theorem 2, two questions mentioned above have been answered. First, there really exist functions Vk = |ek | p , k = 1, 2, · · · , l to satisfy the conditions of Theorem 1. Second, we have given the relationship between the new weighted matrix (bkh )l×l and s kinds of weighted matrices (aikh )l×l , i = 1, 2, · · · , s. That is, s bkh = δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh , k, h = 1, 2, · · · , l. In other words, we do not need the topological property i of s sub-networks (G, (akh )l×l ), (i = 1, 2, · · · , s) to ensure the exponential synchronization of drive-response networks (1) and (2). It is enough to know the topological property of the new digraph (G, (bkh )l×l ).
t→∞
1 log(E|e(t; t0 , e0 )|2 ) = δ < 0, t0 ∈ R1+ , e0 ∈ Rnl . t
In other words, for the given δ, we can find ε > 0 such that
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lim sup
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In fact, drive-response networks (1) and (2) are mean square exponential synchronization under the conditions of Theorem 2. That is, there exists δ < 0 such that
E|e(t; t0 , e0 )|2 ≤ εeδ(t−t0 ) |e0 |2 , t ≥ t0 , e0 ∈ Rnl . For some 0 < p < 2, by Jensen’s inequality, it follows that
Hence, for 0 < p < 2, it holds that lim sup t→∞
Pr e-
[ ]p [ ]p p pδ E|e(t; t0 , e0 )| p ≤ E|e(t; t0 , e0 )|2 2 ≤ εeδ(t−t0 ) |e0 |2 2 = ε 2 e 2 (t−t0 ) |e0 | p . 1 p log(E|e(t; t0 , e0 )| p ) = δ < 0, t0 ∈ R1+ , e0 ∈ Rnl . t 2
This result is written as the following corollary.
Corollary 1. If all the assumptions of Theorem 2 hold, then drive-response networks (1) and (2) can be p-th moment exponentially synchronized for 0 < p < 2.
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Remark 3. In this paper, L´evy noise has been considered. From condition (B3), we know that drive-response networks (1) and (2) can maintain the exponential synchronization if the perturbation intensity of noise and the control gain constant can satisfy the inequality (6). In the real applications, for a given drive network, its dynamics has been given. Based on the concept of drive-response, we can adjust the control gain constant νk , k = 1, 2, · · · , l and the intensity of noise in the response network to make sure the synchronization of driveresponse networks.
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Clearly, if there is no jumping part considered in the response network, then we can get a new response network with white noise as follows: l l l ∑ ∑ ∑ s dyk (t) = fk (yk (t), t) + δ1 a1kh H1 yh (t) + δ2 a2kh H2 yh (t) + · · · + δ s akh H s yh (t) + uk (t) dt (7) h=1 h=1 h=1 + gk (yk (t) − xk (t), t)dW(t), k = 1, 2, · · · , l.
In this case, we can get the exponential synchronization of drive-response networks with white noise, which is stated in the following corollary. 8
Journal Pre-proof Corollary 2. Assume that the condition (B1) and (1 + νk )αk >
l ∑
1 bkh + (p − 1)βk , 2 h=1
(8)
hold for some p > 0. If (G, (bkh )l×l ) is strongly connected, then drive-response networks with white noise (1) and (7) are p-th moment exponentially synchronized under the control law (3).
l ∑
a1kh H1 yh (t) + δ2
l ∑ h=1
h=1
a2kh H2 yh (t) + · · · + δ s
l ∑ h=1
s akh H s yh (t) + uk (t), k = 1, 2, · · · , l. (9)
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y˙ k (t) = fk (yk (t), t) + δ1
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Moreover, if there is no noise considered in the response network, then we can get a new response network without noise as follows:
In this case, we can get the exponential synchronization of determined drive-response networks, which is stated as the following corollary. Corollary 3. Assume that the conditions
Pr e-
(y − x)T ( fk (y, t) − fk (x, t)) ≤ −αk |y − x|2 , x, y ∈ Rn , and
(1 + νk )αk >
l ∑
bkh ,
(10)
h=1
hold for some p > 0. If (G, (bkh )l×l ) is strongly connected, then the determined drive-response networks (1) and (9) are p-th moment exponentially synchronized under the control law (3).
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Obviously, Corollaries 2 and 3 can be easily obtained from Theorem 2 and Corollary 1. We omit the proof for brevity. In fact, condition (A1) is used to get the p-th moment exponential synchronization of drive-response networks (1) and (2). It can show that the decay of p-th moment of the solution to the error network (4) between driveresponse networks (1) and (2) is exponential. In the following, some weaker conditions than (A1) are provided, which are applicable efficiently to show the stochastic synchronization of drive-response networks [32]. (C1). For any k ∈ {1, 2, · · · , l}, there exists a continuous nondecreasing function µk (e) on [0, +∞) such that µk (0) = 0, µk (e) > 0 if e > 0. Assume that there exist functions Vk (ek , t), Fkh (ek , eh , t) and a matrix B = (bkh )l×l such that (G, B) is strongly connected and
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Vk (0, t) = 0, Vk (ek , t) ≥ µk (|ek |),
LVk (ek , t) ≤
l ∑
bkh Fkh (ek , eh , t).
h=1
(C2). For any k ∈ {1, 2, · · · , l}, there exist continuous nondecreasing functions µk (e) and ψk (e) on [0, +∞) such that µk (0) = ψk (0) = 0, µk (e) > 0 and ψk (e) > 0 if e > 0. Additionally, there exist functions Vk (ek , t) and Fkh (ek , eh , t), a matrix B = (bkh )l×l and a positive constant dk such that (G, B) is strongly connected and 9
Journal Pre-proof
µk (|ek |) ≤ Vk (ek , t) ≤ ψk (|ek |), l ∑ LVk (ek , t) ≤ −dk Vk (ek , t) + bkh Fkh (ek , eh , t). h=1
Lemma 2. ([32]) Assume that conditions (A2) and (C1) hold. Then drive-response networks (1) and (2) are stochastic synchronized under the control law (3) . Furthermore, if conditions (A2) and (C2) hold, then driveresponse networks (1) and (2) are stochastic asymptotically synchronized under the control law (3) .
Proof. Define
From the proof of Theorem 2, we can see that
p
LVk (ek (t), t) ≤ − δk |ek (t)| +
p ro
Vk = |ek | p , k = 1, 2, · · · , l. l ∑
bkh (|eh (t)| p − |ek (t)|P )
h=1 l ∑
bkh (|eh (t)| p − |ek (t)|P )
Pr e-
= − δk Vk (ek , t) +
of
Theorem 3. Under the same conditions with Theorem 2, drive-response networks (1) and (2) are stochastic asymptotically synchronized under the control law (3) .
h=1
Along each directed cycle C of the weighted digraph (G, (bkh )l×l ), it holds that ∑ (|eh (t)| p − |ek (t)| p ) = 0. (h,k)∈E(C)
Till here, conditions (A2) and (C2) have been verified, then drive-response networks (1) and (2) are stochastic asymptotically synchronized under the control law (3) .
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Remark 4. Based on the results in graph theory and state feedback control technique, the criteria about two kinds of synchronization (i.e. p-th moment exponential synchronization and stochastic synchronization) are shown. The obtained synchronization results are shown in two different kinds of forms, one is shown in the form of Lyapunov functions and topological properties of weighted graph (i.e. Theorem 1 and Lemma 2), the other is shown in the form of the coefficients of drive-response networks (i.e. Theorems 2 and 3).
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Remark 5. We compare the results of this paper with [32, 35, 44] and other relevant papers from the following aspects:
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(1) The model is much more general than other relevant papers. L´evy noise has been incorporated into the multi-weighted complex networks in this paper, while references [35, 44] did not consider this kind of important noise. Although L´evy noise has been included in the complex networks in the reference [32], it mainly aimed at the stability of complex networks with single weight. Stochastic systems with L´evy noise can better model the very rare yet extreme sudden event, such as earthquakes and epidemics. Deterministic systems or stochastic systems with white noise or Markovian jumps cannot explain the sudden shocks. In fact, if there is no jumping part considered in the response network, then we can get a new response network with white noise. In this case, we can get the exponential synchronization of drive-response networks with white noise, which is stated in Corollary 2. Corollary 2 is similar with Theorem 2 in [44]. So, from this point, our paper is the generalization of [44]. 10
Journal Pre-proof (2) The research content is different. References [32, 35] mainly studied the stability of stochastic complex networks, while this paper mainly focuses on the synchronization of stochastic complex networks. 4. Numerical example In this section, a numerical example with multi-weights and L´evy noise is provided to illustrate the theoretical results. First, we consider the drive network as follows: 6 ∑
a1kh H1 xh (t) + δ2
h=1
6 ∑
a2kh H2 xh (t) + δ3
h=1
6 ∑
a3kh H3 xh (t), k = 1, 2, · · · , 6,
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x˙k (t) = fk (xk (t), t) + δ1
h=1
(11)
p ro
where xk ∈ R3 , fk (xk , t) = (−2.5xk1 + sin xk2 , sin xk1 − 2xk2 , xk1 − 1.5xk3 )T , δ1 = 0.5, δ2 = 0.4, δ3 = 0.3 and the inner coupling networks are as follows: 0.3 0 0.5 0 0.1 0 0 0 0 H1 = 0 0.4 0 , H2 = 0 0.1 0 , H3 = 0 0.05 0 . 0 0 0.5 0 0 0.2 0 0 0.1
h=1
Pr e-
By the concept of drive-response, the response network is designed as 6 6 6 ∑ ∑ ∑ 3 2 1 akh H3 yh (t) + uk (t) dt akh H2 yh (t) + δ3 dyk (t) = fk (yk (t), t) + δ1 akh H1 yh (t) + δ2 +gk (yk (t) − xk (t), t)dW(t) +
∫
h=1
Y
(12)
h=1
˜ γk (yk (t) − xk (t), t, σ)Π(dt, dσ), k = 1, 2, · · · , 6,
where γk (xk , t, σ) = ηk (σ)(xk1 , xk2 , xk3 )T , η1 (σ) = η2 (σ) = η3 (σ) = 0.4, η4 (σ) = η5 (σ) = η6 (σ) = 0.6 and gk (xk , t) = (0.5 sin xk1 , 0.5 sin xk2 , 0.5 sin xk3 )T , k = 1, 2, 3,
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gk (xk , t) = (0.4 arctan xk1 , 0.4 arctan xk2 , 0.4 arctan xk3 )T , k = 4, 5, 6. Let λ(Y) = 1 and p = 2. Three kinds of weighted matrices A(i) = (aikh )6×6 , (i = 1, 2, 3) are given as follows. 0 0 1 2 0 0
2 0 0 0 1 0
0 0 1 0 0 2
0 0 0 0 0 1
0 0 0 1 0 0
0 2 (2) , A = 0 0 0 0
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A(1)
0 1 0 = 0 0 0
0 0 1 1 0 0
1 0 0 0 1 1
0 0 1 0 0 2
0 0 1 0 0 1
0 0 0 1 0 0
0 0 (3) , A = 0 0 0 0
0 0 1 2 0 0
1 0 0 0 2 1
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In the following, we show the simulations under the following feedback control law: uk (t) = νk ( fk (yk (t), t) − fk (xk (t), t)), k = 1, 2, · · · , 6,
0 0 2 0 0 2
0 0 0 0 0 0
0 0 0 1 0 0
. (13)
where νk , −1 is the gain constant to be designed later. Now, we verify the conditions of Theorem 2. By the direct calculation, we can get that αk = 1 (k = 1, 2, · · · , 6), βk = 0.5 (k = 1, 2, 3), βk = 0.4 (k = 4, 5, 6), ηk = 11
Journal Pre-proof 0.4 (k = 1, 2, 3), ηk = 0.6 (k = 4, 5, 6) and a new weighted matrix B as 0 0.73 0 0 0 0 0.65 0 0 0 0 0 0 0.48 0 0.51 0.2 0 B = (bkh )6×6 = . 0 0 0.28 0 0.76 0 0 0 0.51 0 0 0 0 0 0.23 0.96 0.45 0 l ∑
1 bkh + (p − 1)βk + η¯ k λ(Y), 2 h=1
p ro
(1 + νk )αk >
of
Clearly, (G, (bkh )l×l ) is strongly connected. Choosing the feedback gain constants as νk = 1 (k = 1, 2, 3), νk = 2 (k = 4, 5, 6), then we can check that
Pr e-
holds for any k (k = 1, 2, · · · , 6). Hence, drive-response networks (11) and (12) are exponentially synchronized under the control law (13) in mean square. Taking the initial values as x11 (0) = 0.1, x12 (0) = −0.5, x13 (0) = −0.1, x21 (0) = 0.2, x22 (0) = 0.2, x23 (0) = −0.5, x31 (0) = 0.6, x32 (0) = −0.5, x33 (0) = 1, x41 (0) = 0.8, x42 (0) = 0.2, x43 (0) = −1, x51 (0) = −0.3, x52 (0) = 0.4, x53 (0) = −0.6, x61 (0) = 0.5, x62 (0) = −0.1, x63 (0) = 1, y11 (0) = 0.6, y12 (0) = 0, y13 (0) = 0.2, y21 (0) = −0.1, y22 (0) = −0.2, y23 (0) = −0.3, y31 (0) = 0.2, y32 (0) = −0.4, y33 (0) = 0.7, y41 (0) = 0.3, y42 (0) = 0, y43 (0) = −0.6, y51 (0) = 0.1, y52 (0) = 0.8, y53 (0) = −0.2, y61 (0) = 0.3, y62 (0) = −0.2, y63 (0) = 0.8. The simulation result about the second moment of synchronization error for drive-response networks (11) and (12) is presented in Figure 1. We also get from Theorem 3 that drive-response networks (11) and (12) are stochastic asymptotically synchronized under the control law (13). The simulations for the sample path of synchronization error for drive-response networks and the sample path of drive-response networks are presented in Figures 2 and 3, respectively. For better showing the numerical simulations in Figure 3, the second component of each state variable in the drive-response networks (11) and (12) has been added 0.1, and the third component of each state variable minus 0.1.
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5. Conclusion
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In this paper, synchronization for a novel class of stochastic multi-weighted complex networks with L´evy noise has been investigated. By the concept of drive-response, the error network has been first obtained. Then the global Lyapunov function for the error network has been systematically constructed by combining algebraic graph theory and stochastic analysis. By the feedback control method, sufficient criteria for p-th moment exponential synchronization and stochastic synchronization of drive-response networks have been obtained. The criteria are closely related to multi-weights and the intensity of L´evy noise. The main synchronization results are formulated by Lyapunov functions and coefficients of drive-response networks, respectively. We also give some numerical simulations to verify the effectiveness of the new theoretical results. In the real applications, systems may suffer from another kind of noise, i.e. telegraph noise. If we consider both telegraph noise and L´evy noise into the multi-weighted complex networks, it is much more difficult to study its dynamics. This challenging task is our future work. Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 11601445,11801470, 31700347) and the Fundamental Research Funds for the Central Universities (No. 2682018CX60). 12
Journal Pre-proof
0.25
0.2
0.2
2
2
E|e11(t)|
0.2
E|e12(t)|2
0.15
2
E|e21(t)|
E|e22(t)|2
0.15
E|e13(t)|2
E|e31(t)|
E|e23(t)|2
0.1
E|e32(t)|2
0.15
E|e33(t)|2
0.1
0.1 0.05
0
0
1
2 3 Time t
4
0
5
0.25
0.05
0
1
2 3 Time t
0.2
E|e51(t)|
2
1
2 3 Time t
p ro
E|e52(t)|
2
2
E|e43(t)|
4
5
0.1
2
E|e62(t)|
2
0.03
E|e53(t)|
0.1
2
E|e61(t)|
0.04
2
0.15
E|e42(t)|
0.15
0
2
E|e41(t)|
0.2
0
5
0.05
2
E|e63(t)|
0.02
0.05
0.05 0
1
2 3 Time t
4
0
5
0.01
0
1
2 3 Time t
4
Pr e-
0
4
of
0.05
5
0
0
1
2 3 Time t
4
Figure 1: The second moment of synchronization error for drive-response networks (11) and (12).
0.4
0.4
e11(t) e12(t)
0.2
e13(t)
e23(t)
−0.2
2 Time t
4
−0.4
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0
0.4
0.2
0
2 Time t
0
2 Time t
4
0.4
e42(t)
e52(t)
4
0.2
e33(t)
0
−0.4
0
2 Time t
4
e53(t)
e61(t)
0
−0.2
−0.2
0
2 Time t
4
e62(t)
0.2
0
−0.4
e32(t)
0.2
0.4 e51(t)
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−0.2
e31(t)
−0.2
e41(t) e43(t)
0
−0.4
0.2
0
−0.2
−0.4
e22(t)
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0
0.4
e21(t)
−0.4
e63(t)
0
2 Time t
Figure 2: Sample path of synchronization error for drive-response networks (11) and (12).
13
4
5
Journal Pre-proof
x11(t) 0.5
x21(t) 0.5
y11(t)
y21(t)
x12(t)+0.1
x22(t)+0.1
y12(t)+0.1
0
x
y22(t)+0.1
0
(t)−0.1
x
13
y23(t)−0.1
−0.5
−0.5 0
2
4
6
8
10
0
2
4
Time t
x31(t)
y32(t)+0.1
0
x33(t)−0.1
−0.5
y33(t)−0.1 4
10
6
8
y41(t)
0
10
−1
0
2
4
x42(t)+0.1 y42(t)+0.1 x43(t)−0.1 y43(t)−0.1 6
8
10
Time t
1
p ro
Time t
of
x32(t)+0.1
2
8
x41(t)
0.5
y31(t)
0
6 Time t
0.5
−0.5
(t)−0.1
23
y13(t)−0.1
1
x
(t)
51
y51(t)
0.5
x
(t)+0.1
y
(t)+0.1
x
(t)−0.1
52
y53(t)−0.1
−0.5 0
2
4
6
8
10
−0.5
0
2
(t)+0.1
y
(t)+0.1
x
(t)−0.1
62 63
y63(t)−0.1 4
6
8
10
Time t
Pr e-
Time t
x
62
0
53
(t)
61
y61(t)
0.5
52
0
x
Figure 3: Sample path of drive-response networks (11) and (12).
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Journal Pre-proof Figure captions
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Figure 1 is the second moment of synchronization error for drive-response networks (11) and (12). Figure 2 is the sample path of synchronization error for drive-response networks (11) and (12). Figure 3 is the sample path of drive-response networks (11) and (12).
18
PII: DOI: Reference:
S0378-4371(19)31951-X https://doi.org/10.1016/j.physa.2019.123496 PHYSA 123496
To appear in:
Physica A
Received date : 25 March 2019 Revised date : 18 September 2019 Please cite this article as: C. Zhang and Y. Yang, Synchronization of stochastic multi-weighted complex networks with Lévy noise based on graph theory, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123496. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Journal Pre-proof
Synchronization of stochastic multi-weighted complex networks with L´evy noise based on graph theory Chunmei Zhang and Yinghui Yang∗
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School of Mathematics, Southwest Jiaotong University, Chengdu 610031, PR China
Abstract
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This paper addresses the synchronization problem of stochastic multi-weighted complex networks with L´evy noise. Based on the drive-response concept and graph theory, global Lyapunov function of the error network between drive-response networks is obtained by the weighted summation of Lyapunov functions of vertex systems. According to the stochastic analysis and state feedback control technique, the rigorous synchronization analysis of drive-response networks in the p-th moment and probability sense is presented. The obtained synchronization criteria are closely related with multi-weights and the intensity of L´evy noise. Finally, some numerical simulations are provided to illustrate the effectiveness of the theoretical results.
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Keywords: Stochastic complex networks; Synchronization; Multi-weights
1. Introduction
∗
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There are many literature results about the dynamics of complex networks or coupled systems on networks. Many of them were about the synchronization of complex networks [1, 2, 3, 4, 5, 6, 7, 8], including pinning synchronization [9, 10], cluster synchronization [11, 12], projective synchronization [13, 14] and so on. Therein, many research results were about the synchronization of complex networks with single weight [15]. However, in some real networks such as public traffic network and social network, there exist multiple kinds of weights among nodes [16, 17, 18]. For instance, in the social network, treating each person as an individual node, people can contact with each other by Phone, Email, MSN et al. Each kind of communication tool has different weights (i.e. efficiency). So, it is better to use coupled systems with multi-weights to model the social network. There are some results in the literature to study the dynamics of coupled systems with multi-weights [19, 20, 21, 22]. Therein, [21, 22] investigated the synchronization of multi-weighted complex dynamical networks. In the real applications, systems are always disturbed by some environmental noise such as white noise. Researchers often use stochastic differential equations driven by Brownion motion to describe the systems perturbed by white noise [23, 24, 25]. It should be noted that complex networks in science and engineering may suffer sudden environmental shocks such as earthquakes and epidemics [26]. However, systems with this sudden shock can not be described by stochastic systems with white noise. Scholars always call this kind of very rare yet extreme shock as L´evy noise or L´evy jump [27, 28, 29] and use stochastic systems driven by compensated Possion random measure to explain this kind of noise [30, 31, 32]. In [33, 34], they investigated the Corresponding author. Tel.: +86 028 66367140. Email address: [email protected], [email protected] (Chunmei Zhang and Yinghui Yang )
Preprint submitted to Physica A: Statistical Mechanics and its Applications
November 9, 2019
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synchronization of stochastic neural networks with L´evy noise. While in [16, 19, 21, 22], scholars studied the synchronization of determined complex networks with multi-weights. However, few papers were about the dynamics of stochastic complex networks with multi-weights [35]. Motivated by the above discussions, our paper aims to study the synchronization of stochastic multi-weighted complex networks with L´evy noise. Lyapunov method is the classical tool to study the dynamics of systems. But it is an open problem that how to construct the Lyapunov function for any given system. There is no systematic method for this construction, especially for some complex systems. In our paper, both multi-weights and L´evy noise are considered into the complex networks, which makes the construction of Lapunov function much more difficult. Fortunately, Michael Y. Li and his collaborators developed a systematic method to construct the global Lyapunov function for determined complex networks [36]. That is, the Lyapunov function of complex networks can be obtained by the weighted summation of Lyapunov functions of vertex systems. This method is always called as graphtheoretic method since some results in algebraic graph theory are used. Following this pioneering work, many researchers used the graph-theoretic method to investigate other kinds of complex networks, such as stochastic complex networks [37, 38, 39], delayed complex networks [40, 41], impulsive complex networks [42], multiweighted complex networks [35] and so on. Naturally, can we use the graph-theoretic method to study the synchronization of stochastic multi-weighted complex networks with L´evy noise? This is the main task of this paper. The contributions of this paper are threefold. Firstly, the model is general and novel. Both multi-weights and L´evy noise are considered in the model. In detail, by the method of network split and drive-response concept, the drive system is designed as the determined complex networks with multi-weights. While in the response system, L´evy noise and the feedback controller are added in the complex networks with multi-weights. The response system is modelled as stochastic coupled systems driven by Brownian motion and compensated Possion random measure. The error network is then obtained by subtracting the drive network from the response network. Secondly, by using the graph-theoretic method, the global Lyapunov function V(e, t) of the error network is ∑ obtained by the weighted summation of Lyapunov functions of vertex systems, i.e., V(e, t) = lk=1 ck Vk (ek , t), ck is the cofactor of the k-th diagonal element of Laplacian matrix L for (G, (bkh )l×l ). bkh = δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + s · · · + δ s hˇ s akh , k, h = 1, 2, · · · , l. Thirdly, according to stochastic analysis and state feedback control technique, the synchronization criteria of drive-response networks in the p-th moment and probability sense are presented, respectively. A numerical example is also given to illustrate the effectiveness of the theoretical results. 2. Model description and preliminaries
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First, we give some basic notations and preliminaries. Let (Ω, F , F, P) be a complete probability space with a filtration F = {Ft }t≥0 satisfying the usual conditions (i.e. the filtration is right continuous and F0 contains all P-null sets). W(·) is a one dimensional standard Brownian motion defined on the given probability space and E(·) is the mathematical expectation with respect to P. Let | · | be the Euclidean norm for vectors or the trace norm for matrices. AT denotes the transpose of a matrix A. Denote Rn+ = {e ∈ Rn : ei > 0, i = 1, 2, · · · , n}. ˜ Π(dt, dσ) = Π(dt, dσ) − λ(dσ)dt is a compensated Poisson random measure, where Π(dt, dσ) is a Poisson counting measure on [0, +∞] × Y, which is independent of Brownian motion W(t). λ is the intensity measure and Y is a measurable subset of Rn+ such that λ(Y) < ∞. Throughout this paper, we simply write the left limit of x(t) as x(t) for brevity. For many basic concepts of graph theory such as unicyclic graph and spanning subgraph, we omit them to highlight the synchronization results of this paper. For the detail, please see [36]. We just list an important lemma about graph theory. 2
Journal Pre-proof Lemma 1. ([36]) Let l ≥ 2. Assume that (G, B), B = (bkh )l×l , is a weighted digraph. Let Q be the set of all spanning unicyclic graphs Q of (G, B), CQ be the cycle of Q, ω(Q) be the weight of Q and ck be the cofactor of the k-th diagonal element of the Laplacian matrix L for (G, B). Then for arbitrary functions Fkh (ek , eh ), 1 ≤ k, h ≤ l, it holds that l ∑ ∑ ∑ ck bkh Fkh (ek , eh ) = ω(Q) Frs (er , e s ). k,h=1
Q∈Q
(s,r)∈E(CQ )
In addition, if (G, B) is strongly connected, then ck > 0 for k = 1, 2, · · · , l.
x˙k (t) = fk (xk (t), t) + δ1
l ∑
a1kh H1 xh (t)
+ δ2
h=1
l ∑
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In the following, we show the model description. A digraph G with l (l ≥ 2) vertices and s kinds of weights is given. Since multiple weights have different natures, the weights with same nature and l vertices compose a sub-network. Hence, by the theory of network split, the complex network with s kinds of weights can be split into s sub-networks with single weight. That is, the multi-weighted digraph G can be splitted into s digraphs with single weight (G, A(i) ), A(i) = (aikh )l×l , i = 1, 2, · · · , s. The drive system can be described as a coupled system of nonlinear equations on G as follows: a2kh H2 xh (t)
h=1
+ · · · + δs
l ∑ h=1
s akh H s xh (t), k = 1, 2, · · · , l,
(1)
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Pr e-
where xk ∈ Rn is the state variable of vertex k in the drive system, fk : Rn × R1+ → Rn is continuously differentiable for each k and represents the dynamics of vertex k. δi (i = 1, 2, · · · , s) is the coupling strength of the i-th digraph with single weight (G, A(i) ). aikh is the i-th weight from h-th node to k-th node if it exists, and 0 otherwise. The inner coupling Hi = diag(hi11 , hi22 , · · · , hinn ) (i = 1, 2, · · · , s) is a positive diagonal n × n matrix. As mentioned in Introduction, the real systems are always perturbed by environmental noise. In this paper, we mainly consider the effect of L´evy noise on the synchronization of multi-weighted complex networks. Based on the concept of drive-response synchronization and state feedback control technique, the response system can be designed as stochastic coupled systems driven by Brownian motion and compensated Possion random measure as follows: l l l ∑ ∑ ∑ 1 2 s dyk (t) = fk (yk (t), t) + δ1 akh H1 yh (t) + δ2 akh H2 yh (t) + · · · + δ s akh H s yh (t) + uk (t) dt h=1 h=1 h=1 (2) ∫ ˜ + gk (yk (t) − xk (t), t)dW(t) + γk (yk (t) − xk (t), t, σ)Π(dt, dσ), k = 1, 2, · · · , l, Y
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where yk ∈ Rn is the state variable of vertex k in the response system. gk : Rn ×R1+ → Rn and γk ∈ Rn ×R1+ ×Y → Rn are the diffusion coefficient and jumping coefficient on the k-th vertex system, respectively. uk ∈ Rn is the state feedback controller to be designed. Assume that the coefficients of drive-response networks (1) and (2) satisfy some usual conditions (such as local Lipschitz condition and linear growth condition). Therefore, by the existence and uniqueness theorem in [43], there exists a unique solution to (1) and (2) for any given initial data x(t0 ) = x0 and y(t0 ) = y0 in Rnl . Denote them as x(t) and y(t), respectively. Here, x(t) = (x1T (t), x2T (t), · · · , xlT (t))T and y(t) = (yT1 (t), yT2 (t), · · · , yTl (t))T . Define ek (t) = yk (t) − xk (t) and denote e(t) = (eT1 (t), eT2 (t), · · · , eTl (t))T as the synchronization error. Considering the linear state feedback controller uk (t) as uk (t) = νk ( fk (yk (t), t) − fk (xk (t), t)), k = 1, 2, · · · , l, 3
(3)
Journal Pre-proof in which νk , −1 is the gain constant to be designed. With the control law (3), we can get the error network: l l l ∑ ∑ ∑ s dek (t) = (1 + νk )( fk (yk (t), t) − fk (xk (t), t)) + δ1 a1kh H1 eh (t) + δ2 a2kh H2 eh (t) + · · · + δ s akh H s eh (t) dt + gk (ek (t), t)dW(t) +
∫
Y
h=1
h=1
h=1
˜ γk (ek (t), t, σ)Π(dt, dσ), k = 1, 2, · · · , l.
(4)
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Through this paper, assume that gk (0, t) = γk (0, t, σ) = 0, k = 1, 2, · · · , l for any t ≥ t0 and σ ∈ Y. So, e(t) = 0 is the trivial solution to the error network (4). For any initial data e(t0 ) = e0 ∈ Rnl , there exists a unique solution of the error network (4), denote it as e(t; t0 , e0 ). We always write it as e(t) for brevity. Denote C 2,1 (Rn × R1+ ; R1+ ) for the family of all nonnegative function V(e, t) on Rn × R1+ that are continuously twice differentiable in e and once in t. If Vk ∈ C 2,1 (Rn ×R1+ ; R1+ ), define an operator LVk from Rn ×R1+ to R1+ associated with the k-th equation of error network (4) by l l ∑ ∑ ∂Vk (ek , t) ∂Vk (ek , t) LVk (ek , t) = + a1kh H1 eh (t) + δ2 a2kh H2 eh (t) + · · · (1 + νk )( fk (yk (t), t) − fk (xk (t), t)) + δ1 ∂t ∂ek h=1 h=1 2 l ∑ 1 ∂V (ek , t) s +δ s akh H s eh (t) + trace gTk (ek (t), t) k(i) ( j) gk (ek (t), t) 2 ∂ek ∂ek n×n h=1 ] ∫ [ ∂Vk (ek , t) γk (ek (t), t, σ) λ(dσ), + Vk (ek + γk (ek (t), t, σ), t) − Vk (ek , t) − ∂ek Y (5) where
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∂Vk (ek , t) ∂Vk (ek , t) ∂Vk (ek , t) ∂Vk (ek , t) = , , ··· , , ∂ek ∂e(1) ∂e(2) ∂e(n) k k k
∂2 Vk (ek ,t) (1) 2 · · · 2 (∂ek ) ∂Vk (ek , t) .. = ... . (i) ( j) ∂ek ∂ek n×n ∂2 Vk (ek ,t) ··· ∂e(n) ∂e(1) k
k
∂2 Vk (ek ,t) ∂e(1) ∂e(n) k k
∂2 Vk (ek ,t) (n) .. .
(∂ek )2
.
n×n
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Finally, we present some definitions about p-th moment exponential synchronization and stochastic asymptotical synchronization of drive-response networks [23, 38]. Definition 1. The drive-response networks (1) and (2) are said to be p-th moment exponentially synchronized if the p-th moment Lyapunov exponent lim sup t→∞
1 log(E|e(t; t0 , e0 )| p ) < 0, t0 ∈ R1+ , e0 ∈ Rnl . t
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When p = 2, they are said to be exponentially synchronized in mean square. Definition 2. The drive-response networks (1) and (2) are said to be stochastically synchronized if for every ε ∈ (0, 1), c > 0, and t0 ≥ 0, there exists a δ = δ(ε, c, t0 ) > 0 such that P(|e(t; t0 , e0 )| < c, ∀t ≥ t0 ) ≥ 1 − ε,
for any e0 ∈ Rnl satisfying |e0 | < δ. 4
Journal Pre-proof Definition 3. The drive-response networks (1) and (2) are said to be stochastically asymptotically synchronized if it is stochastically synchronized and, furthermore, for every ε ∈ (0, 1) and t0 ≥ 0, there exists a δ = δ(ε, t0 ) > 0 such that P(lim e(t; t0 , e0 ) = 0) ≥ 1 − ε, t→∞
nl
for any e0 ∈ R satisfying |e0 | < δ. 3. Synchronization analysis for drive-response networks (1) and (2)
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In this section, based on the drive-response concept and state feedback control technique, some synchronization results will be provided. Both the exponential synchronization criterion and stochastic synchronization criterion of drive-response networks (1) and (2) will be presented by stochastic analysis and graph theory [35, 36]. We first give some assumptions. (A1). For some p > 0, there exist positive constants αk , βk , ηk and functions Vk , Fkh such that αk |ek | p ≤ Vk (ek , t) ≤ βk |ek | p , p
h=1
bkh Fkh (ek , eh , t), ek , eh ∈ Rn , t ≥ t0 .
Pr e-
LVk (ek , t) ≤ −ηk |ek | +
l ∑
(A2). Along each directed cycle C of weighted digraph (G, (bkh )l×l ), it holds that ∑ Fkh (ek , eh , t) ≤ 0, ek , eh ∈ Rn , t ≥ t0 . (h,k)∈E(C)
Theorem 1. If conditions (A1)-(A2) hold and (G, (bkh )l×l ) is strongly connected, then drive-response networks (1) and (2) are p-th moment exponentially synchronized under the control law (3). Proof. Define
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V(e, t) =
l ∑
ck Vk (ek , t),
k=1
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in which ck is the cofactor the k-th diagonal element of the Laplacian matrix L for (G, (bkh )l×l ). From condition (A1), it follows easily that there are positive constants α¯ and β¯ such that ¯ p , t ≥ t0 . α|e| ¯ p ≤ V(e, t) ≤ β|e|
From condition (A2) and Lemma 1, we can get that
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LV(e, t) = ≤
l ∑
k=1 l ∑ k=1
=−
ck LVk (ek , t) l ∑ p ck −ηk |ek | + bkh Fkh (ek , eh , t)
l ∑ k=1
ck ηk |ek | p + 5
h=1 l ∑
k,h=1
ck bkh Fkh (ek , eh , t)
Journal Pre-proof =− ≤−
l ∑ k=1
l ∑ k=1
∑
ck ηk |ek | p +
Q∈Q
ω(Q)
∑
Fkh (ek , eh , t)
(h,k)∈E(CQ )
ck ηk |ek | p p
≤ − min {ck ηk }l(1− 2 )∧0 |e| p , −η¯ |e| p , 1≤k≤l
where a ∧ b , min{a, b}. Hence, with the similar discussion as [38], it follows that β¯ − βη¯¯ (t−t0 ) p |e0 | . e α¯
of
E|e(t; t0 , e0 )| p ≤ Clearly,
1 η¯ log(E|e(t; t0 , e0 )| p ) = − < 0. t β¯ t→∞ Therefore, drive-response networks (1) and (2) are p-th moment exponentially synchronized by Definition 1.
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Remark 1. It is always known that Lyapunov method is an efficient tool to study the dynamics of systems. In this paper, multi-weights and stochastic perturbations are considered into complex networks. Hence, it is difficult to construct the Lyapunov function for the error network (4) directly. Fortunately, Theorem 1 shows an useful method to solve this difficulty. That is, we only need to construct the Lyapunov function Vk (ek , t) for each vertex system e˙ k (t) = (1 + νk )( fk (yk (t), t) − fk (xk (t), t)) (k = 1, 2, · · · , l) and then use the weighted summation ∑l k=1 ck Vk (ek , t) to get the Lyapunov function V of the error network (4) indirectly. Here, the weight ck is the key point. ck is the cofactor of the k-th diagonal element of Laplacian matrix L for (G, (bkh )l×l ). What is the relationship between the new weighted matrix (bkh )l×l and s kinds of weighted matrices (aikh )l×l , i = 1, 2, · · · , s? We tend to find the answer in the next theorem.
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In Theorem 1, some conditions for vertex Lyapunov functions Vk , k = 1, 2, · · · , l have been given. There is a natural question: can we find some functions to satisfy those conditions in Theorem 1? In the following, we will answer this question. Some assumptions for the coefficients of drive-response networks (1) and (2) are given as follows.
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(B1). There exist positive constants αk and βk such that (y − x)T ( fk (y, t) − fk (x, t)) ≤ −αk |y − x|2 , |gk (x, t)|2 ≤ βk |x|2 , x, y ∈ Rn , k = 1, 2, · · · , l.
(B2). There exists a positive constant ηk such that
(B3). It holds that
|γk (e, t, σ)| ≤ ηk |e|, e ∈ Rn , σ ∈ Y, t ≥ t0 , k = 1, 2, · · · , l.
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(1 + νk )αk >
for some p ≥ 2, where η¯ k =
l ∑
1 bkh + (p − 1)βk + η¯ k λ(Y), k = 1, 2, · · · , l, 2 h=1
1 s , hˇ i = max hij j , i = 1, 2, · · · , s. ((1 + ηk ) p − 1 − pηk ), bkh = δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh 1≤ j≤n p 6
(6)
Journal Pre-proof Theorem 2. Assume that assumptions (B1)−(B3) hold. If (G, (bkh )l×l ) is strongly connected, then drive-response networks (1) and (2) are p-th moment exponentially synchronized under the control law (3). Proof. Define Vk = |ek | p , k = 1, 2, · · · , l. By using the definition of operator (5), it follows that LVk (ek (t), t)
l l ∑ ∑ p−2 T 1 a2kh H2 eh (t) =p|ek | ek (t) (1 + νk )( fk (yk (t), t) − fk (xk (t), t)) + δ1 akh H1 eh (t) + δ2 h=1
h=1
( ) 1 s akh H s eh (t) + trace gTk (ek (t), t)(p|ek (t)| p−2 In + p(p − 2)|ek (t)| p−4 ek (t)eTk (t))gk (ek (t), t) + · · · + δs 2 h=1 ∫ ( ) |ek (t) + γk (ek (t), t, σ)| p − |ek (t)| p − p|ek (t)| p−2 eTk (t)γk (ek (t), t, σ) λ(dσ) + Y
ˇ1
p
≤ − p(1 + νk )αk |ek (t)| + pδ1 h |ek (t)| + pδ2 h |ek (t)|
p−2
a2kh
h=1
1 + p(p − 1)βk |ek | p + 2
∫
(
Y
l ∑
a1kh
h=1
(
|ek (t)|2 |eh (t)|2 + 2 2
)
) ( ) l 2 2 ∑ |ek (t)| |eh (t)|2 |e (t)| |e (t)| k h s p−2 s + akh + + · · · + pδ s hˇ |ek (t)| 2 2 2 2 h=1 2
((1 + ηk ) p − 1 − pηk ) |ek | p λ(dσ)
Pr e-
ˇ2
l ∑
p−2
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l ∑
l
∑( ) p s ≤ − p(1 + νk )αk |ek (t)| p + |ek (t)| p δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh 2 h=1 l
l
+
∑( ∑( ) ) p−2 s s δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh + |eh (t)| p |ek (t)| p 2 h=1 h=1
, − δk |ek (t)| p +
l ∑ h=1
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[ ] 1 + p(p − 1)βk |ek (t)| p + (1 + ηk ) p − 1 − pηk |ek (t)| p λ(Y) 2 l l ∑ ∑ 1 = − p(1 + νk )αk |ek (t)| p + p bkh |ek (t)| p + bkh (|eh (t)| p − |ek (t)| p ) + p(p − 1)βk |ek (t)| p + pη¯ k λ(Y)|ek (t)| p 2 h=1 h=1 l l ∑ ∑ 1 = − p (1 + νk )αk − bkh − (p − 1)βk − η¯ k λ(Y) |ek (t)| p + bkh (|eh (t)| p − |ek (t)| p ) 2 h=1 h=1 bkh (|eh (t)| p − |ek (t)| p ),
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[ ] ∑ where δk = p (1 + νk )αk − lh=1 bkh − (p − 1)βk /2 − η¯ k λ(Y) > 0 and In is the identity matrix of size n. Clearly, along each directed cycle C of the weighted digraph (G, (bkh )l×l ), it holds that ∑ (|eh (t)| p − |ek (t)| p ) = 0. (h,k)∈E(C)
Hence, Theorem 1 tells us that drive-response networks (1) and (2) can be p-th moment exponentially synchronized. 7
Journal Pre-proof Remark 2. In Theorem 2, two questions mentioned above have been answered. First, there really exist functions Vk = |ek | p , k = 1, 2, · · · , l to satisfy the conditions of Theorem 1. Second, we have given the relationship between the new weighted matrix (bkh )l×l and s kinds of weighted matrices (aikh )l×l , i = 1, 2, · · · , s. That is, s bkh = δ1 hˇ 1 a1kh + δ2 hˇ 2 a2kh + · · · + δ s hˇ s akh , k, h = 1, 2, · · · , l. In other words, we do not need the topological property i of s sub-networks (G, (akh )l×l ), (i = 1, 2, · · · , s) to ensure the exponential synchronization of drive-response networks (1) and (2). It is enough to know the topological property of the new digraph (G, (bkh )l×l ).
t→∞
1 log(E|e(t; t0 , e0 )|2 ) = δ < 0, t0 ∈ R1+ , e0 ∈ Rnl . t
In other words, for the given δ, we can find ε > 0 such that
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In fact, drive-response networks (1) and (2) are mean square exponential synchronization under the conditions of Theorem 2. That is, there exists δ < 0 such that
E|e(t; t0 , e0 )|2 ≤ εeδ(t−t0 ) |e0 |2 , t ≥ t0 , e0 ∈ Rnl . For some 0 < p < 2, by Jensen’s inequality, it follows that
Hence, for 0 < p < 2, it holds that lim sup t→∞
Pr e-
[ ]p [ ]p p pδ E|e(t; t0 , e0 )| p ≤ E|e(t; t0 , e0 )|2 2 ≤ εeδ(t−t0 ) |e0 |2 2 = ε 2 e 2 (t−t0 ) |e0 | p . 1 p log(E|e(t; t0 , e0 )| p ) = δ < 0, t0 ∈ R1+ , e0 ∈ Rnl . t 2
This result is written as the following corollary.
Corollary 1. If all the assumptions of Theorem 2 hold, then drive-response networks (1) and (2) can be p-th moment exponentially synchronized for 0 < p < 2.
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Remark 3. In this paper, L´evy noise has been considered. From condition (B3), we know that drive-response networks (1) and (2) can maintain the exponential synchronization if the perturbation intensity of noise and the control gain constant can satisfy the inequality (6). In the real applications, for a given drive network, its dynamics has been given. Based on the concept of drive-response, we can adjust the control gain constant νk , k = 1, 2, · · · , l and the intensity of noise in the response network to make sure the synchronization of driveresponse networks.
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Clearly, if there is no jumping part considered in the response network, then we can get a new response network with white noise as follows: l l l ∑ ∑ ∑ s dyk (t) = fk (yk (t), t) + δ1 a1kh H1 yh (t) + δ2 a2kh H2 yh (t) + · · · + δ s akh H s yh (t) + uk (t) dt (7) h=1 h=1 h=1 + gk (yk (t) − xk (t), t)dW(t), k = 1, 2, · · · , l.
In this case, we can get the exponential synchronization of drive-response networks with white noise, which is stated in the following corollary. 8
Journal Pre-proof Corollary 2. Assume that the condition (B1) and (1 + νk )αk >
l ∑
1 bkh + (p − 1)βk , 2 h=1
(8)
hold for some p > 0. If (G, (bkh )l×l ) is strongly connected, then drive-response networks with white noise (1) and (7) are p-th moment exponentially synchronized under the control law (3).
l ∑
a1kh H1 yh (t) + δ2
l ∑ h=1
h=1
a2kh H2 yh (t) + · · · + δ s
l ∑ h=1
s akh H s yh (t) + uk (t), k = 1, 2, · · · , l. (9)
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y˙ k (t) = fk (yk (t), t) + δ1
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Moreover, if there is no noise considered in the response network, then we can get a new response network without noise as follows:
In this case, we can get the exponential synchronization of determined drive-response networks, which is stated as the following corollary. Corollary 3. Assume that the conditions
Pr e-
(y − x)T ( fk (y, t) − fk (x, t)) ≤ −αk |y − x|2 , x, y ∈ Rn , and
(1 + νk )αk >
l ∑
bkh ,
(10)
h=1
hold for some p > 0. If (G, (bkh )l×l ) is strongly connected, then the determined drive-response networks (1) and (9) are p-th moment exponentially synchronized under the control law (3).
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Obviously, Corollaries 2 and 3 can be easily obtained from Theorem 2 and Corollary 1. We omit the proof for brevity. In fact, condition (A1) is used to get the p-th moment exponential synchronization of drive-response networks (1) and (2). It can show that the decay of p-th moment of the solution to the error network (4) between driveresponse networks (1) and (2) is exponential. In the following, some weaker conditions than (A1) are provided, which are applicable efficiently to show the stochastic synchronization of drive-response networks [32]. (C1). For any k ∈ {1, 2, · · · , l}, there exists a continuous nondecreasing function µk (e) on [0, +∞) such that µk (0) = 0, µk (e) > 0 if e > 0. Assume that there exist functions Vk (ek , t), Fkh (ek , eh , t) and a matrix B = (bkh )l×l such that (G, B) is strongly connected and
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Vk (0, t) = 0, Vk (ek , t) ≥ µk (|ek |),
LVk (ek , t) ≤
l ∑
bkh Fkh (ek , eh , t).
h=1
(C2). For any k ∈ {1, 2, · · · , l}, there exist continuous nondecreasing functions µk (e) and ψk (e) on [0, +∞) such that µk (0) = ψk (0) = 0, µk (e) > 0 and ψk (e) > 0 if e > 0. Additionally, there exist functions Vk (ek , t) and Fkh (ek , eh , t), a matrix B = (bkh )l×l and a positive constant dk such that (G, B) is strongly connected and 9
Journal Pre-proof
µk (|ek |) ≤ Vk (ek , t) ≤ ψk (|ek |), l ∑ LVk (ek , t) ≤ −dk Vk (ek , t) + bkh Fkh (ek , eh , t). h=1
Lemma 2. ([32]) Assume that conditions (A2) and (C1) hold. Then drive-response networks (1) and (2) are stochastic synchronized under the control law (3) . Furthermore, if conditions (A2) and (C2) hold, then driveresponse networks (1) and (2) are stochastic asymptotically synchronized under the control law (3) .
Proof. Define
From the proof of Theorem 2, we can see that
p
LVk (ek (t), t) ≤ − δk |ek (t)| +
p ro
Vk = |ek | p , k = 1, 2, · · · , l. l ∑
bkh (|eh (t)| p − |ek (t)|P )
h=1 l ∑
bkh (|eh (t)| p − |ek (t)|P )
Pr e-
= − δk Vk (ek , t) +
of
Theorem 3. Under the same conditions with Theorem 2, drive-response networks (1) and (2) are stochastic asymptotically synchronized under the control law (3) .
h=1
Along each directed cycle C of the weighted digraph (G, (bkh )l×l ), it holds that ∑ (|eh (t)| p − |ek (t)| p ) = 0. (h,k)∈E(C)
Till here, conditions (A2) and (C2) have been verified, then drive-response networks (1) and (2) are stochastic asymptotically synchronized under the control law (3) .
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Remark 4. Based on the results in graph theory and state feedback control technique, the criteria about two kinds of synchronization (i.e. p-th moment exponential synchronization and stochastic synchronization) are shown. The obtained synchronization results are shown in two different kinds of forms, one is shown in the form of Lyapunov functions and topological properties of weighted graph (i.e. Theorem 1 and Lemma 2), the other is shown in the form of the coefficients of drive-response networks (i.e. Theorems 2 and 3).
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Remark 5. We compare the results of this paper with [32, 35, 44] and other relevant papers from the following aspects:
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(1) The model is much more general than other relevant papers. L´evy noise has been incorporated into the multi-weighted complex networks in this paper, while references [35, 44] did not consider this kind of important noise. Although L´evy noise has been included in the complex networks in the reference [32], it mainly aimed at the stability of complex networks with single weight. Stochastic systems with L´evy noise can better model the very rare yet extreme sudden event, such as earthquakes and epidemics. Deterministic systems or stochastic systems with white noise or Markovian jumps cannot explain the sudden shocks. In fact, if there is no jumping part considered in the response network, then we can get a new response network with white noise. In this case, we can get the exponential synchronization of drive-response networks with white noise, which is stated in Corollary 2. Corollary 2 is similar with Theorem 2 in [44]. So, from this point, our paper is the generalization of [44]. 10
Journal Pre-proof (2) The research content is different. References [32, 35] mainly studied the stability of stochastic complex networks, while this paper mainly focuses on the synchronization of stochastic complex networks. 4. Numerical example In this section, a numerical example with multi-weights and L´evy noise is provided to illustrate the theoretical results. First, we consider the drive network as follows: 6 ∑
a1kh H1 xh (t) + δ2
h=1
6 ∑
a2kh H2 xh (t) + δ3
h=1
6 ∑
a3kh H3 xh (t), k = 1, 2, · · · , 6,
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x˙k (t) = fk (xk (t), t) + δ1
h=1
(11)
p ro
where xk ∈ R3 , fk (xk , t) = (−2.5xk1 + sin xk2 , sin xk1 − 2xk2 , xk1 − 1.5xk3 )T , δ1 = 0.5, δ2 = 0.4, δ3 = 0.3 and the inner coupling networks are as follows: 0.3 0 0.5 0 0.1 0 0 0 0 H1 = 0 0.4 0 , H2 = 0 0.1 0 , H3 = 0 0.05 0 . 0 0 0.5 0 0 0.2 0 0 0.1
h=1
Pr e-
By the concept of drive-response, the response network is designed as 6 6 6 ∑ ∑ ∑ 3 2 1 akh H3 yh (t) + uk (t) dt akh H2 yh (t) + δ3 dyk (t) = fk (yk (t), t) + δ1 akh H1 yh (t) + δ2 +gk (yk (t) − xk (t), t)dW(t) +
∫
h=1
Y
(12)
h=1
˜ γk (yk (t) − xk (t), t, σ)Π(dt, dσ), k = 1, 2, · · · , 6,
where γk (xk , t, σ) = ηk (σ)(xk1 , xk2 , xk3 )T , η1 (σ) = η2 (σ) = η3 (σ) = 0.4, η4 (σ) = η5 (σ) = η6 (σ) = 0.6 and gk (xk , t) = (0.5 sin xk1 , 0.5 sin xk2 , 0.5 sin xk3 )T , k = 1, 2, 3,
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gk (xk , t) = (0.4 arctan xk1 , 0.4 arctan xk2 , 0.4 arctan xk3 )T , k = 4, 5, 6. Let λ(Y) = 1 and p = 2. Three kinds of weighted matrices A(i) = (aikh )6×6 , (i = 1, 2, 3) are given as follows. 0 0 1 2 0 0
2 0 0 0 1 0
0 0 1 0 0 2
0 0 0 0 0 1
0 0 0 1 0 0
0 2 (2) , A = 0 0 0 0
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A(1)
0 1 0 = 0 0 0
0 0 1 1 0 0
1 0 0 0 1 1
0 0 1 0 0 2
0 0 1 0 0 1
0 0 0 1 0 0
0 0 (3) , A = 0 0 0 0
0 0 1 2 0 0
1 0 0 0 2 1
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In the following, we show the simulations under the following feedback control law: uk (t) = νk ( fk (yk (t), t) − fk (xk (t), t)), k = 1, 2, · · · , 6,
0 0 2 0 0 2
0 0 0 0 0 0
0 0 0 1 0 0
. (13)
where νk , −1 is the gain constant to be designed later. Now, we verify the conditions of Theorem 2. By the direct calculation, we can get that αk = 1 (k = 1, 2, · · · , 6), βk = 0.5 (k = 1, 2, 3), βk = 0.4 (k = 4, 5, 6), ηk = 11
Journal Pre-proof 0.4 (k = 1, 2, 3), ηk = 0.6 (k = 4, 5, 6) and a new weighted matrix B as 0 0.73 0 0 0 0 0.65 0 0 0 0 0 0 0.48 0 0.51 0.2 0 B = (bkh )6×6 = . 0 0 0.28 0 0.76 0 0 0 0.51 0 0 0 0 0 0.23 0.96 0.45 0 l ∑
1 bkh + (p − 1)βk + η¯ k λ(Y), 2 h=1
p ro
(1 + νk )αk >
of
Clearly, (G, (bkh )l×l ) is strongly connected. Choosing the feedback gain constants as νk = 1 (k = 1, 2, 3), νk = 2 (k = 4, 5, 6), then we can check that
Pr e-
holds for any k (k = 1, 2, · · · , 6). Hence, drive-response networks (11) and (12) are exponentially synchronized under the control law (13) in mean square. Taking the initial values as x11 (0) = 0.1, x12 (0) = −0.5, x13 (0) = −0.1, x21 (0) = 0.2, x22 (0) = 0.2, x23 (0) = −0.5, x31 (0) = 0.6, x32 (0) = −0.5, x33 (0) = 1, x41 (0) = 0.8, x42 (0) = 0.2, x43 (0) = −1, x51 (0) = −0.3, x52 (0) = 0.4, x53 (0) = −0.6, x61 (0) = 0.5, x62 (0) = −0.1, x63 (0) = 1, y11 (0) = 0.6, y12 (0) = 0, y13 (0) = 0.2, y21 (0) = −0.1, y22 (0) = −0.2, y23 (0) = −0.3, y31 (0) = 0.2, y32 (0) = −0.4, y33 (0) = 0.7, y41 (0) = 0.3, y42 (0) = 0, y43 (0) = −0.6, y51 (0) = 0.1, y52 (0) = 0.8, y53 (0) = −0.2, y61 (0) = 0.3, y62 (0) = −0.2, y63 (0) = 0.8. The simulation result about the second moment of synchronization error for drive-response networks (11) and (12) is presented in Figure 1. We also get from Theorem 3 that drive-response networks (11) and (12) are stochastic asymptotically synchronized under the control law (13). The simulations for the sample path of synchronization error for drive-response networks and the sample path of drive-response networks are presented in Figures 2 and 3, respectively. For better showing the numerical simulations in Figure 3, the second component of each state variable in the drive-response networks (11) and (12) has been added 0.1, and the third component of each state variable minus 0.1.
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5. Conclusion
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In this paper, synchronization for a novel class of stochastic multi-weighted complex networks with L´evy noise has been investigated. By the concept of drive-response, the error network has been first obtained. Then the global Lyapunov function for the error network has been systematically constructed by combining algebraic graph theory and stochastic analysis. By the feedback control method, sufficient criteria for p-th moment exponential synchronization and stochastic synchronization of drive-response networks have been obtained. The criteria are closely related to multi-weights and the intensity of L´evy noise. The main synchronization results are formulated by Lyapunov functions and coefficients of drive-response networks, respectively. We also give some numerical simulations to verify the effectiveness of the new theoretical results. In the real applications, systems may suffer from another kind of noise, i.e. telegraph noise. If we consider both telegraph noise and L´evy noise into the multi-weighted complex networks, it is much more difficult to study its dynamics. This challenging task is our future work. Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 11601445,11801470, 31700347) and the Fundamental Research Funds for the Central Universities (No. 2682018CX60). 12
Journal Pre-proof
0.25
0.2
0.2
2
2
E|e11(t)|
0.2
E|e12(t)|2
0.15
2
E|e21(t)|
E|e22(t)|2
0.15
E|e13(t)|2
E|e31(t)|
E|e23(t)|2
0.1
E|e32(t)|2
0.15
E|e33(t)|2
0.1
0.1 0.05
0
0
1
2 3 Time t
4
0
5
0.25
0.05
0
1
2 3 Time t
0.2
E|e51(t)|
2
1
2 3 Time t
p ro
E|e52(t)|
2
2
E|e43(t)|
4
5
0.1
2
E|e62(t)|
2
0.03
E|e53(t)|
0.1
2
E|e61(t)|
0.04
2
0.15
E|e42(t)|
0.15
0
2
E|e41(t)|
0.2
0
5
0.05
2
E|e63(t)|
0.02
0.05
0.05 0
1
2 3 Time t
4
0
5
0.01
0
1
2 3 Time t
4
Pr e-
0
4
of
0.05
5
0
0
1
2 3 Time t
4
Figure 1: The second moment of synchronization error for drive-response networks (11) and (12).
0.4
0.4
e11(t) e12(t)
0.2
e13(t)
e23(t)
−0.2
2 Time t
4
−0.4
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0
0.4
0.2
0
2 Time t
0
2 Time t
4
0.4
e42(t)
e52(t)
4
0.2
e33(t)
0
−0.4
0
2 Time t
4
e53(t)
e61(t)
0
−0.2
−0.2
0
2 Time t
4
e62(t)
0.2
0
−0.4
e32(t)
0.2
0.4 e51(t)
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−0.2
e31(t)
−0.2
e41(t) e43(t)
0
−0.4
0.2
0
−0.2
−0.4
e22(t)
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0
0.4
e21(t)
−0.4
e63(t)
0
2 Time t
Figure 2: Sample path of synchronization error for drive-response networks (11) and (12).
13
4
5
Journal Pre-proof
x11(t) 0.5
x21(t) 0.5
y11(t)
y21(t)
x12(t)+0.1
x22(t)+0.1
y12(t)+0.1
0
x
y22(t)+0.1
0
(t)−0.1
x
13
y23(t)−0.1
−0.5
−0.5 0
2
4
6
8
10
0
2
4
Time t
x31(t)
y32(t)+0.1
0
x33(t)−0.1
−0.5
y33(t)−0.1 4
10
6
8
y41(t)
0
10
−1
0
2
4
x42(t)+0.1 y42(t)+0.1 x43(t)−0.1 y43(t)−0.1 6
8
10
Time t
1
p ro
Time t
of
x32(t)+0.1
2
8
x41(t)
0.5
y31(t)
0
6 Time t
0.5
−0.5
(t)−0.1
23
y13(t)−0.1
1
x
(t)
51
y51(t)
0.5
x
(t)+0.1
y
(t)+0.1
x
(t)−0.1
52
y53(t)−0.1
−0.5 0
2
4
6
8
10
−0.5
0
2
(t)+0.1
y
(t)+0.1
x
(t)−0.1
62 63
y63(t)−0.1 4
6
8
10
Time t
Pr e-
Time t
x
62
0
53
(t)
61
y61(t)
0.5
52
0
x
Figure 3: Sample path of drive-response networks (11) and (12).
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Journal Pre-proof Figure captions
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Pr e-
p ro
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Figure 1 is the second moment of synchronization error for drive-response networks (11) and (12). Figure 2 is the sample path of synchronization error for drive-response networks (11) and (12). Figure 3 is the sample path of drive-response networks (11) and (12).
18