Stability analysis of stochastic delayed complex networks with multi-weights based on Razumikhin technique and graph theory

Physica A 538 (2020) 122827

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Stability analysis of stochastic delayed complex networks with multi-weights based on Razumikhin technique and graph theory Chunmei Zhang, Bang-Sheng Han

∗

School of Mathematics, Southwest Jiaotong University, Chengdu 610031, PR China

article

info

a b s t r a c t In this paper, a novel kind of stochastic delayed complex networks with multi-weights (SDCNMW) is considered. We first split SDCNMW into multiple stochastic complex networks with single weight. This paper is mainly concerned with the issues of exponential stability for SDCNMW. A global Lyapunov functional is constructed by a graph-theoretical approach based on Kirchhoff’s matrix tree theorem. Razumikhin-type theorem is presented by stochastic stability theory and Razumikhin technique. An easyverified coefficients-type theorem is also given. Finally, theoretical results are illustrated by stochastic oscillators network successfully. © 2019 Elsevier B.V. All rights reserved.

Article history: Received 2 June 2019 Received in revised form 25 August 2019 Available online 24 September 2019 Keywords: Stochastic complex networks Multi-weights Exponential stability Lyapunov function Razumikhin technique

1. Introduction There are many extensive studies in the literature concerned with the stability analysis of complex networks with time delays, see for example [1–6]. Therein, weighted complex networks with time delays have attracted the attention of researchers in many fields. In particular, most papers are aimed at the stability of single weighted complex networks [7–9]. However, in many real-world networks such as public traffic networks and communication networks, there are different kinds of weights among nodes on networks [10–15]. These networks cannot be modeled by single weighted networks mentioned above. Hence, it is necessary to introduce delayed complex networks with multi-weights. There is an important addition should be taken into account in the complex networks, that is the effect of random perturbations [16–19], which may change or destroy the stability of complex networks. The stability of stochastic complex networks has been a focal subject for research [20–23]. So, this paper aims to study the exponential stability of stochastic delayed complex networks with multi-weights (SDCNMW) as follow:

[ (k) dxi (t)

(k)

= fi (x (t), x (t − τk ), t) + (k)

(k)

l ∑

(kh) Hi

(

(h) xi (t

− τh )

] )

dt

h=1

[ +

(k) gi (x(k) (t)

, x (t − τk ), t) + (k)

l ∑

(kh) Ni

(

(h) xi (t

− τh )

] )

dW (t), 1 ≤ i ≤ n, 1 ≤ k ≤ l.

h=1

There are l nodes and n weights among nodes in this system. For the detailed description of SDCNMW, please see Section 2. ∗ Corresponding author. E-mail address: [email protected] (B.-S. Han). https://doi.org/10.1016/j.physa.2019.122827 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

As is well known, Lyapunov method is an efficient tool to study the stability of systems. But the classical method is not applicable any more for large-scale complex networks. Recently, there is a significant progress on Lyapunov method. That is, a global Lyapunov function for large-scale complex networks can be constructed by combining graph theory and Lyapunov functions of each vertex system in [24,25]. Following this pioneering work, there are lots of results existed for various complex networks, including delayed complex networks [26–28], stochastic complex networks [29,30], discrete complex networks [31,32] and the references cited therein. However, few papers in the literature considered the complex networks with multi-weights [11,12], let alone SDCNMW. It is worth pointing out that multiple weights, time delays and stochastic perturbation are all included in the SDCNMW, which greatly increase the complexity of its stability analysis. In fact, complex networks with multi-weights can be split into multiple sub-networks by the method of network split. But these sub-networks are not isolated, i.e., they are interacting with each other. Therefore, the single-graph theoretic method appeared in the aforementioned work cannot be applicable for complex networks with multi-weights. Hence, how to seek the stability criterion about stochastic complex networks with multi-weights is a hot spot issue. So, this paper attempts to present the corresponding multi-graph theoretic method based on Kirchhoff’s matrix tree theorem to show the stability of SDCNMW. Moreover, Razumikhin technique [33] has a well-known advantage, i.e., its Lyapunov functional needs to be decreasing in the case that the current value of the function dominates other values over the interval of time delay [34]. This technique is less conservative and thus has been widely utilized to deal with the stability of various delayed equations with or without disturbance [28,35–38]. Motivated by the above discussions, this paper aims to study the exponential stability of SDCNMW. The method of network split is used to split SDCNMW into multiple stochastic delayed complex networks with single weight. Then, Razumikhin technique, graph theory together with stochastic stability theory are used to obtain stability principles of SDCNMW. Some basic preliminaries and model descriptions are arranged in Section 2. We show that two kinds of stability theorems: Razumikhin-type theorem and coefficients-type theorem in Section 3. Section 4 is the application of theoretic results to stochastic oscillator networks. Finally, we make a conclusion in Section 5. 2. Preliminaries Some useful notations, definitions and lemmas about graph theory are stated in Section 2.1. Then the model formulation and definition are presented in Section 2.2. 2.1. Mathematical preliminaries Let (Ω , F , F, P) be a complete probability space with a filtration F = {Ft }t ≥0 satisfying the usual conditions (the filtration is right continuous and F0 contains all P-null sets). W (t) is a scalar Brownian motion defined on the space and E(·) is the mathematical expectation with respect to P. Denote |·| as the Euclidean norm for vectors or the trace norm for matrices. Denote Z+ = {1, 2, . . .} and R1+ = [0, +∞). We write C 2,1 (Rn × R1+ ; R1+ ) for the family of all nonnegative functions V (x, t) on Rn × R1+ that are continuously twice differentiable in x and once in t. As usual, C ([−τ , 0]; Rn ) and p LF0 ([−τ , 0]; Rn ) represent the space of continuous functions x : [−τ , 0] → Rn with norm ∥x∥ = sup−τ ≤u≤0 |x(u)| and the family of F0 -measurable C ([−τ , 0]; Rn )-valued random variables y such that E∥y∥p < ∞, respectively. Let us recall some essential concepts and theorems on graph theory from [39]. A digraph G = (U , E) contains a set U = {1, 2, . . . , l} of vertices and a set E of arcs (k, h) leading from initial vertex k to terminal vertex h. A subgraph H of G is said to be spanning if H and G have the same vertex set. A digraph G is weighted if each arc (h, k) is assigned a positive weight akh . Here akh > 0 if and only if there exists an arc from vertex h to vertex k in G , and we call A = (akh )l×l as the weight matrix. The weight ω(G ) of G is the product of the weights on all its arcs. A directed path P in G is a subgraph with distinct vertices {k1 , k2 , . . . , ks } such that its set of arcs is {(ki , ki+1 ) : i = 1, 2, . . . , s − 1}. If ks = k1 , we call P a directed cycle. A connected subgraph T is a tree if it contains no cycles. A tree T is rooted at vertex k, called the root, if k is not a terminal vertex of any arcs, and each of the remaining vertices is a terminal vertex of exactly one arc. A subgraph Q is unicyclic if it is a disjoint union of rooted trees whose roots form a directed cycle. A digraph G is strongly connected if, for any pair of distinct vertices, there exists a directed path from one to the other. Denote the digraph with weight matrix A as (G , A). A weighted digraph (G , A) is said to be balanced if ω(C ) = ω(−C ) for all directed cycles C . Here, −C denotes the reverse of C and is constructed by reversing the direction of all arcs in C . For a unicyclic graph Q with cycle ˜ be the unicyclic graph obtained by replacing CQ with −CQ . Suppose that (G , A) is balanced, CQ , let Q then ω(Q) = ω(Q˜ ). ∑ The Laplacian matrix of (G , A) is defined as L = (pkh )l×l , where pkh = −akh for h ̸ = k and pkh = j̸ =k akj for h = k. We now state some important results in graph theory without proof [25], which will be used in the proofs of our main results. Lemma 1 (Kirchhoff’s Matrix Tree Theorem). Assume l ≥ 2 and let ck be the cofactor of the kth diagonal element of L = (pkh )l×l . Then ck =

∑

ω (T ) ,

1 ≤ k ≤ l,

T ∈Tk

where Tk is the set of all spanning trees T of (G , A) that are rooted at vertex k, and ω(T ) is the weight of T . In particular, if (G , A) is strongly connected, then ck > 0 for 1 ≤ k ≤ l.

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

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Fig. 1. The topology map of complex networks with three weights and its split.

Lemma 2. Let l ≥ 2 and ck be the cofactor of the kth diagonal element of L = (pkh )l×l . Then the following identity holds: l ∑

ck akh Fkh (xk , xh ) =

k,h=1

∑ Q∈Q

ω(Q)

∑

Frs (xr , xs ).

(s,r)∈E(CQ )

Here for any 1 ≤ k, h ≤ l, Fkh (xk , xh ) is an arbitrary function, Q is the set of all spanning unicyclic graphs of (G , A), ω(Q) is the weight of Q, and CQ denotes the directed cycle of Q. 2.2. Model formulations In this paper, we describe the SDCNMW as a coupled system on a complex network with n weights. Given a weighted (kh) (kh) (kh) (kh) digraph G with l (≥ 2) nodes and n weights (a1 , a2 , . . . , an ), k, h = 1, 2, . . . , l. ai is the ith weight at the network (kh) (kh) edge from the node h to node k, ai ≥ 0 and ai = 0 if and only if there exists no ith weight from the node h to node k. Since different weights have different natures, the weights with the same nature along with l nodes form a sub-network. Hence, a complex network with n weights can be split into n sub-networks with single weight by the theory of network split. We take 4 coupled nodes as an example and the architecture for such complex network with three weights and its split is shown in Fig. 1. The SDCNMW on G can be built as follows. Since time delay is often encountered in many practical systems, each node k(1 ≤ k ≤ l) is assigned a delayed dynamical system described by (k)

[

(k)

]

dxi (t) = fi (x(k) (t), x(k) (t − τk ), t) dt , t ≥ 0, 1 ≤ i ≤ n,

(1)

4

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827 (k)

(k)

(k)

(k)

(k)

where τk is a constant delay, xi (t) ∈ Rmi , fi : Rm × Rm × R1+ → Rmi and x(k) (t) = (x1 (t), x2 (t), . . . , xn (t))T ∈ Rm . ∑n + Hereafter, m = i=1 mi for mi ∈ Z . Coupling delay is an important type of delay caused by information transmission between subsystems. Hence, the inclusion of delayed coupling into coupled systems is natural and reasonable in the realistic modeling. Suppose that (kh) ¯ (kh) (h) ¯ (kh) (x(h) (t − τh )) : the influence from vertex h to vertex k is provided in the form ai H (xi (t − τh )), in which H i i i Rmi → Rmi describes the ith sub-network inner delayed coupling. Hence, after considering n weights, we replace ∑ (k) (k) (kh) ¯ (kh) (h) (kh) ¯ (kh) (h) l fi (x(k) (t), x(k) (t − τk ), t) by fi (x(k) (t), x(k) (t − τk ), t) + h=1 ai H (x (t − τ )). For brevity, write a H (x h i i (t − τh )) i i i (kh) (h) as Hi (xi (t − τh )) throughout this paper. Thus, we arrive at a coupled systems of m × l-dimensional delayed equations on G as follows:

[ (k) dxi (t)

(k)

= fi (x (t), x (t − τk ), t) + (k)

(k)

l ∑

(kh) Hi

(

(h) xi (t

)

]

− τh )

dt , 1 ≤ i ≤ n, 1 ≤ k ≤ l.

(2)

h=1

There are many external perturbations from the circumstance of network itself or any other probabilistic reasons in reality that affect network. After taking stochastic perturbation into account, we can get the stochastic counterpart corresponding to (2) shown as:

[ (k) dxi (t)

(k)

= fi (x (t), x (t − τk ), t) + (k)

(k)

l ∑

(kh) Hi

(

(h) xi (t

− τh )

] )

dt

h=1

[ +

(k) gi (x(k) (t)

, x(k) (t − τk ), t) +

l ∑

(kh) Ni

(

(h) xi (t

− τh )

] )

(3) dW (t), 1 ≤ i ≤ n, 1 ≤ k ≤ l,

h=1 (k)

(kh)

where gi : Rm × Rm × R1+ → Rmi and Ni : Rmi → Rmi are the intensity of perturbation. By using the method of network split, SDCNMW (3) can be split into l different stochastic delayed complex networks with single weight. But, it should be pointed out that these complex networks with single weight are not isolated and they are interacted with each other. So, this paper is totally different with the extensive investigations for general stochastic delayed complex networks with single weight. Example 1. We consider a stochastic delayed predator–prey model with dispersal among l patches, which is described by the following SDCNMW with l nodes and two weights.

[ ] ⎧ ∑ ⎨ dxk (t) = xk (t)(rk − bk xk (t) − ek yk (t)) + lh=1 dkh (xh (t) − αkh xk (t)) dt + ρk xk (t)dW (t), [ ] ⎩ dy (t) = y (t)(−γ − δ y (t) + ε x (t − τ )) + ∑l p (y (t) − β y (t)) dt + σ y (t)dW (t), 1 ≤ k ≤ l. k k k k k k k k kh h kh k k k h=1 (4) The parameters in the model (4) are nonnegative constants. τk is the time delay due to converting prey biomass into predator biomass. dkh is the first kind of weight, which refers to the diffusion rate of preys between patches k and h. pkh is the second kind of weight, which refers to the diffusion rate of predators between patches k and h. For the meaning of other parameters, please see [40,41] and the references cited therein. (k)

(k)

Clearly, compared to (3), there are only two parts xk and yk in each patch. More precisely, x1 = xk , x2 = (k) (k) (k) (k) yk , f1 (xk , yk , t) = xk (rk −bk xk −ek yk ), f2 (xk , yk , t −τk ) = yk (−γk −δk yk +εk xk (t −τk )), g1 (xk , yk , t) = ρk xk , g2 (xk , yk , t) = (kh) (kh) σk yk , H1 (xk , yk ) = dkh (xh −αkh xk ), H2 (xk , yk ) = pkh (yh −βkh yk ). Note that dkh = 0 if and only if there is no prey dispersal from patch h to patch k, pkh = 0 if and only if predator does not disperse from patch h to patch k. All of these can show that (4) is a special example of our model (3). Take l = 3, we have a diagram for the topology map of (4) and its split in Fig. 2. In this case, d12 = d13 = p23 = p31 = 0. (k) (k) Through the paper, τk ∈ [−τ , 0] for any 1 ≤ k ≤ l. For the aim of this paper, we suppose that functions fi , gi , (kh) (kh) Hi and Ni (1 ≤ k, h ≤ l, 1 ≤ i ≤ n) are such that initial value problem (3) has a unique solution x(t) with p initial value x(t) = ϕ (t), t ∈ [−τ , 0], where ϕ (t) ∈ LF0 ([−τ , 0]; Rlm ). Denote the trivial solution (or the origin) as (1)

(1)

(1)

(l)

(l)

(l)

x(t ; ϕ ) ≜ x(t) = ((x(1) (t))T , . . . , (x(l) (t))T )T = (x1 (t), x2 (t), . . . , xn (t), . . . , x1 (t), x2 (t), . . . , xn (t))T = 0. Next, we give the definition of exponential stability for SDCNMW (3).

Definition 1. Let p > 0. The origin of SDCNMW (3) is said to be exponentially stable in pth moment if there are positive constants C and σ such that

E|x(t)|p ≤ C exp(−σ t), t ≥ 0, p

for any initial value ϕ (t) ∈ LF0 ([−τ , 0]; Rlm ). Particularly, when p = 2, it is said to be exponentially stable in mean square.

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

5

Fig. 2. The topology map for (4) in three patches with two weights and the diagram for its split into two sub-networks with single weight.

Differential operator associated with the ith part in the kth node of SDCNMW (3) is as follow: (k) (k) LVi (xi (t)

[ ] l ( ) ( (k) ) ∑ ∂ Vi(k) (x(k) ∂ Vi(k) (x(k) (k) (kh) (h) i , t) i , t) (k) + , t) = fi x (t), x (t − τk ), t + Hi xi (t − τh ) ∂t ∂ x(k) i h=1 [ ] l ( ) T ∂ 2 V (k) (x(k) , t) ( (k) ) ∑ 1 (k) (kh) (h) i i (k) + Trace gi x (t), x (t − τk ), t + Ni xi (t − τh ) (k) 2 2 ∂ (x i ) h=1 [ ] l ( ) ( ) ∑ (kh) (h) × gi(k) x(k) (t), x(k) (t − τk ), t + Ni xi (t − τh ) ,

(5)

h=1

where the trace of an n × n matrix A = (aij )n×n is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A. 3. Global exponential stability analysis for general SDCNMW In this section, by using graph-theoretic method [22,25], Razumikhin technique together with Lyapunov method [28,36,37], we present the multi-graph theoretic method to analyze the stability of SDCNMW and show our main results in two distinct forms: Razumikhin-type theorem and coefficients-type theorem. Through the paper, we always assume that (G , Ai ) is strongly connected for 1 ≤ i ≤ n. If we remove this restriction, we can also get the stability criteria by the hierarchical method along with the theory of asymptotically autonomous systems [20,42]. (k) (k) First, we present some assumptions for set {Vi (xi , t), 1 ≤ k ≤ l, 1 ≤ i ≤ n}.

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C. Zhang and B.-S. Han / Physica A 538 (2020) 122827 (k)

(A1) There are positive constants αi (k) i

α |

(k) p xi

| ≤

(k) (k) Vi (xi

(k)

and βi (k) i

such that

(k) p xi

, t) ≤ β |

|. (k)

(kh)

(A2) There exist positive constants q > 1 and γi , a function Fi that n ∑

(k)

(k)

(k)

ci LVi (xi (t), t) ≤ −

i=1

n ∑

(k)

(k)

(k)

(k)

ci γi Vi (xi (t), t) +

(kh)

and a matrix Ai = (ai

n ∑

i=1

(k)

ci

l ∑

i=1

(kh) (kh) (k) Fi (xi (t

ai

(kh)

)l×l , in which ai

≥ 0, such

− τk ), x(h) i (t − τh )),

(6)

h=1

(k)

for all t ≥ 0 and those xi (t) satisfying (k)

(k)

(k)

(k)

Vi (xi (t + θ ), t + θ ) < qVi (xi (t), t), − τ ≤ θ ≤ 0, (k)

where ci

is the cofactor of the kth diagonal element of Laplacian matrix of (G , Ai ).

(A3) Along each directed cycle C (i) of weighted digraph (G , Ai ), there is

∑

(kh)

Fi

(k)

(h)

(xi (t − τk ), xi (t − τh )) ≤ 0,

(7)

(h,k)∈E(C (i)) (h)

(k)

for all xi , xi

∈ Rmi .

3.1. Razumikhin-type theorem Before we present our main theorem, a vital lemma should be introduced first, which plays an important role in the p proof of our main theorem. For simplicity, fix any ϕ ∈ LF0 ([−τ , 0]; Rlm ) and write x(t ; ϕ ) = x(t), therein, denote the

solution of the ith part in node k as xi . For Vi (xi , t) ∈ C 2,1 (Rmi × R1+ ; R1+ ) and constant σ > 0, define (k)

(k)

{

(k)

Φi (t) = max

−τ ≤θ ≤0

(k)

(k)

}

(k)

exp(σ (t + θ ))EVi (xi (t + θ ), t + θ ) , t ≥ 0,

(8)

and

( D

+

n l ∑ ∑

) (k) (k) ci Φi (t)

≜

i=1 k=1

n l ∑ ∑

( (k) ci

i=1 k=1

(k)

(k)

lim sup

Φi (t + s) − Φi (t) s

s→0+ (k)

) .

(k)

Lemma 3. Suppose that SDCNMW (3) admits a set {Vi (xi , t), 1 ≤ k ≤ l, 1 ≤ i ≤ n} satisfying (A1) − (A3). Then it follows that D+

( n l ∑∑

) (k) (k) ci Φi (t)

≤ 0.

(9)

i=1 k=1 (k)

(k)

(k)

Proof. In (8), taking σ < min1≤i≤n,1≤k≤l {log(q)/τ , γi } for some q > 1. We can easily check that EVi (xi (t), t) is (k) (k) (k) (k) continuous since xi (t) and Vi (xi (t), t) are continuous for any 1 ≤ i ≤ n, 1 ≤ k ≤ l. Then we can know that Φi (t) is continuous for 1 ≤ i ≤ n, 1 ≤ k ≤ l. In what follows, we just need to claim that there exists sufficiently small constant s∗ > 0 such that (k)

(k)

Φi (t + s) ≤ Φi (t), 0 < s < s∗ .

(10)

Fix t ≥ 0 and for 1 ≤ i ≤ n, 1 ≤ k ≤ l, define (k) θ¯i(k) = max{θ ∈ [−τ , 0] : exp(σ (t + θ ))EVi(k) (x(k) i (t + θ ), t + θ ) = Φi (t)}. (k)

(k)

(k)

(k)

(k)

(k)

(k)

It is trivial to find that θ¯i ∈ [−τ , 0] and exp(σ (t + θ¯i ))EVi (xi (t + θ¯i ), t + θ¯i ) = Φi (t). In the following, we (k) (k) (k) divide the discussion into three cases: θ¯i = −τ , θ¯i ∈ (−τ , 0) and θ¯i = 0. Fix 1 ≤ i ≤ n, for simplicity, denote three sets and their corresponding indicator functions as (k)

Ωi1 = {k ∈ L : θ¯i

= −τ }, Ωi2 = {k ∈ L : θ¯i(k) ∈ (−τ , 0)}, Ωi3 = {k ∈ L : θ¯i(k) = 0}

and

{ IΩ j = i

1, 0,

j

k ∈ Ωi , k∈ / Ωij ;

Clearly, IΩ 1 + IΩ 2 + IΩ 3 = 1. i

i

i

j = 1, 2, 3.

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

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Case I: k ∈ Ωi1 . We have that for any −τ < θ ≤ 0, it holds that (k)

(k)

(k)

(k)

exp(σ (t + θ ))EVi (xi (t + θ ), t + θ ) < exp(σ (t − τ ))EVi (xi (t − τ ), t − τ ). Taking θ = 0 implies that (k)

(k)

(k)

(k)

exp(σ t)EVi (xi (t), t) < exp(σ (t − τ ))EVi (xi (t − τ ), t − τ ). (k)

(k)

By virtue of the continuity of EVi (xi (t), t), we have that (k)

(k)

lim exp(σ (t + s))EVi (xi (t + s), t + s)

s→0+

< exp(σ (t − τ ))EVi(k) (x(k) (t − τ ), t − τ ) ( i [∫ ) ] l t +s ∑ (kh) (kh) (k) (h) + lim exp(σ r)E ai Fi (xi (r − τk ), xi (r − τh )) dr . s→0+

t

h=1

Hence, there exists sufficiently small s1 > 0 such that for 0 < s < s1 , it holds that (k)

(k)

exp(σ (t + s))EVi (xi (t + s), t + s)

< exp(σ (t − τ ))EVi(k) (xi(k) (t − τ ), t − τ ) ) ∫ t +s ( l ∑ (kh) (kh) (k) (h) exp(σ r) ai Fi (xi (r − τk ), xi (r − τh )) dr . +E t

(11)

h=1

Case II: k ∈ Ωi2 . (k) (k) Clearly, for θ¯i < θi ≤ 0, it follows that (k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

exp(σ (t + θi ))EVi (xi (t + θi ), t + θi ) < exp(σ (t + θ¯i ))EVi (xi (t + θ¯i ), t + θ¯i ). (k)

Letting θi

= 0 yields (k)

(k)

(k)

(k)

(k)

(k)

(k)

exp(σ t)EVi (xi (t), t) < exp(σ (t + θ¯i ))EVi (xi (t + θ¯i ), t + θ¯i ). As the similar discussion as case I, we have that (k)

(k)

lim exp(σ (t + s))EVi (xi (t + s), t + s)

s→0+

¯ (k) ¯ (k) < exp(σ (t + θ¯i(k) ))EVi(k) (x(k) i (t + θi ), t + θi ) ( [∫ ) ] l t +s ∑ (kh) (kh) (k) (h) exp(σ r)E ai Fi (xi (r − τk ), xi (r − τh )) dr . + lim s→0+

t

h=1

Hence, there exists sufficiently small 0 < s2 < s1 such that for 0 < s < s2 , it holds that (k)

(k)

exp(σ (t + s))EVi (xi (t + s), t + s)

¯ (k) ¯ (k) < exp(σ (t + θ¯i(k) ))EVi(k) (x(k) i (t + θi ), t + θi ) ( ) ∫ t +s l ∑ (kh) (kh) (k) (h) +E exp(σ r) ai Fi (xi (r − τk ), xi (r − τh )) dr . t

h=1

Case III: k ∈ Ωi3 . (k) For −τ ≤ θi ≤ 0, it follows that (k)

(k)

(k)

(k)

(k)

(k)

(k)

exp(σ (t + θi ))EVi (xi (t + θi ), t + θi ) ≤ exp(σ t)EVi (xi (t), t), (k)

which implies that for −τ ≤ θi (k) (k) EVi (xi (t

(k) i )

+θ

(k) i )

,t + θ

≤ 0, (k) (k) ≤ exp(σ τ )EVi(k) (x(k) i (t), t) < qEVi (xi (t), t).

By condition (A2), we can have n ∑

(k)

(k)

(k)

ci ELVi (xi (t), t)

i=1

≤−

n ∑ i=1

(k)

(k)

(k)

(k)

ci γi EVi (xi (t), t) +

n ∑ i=1

(k)

ci E

l ∑ h=1

(kh) (kh) (k) Fi (xi (t

ai

− τk ), x(h) i (t − τh )).

(12)

8

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

Then, by Itô formula, there exists 0 < s3 < s2 such that for any 0 < s < s3 , it holds that exp(σ (t + s))

n ∑

(k)

(k)

(k)

ci EVi (xi (t + s), t + s)

i=1 n ∑

= exp(σ t)

(k)

(k)

(k)

ci EVi (xi (t), t) + E

n

∑

n ∑

(k)

ci

n ∑

(k)

(k)

(k)

ci EVi (xi (t), t) +

exp(σ r) − t

l ∑

i=1

(kh) (kh) (k) Fi (xi (r

n ∑

h=1 n ∑

≤ exp(σ t)

[

(k)

(k)

(k)

]

(k)

LVi (xi (r), r) + σ Vi (xi (r), r) dr

(k)

(k)

(k)

(k)

ci γi EVi (xi (r), r)

(13)

i=1

− τk ), xi(h) (r − τh )) + σ

ai

(k)

ci

i=1

[

t +s

∫

i=1

+E

exp(σ r)

t

i=1

≤ exp(σ t)

t +s

∫

n ∑

] (k)

(k)

(k)

ci EVi (xi (r), r) dr

i=1

(k)

(k)

(k)

ci EVi (xi (t), t) + E

t +s

∫

exp(σ r)

n ∑

t

i=1

(k)

ci

l ∑

i=1

(kh) (kh) (k) Fi (xi (r

ai

− τk ), x(h) i (r − τh ))dr .

h=1

(k) i

Above, we have used the inequality γ − σ > 0 by the definition of σ . Hence, (11), (12), (13) together with (7) can show that for any 0 < s < s3 , we have n l ∑ ∑

(k)

(k)

(k)

(k)

(k)

exp(σ (t + s))EVi (xi (t + s), t + s)

(k)

exp(σ (t + s))EVi (xi (t + s), t + s)(IΩ 1 + IΩ 2 + IΩ 3 )

ci

i=1 k=1

=

n l ∑ ∑

ci

i

i

i

i=1 k=1

≤

n l ∑ ∑

(k)

ci

[

(k)

(k)

(k)

(k)

(k)

(k)

(k)

exp(σ (t − τ ))EVi (xi (t − τ ), t − τ )IΩ 1 + exp(σ (t + θ¯i ))EVi (xi (t + θ¯i ), t + θ¯i )IΩ 2 i

i

i=1 k=1

+ exp(σ

]

(k) (k) t)EVi (xi (t)

, t)IΩ 3 +

n l ∑ ∑

i

(k) ci E

n l ∑ ∑

(k)

(k)

ci Φi (t) +

∫ n ∑

i=1 k=1 n

≤

t +s t

i=1

∑

exp(σ r)

E

exp(σ r) t

i=1 k=1

=

t +s

∫

ω (Q i )

Qi ∈Qi

(14)

l ∑

(kh) (kh) (k) ai Fi (xi (r

− τk ),

(h) xi (r

− τh ))dr

h=1

∑

(kh)

Fi

(k)

(h)

(xi (r − τk ), xi (r − τh ))dr

(h,k)∈E(CQi )

l

∑∑

(k)

(k)

ci Φi (t).

i=1 k=1

Above, Qi is the set of all spanning unicyclic graphs of (G , Ai ), ω(Qi ) is the weight of Qi , and CQi denotes the directed (k) cycle of Qi . Therefore, by the definition of Φi (t), it can easily deduce that for 0 < s < s3 , n l ∑ ∑

(k)

(k)

ci Φi (t + s) ≤

i=1 k=1

n l ∑ ∑

(k)

(k)

ci Φi (t),

i=1 k=1

+

which implies that D

(∑ ∑ n l i=1

(k) (k) k=1 ci Φi (t)

)

≤ 0. The proof is completed. □

Theorem 1. Suppose that the conditions in Lemma 3 are satisfied. Then the origin of SDCNMW (3) is exponentially stable in pth moment. (k)

Proof. We obtain from Lemma 3 that there exists q > 1 such that σ < min1≤i≤n,1≤k≤l {log(q)/τ , γi } for some σ > 0 and

( D+

n l ∑ ∑

) (k)

(k)

ci Φi (t)

≤ 0,

i=1 k=1 (k)

where Φi (t) is defined in (8). Then it follows easily that n l ∑ ∑ i=1 k=1

(k)

(k)

ci Φi (t) ≤

n l ∑ ∑ i=1 k=1

(k)

(k)

ci Φi (0), t ≥ 0.

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

9

By condition (A1), we obtain that n l ∑ ∑

(k)

(k)

(k)

(k)

(k)

ci exp(σ t)Eαi |xi (t)|

p

i=1 k=1

≤

n l ∑ ∑

(k)

ci

exp(σ t)EVi (xi (t), t)

i=1 k=1

≤

n l ∑ ∑

(k)

ci

−τ ≤θ≤0

i=1 k=1

=

n l ∑ ∑

{

max

(k) (k) ci Φi (t)

(k)

≤

i=1 k=1

=

n l ∑ ∑

(15) (k) (k) ci Φi (0)

i=1 k=1

n l ∑ ∑

(k)

ci

n l ∑ ∑

{

max −τ ≤θ≤0

i=1 k=1

≤

}

(k)

exp(σ (t + θ ))EVi (xi (t + θ ), t + θ )

(k)

(k)

ci βi

i=1 k=1

(k)

(k)

exp(σ θ )EVi (xi (θ ), θ )

}

p

(k)

max |xi (θ )| .

−τ ≤θ ≤0

(k)

Since (G , Ai ) is strongly connected, ci p

(nl)(1− 2 )∧0 E|x(t)|p ≤

n l ∑ ∑

> 0 for any 1 ≤ i ≤ n, 1 ≤ k ≤ l. We therefore must have

(k)

E|xi (t)|

p

i=1 k=1

[∑

n i=1

≤

∑l

(k)

(k)

k=1 ci βi

(k)

p

max−τ ≤θ ≤0 |xi (θ )| (k)

(k)

min1≤i≤n,1≤k≤l {ci αi }

] exp(−σ t).

So, there is a constant C > 0 such that

E|x(t)|p ≤ C exp(−σ t). Then, the origin of SDCNMW (3) is exponentially stable in pth moment by Definition 1.

□

(k)

Remark 1. The advantage of considering the parameter ci in the inequality in (A2) is that global Lyapunov function V (k) of SDCNMW (3) can be constructed as the weighted summation of Vi , i.e., V (x, t) =

n l ∑ ∑

(k)

(k)

(k)

ci Vi (xi , t),

i=1 k=1

where (k)

ci

=

∑

ω (Ti ) ,

1 ≤ k ≤ l.

Ti ∈Tki

Here, Tki is the set of all spanning trees Ti of (G , Ai ) that are rooted at vertex k, and ω(Ti ) is the weight of Ti . Additionally, (k) (kh) ci is the cofactor of the kth diagonal element of Laplacian matrix of (G , Ai ), Ai = (ai )l×l . (k) From condition (A2), we can also see that (6) is satisfied for those xi (t) satisfying (k)

(k)

(k)

(k)

Vi (xi (t + θ ), t + θ ) < qVi (xi (t), t), − τ ≤ θ ≤ 0, q > 1, (k)

not for any xi (t). In other words, it is not necessary for global Lyapunov function V to be decreasing on the whole range. Function V can increase on the local scale, but it should be decreasing when it reaches the certain limit. This can guarantee that Lyapunov function V has degression tendency, which is the advantage of Razumikhin-type technique. Furthermore, Lemma 3 and Theorem 1 can tell us that time delays cannot affect the stabilization of main system (3), but can affect the rate of convergence.

10

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

We next give another assumption for any 1 ≤ k ≤ l, 1 ≤ i ≤ n. (k)

(k)

(B2) There exist positive constants γi n ∑

(k)

(k)

(kh)

(kh)

and δi , a function Fi

(k)

ci LVi (xi (t), t) ≤ −

i=1

n ∑

(k)

(k)

(k)

and a matrix Ai = (ai

(k)

ci γi Vi (xi (t), t) +

i=1

(k)

(k)

(16)

l

∑

(k) ci

i=1 (k)

(k) (k)

∑

(kh) (kh) (k) ai Fi (xi (t

≥ 0, such that

ci δi Vi (xi (t − τk ), t − τk )

i=1

n

+

n ∑

(kh)

)l×l , in which ai

− τk ),

(h) xi (t

− τh )),

h=1

is the cofactor of the kth diagonal element of Laplacian matrix of (G , Ai ).

where ci

(k)

(k)

Corollary 1. Let γi > δi for 1 ≤ i ≤ n, 1 ≤ k ≤ l. Then the conclusion of Theorem 1 holds if (A2) is replaced by (B2), i.e., the origin of SDCNMW (3) is exponentially stable in pth moment. (k)

Proof. Since γi

> δi(k) for 1 ≤ i ≤ n, 1 ≤ k ≤ l, then there is a constant q > 1 such that

γi(k) > qδi(k) , 1 ≤ i ≤ n, 1 ≤ k ≤ l. Assuming that (k)

(k) xi (t)

(17)

satisfying

(k)

(k)

(k)

Vi (xi (t + θ ), t + θ ) < qVi (xi (t), t), − τ ≤ θ ≤ 0, for the q > 1 mentioned in (17). Then it follows easily by (16) that n ∑

(k)

(k)

(k)

ci LVi (xi (t), t) ≤ −

n ∑

(k)

(k)

(k)

(k)

ci γi Vi (xi (t), t) +

n ∑

+

n ∑

l ∑ (k)

ci

i=1

=−

(k)

(k)

(k)

i=1

i=1

i=1

(k)

ci qδi Vi (xi (t), t)

n ∑

(kh) (kh) (k) Fi (xi (t

ai

− τk ), x(h) i (t − τh ))

h=1

(18)

) ( (k) ci γi(k) − qδi(k) Vi(k) (x(k) i (t), t)

i=1

+

n ∑

(k)

ci

i=1

l ∑

(kh) (kh) (k) Fi (xi (t

ai

− τk ), x(h) i (t − τh )).

h=1

Then, the origin of SDCNMW (3) is exponentially stable in pth moment by using Theorem 1. This proof is completed. □ Remark 2. Theorem 1 is shown in the form of Razumikhin-type conditions, thus we call it as Razumikhin-type theorem. Theorem 1 demands the derivative of Lyapunov function V less than zero under some restrictions. While Corollary 1 generally requires the derivative of Lyapunov function V less than zero. It means that the conditions in Corollary 1 is much stronger than which in Theorem 1. (k)

(h)

Remark 3. In general, inequality (7) is not easy to be verified since they hold for any xi , xi ∈ Rmi and each directed (k) (h) cycle of weighted digraph (G , Ai ). The number of directed cycles may be very large and xi , xi are arbitrary, so we must check (7) for infinite times. It seems to be so complicated. However, this problem can be successfully solved if we find (kh) (kh) some appropriate functions Fi , 1 ≤ k, h ≤ l, 1 ≤ i ≤ n. For example, for every Fi , 1 ≤ k, h ≤ l, 1 ≤ i ≤ n, there exists (k) a function Pi (·), such that (kh)

Fi

(

(k)

(h)

)

(k)

xi (t − τk ), xi (t − τh ) ≤ Pi

(

(k)

)

(h)

xi (t − τk ) − Pi

(

(h)

)

xi (t − τh ) , 1 ≤ k, h ≤ l, 1 ≤ i ≤ n.

(19)

Then, we have

∑

(kh)

Fi

(

(k)

(h)

xi (t − τk ), xi (t − τh )

)

(h,k)∈E(C (i))

≤

∑

(

(k)

Pi

(

(k)

)

(h)

xi (t − τk ) − Pi

(

(h)

xi (t − τh )

))

= 0,

(h,k)∈E(C (i))

which shows that (7) is satisfied. So, we can easily get that Theorem 1 and Corollary 1 also hold if (7) is replaced by (19).

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

11

With the help of some properties in graph theory, some other simple conditions are discussed. Note that if (G , Ai ) = (kh) (G , (ai )l×l ) is balanced, then l ∑

(k) (kh) (kh) Fi

ci ai

(

(k)

(h)

xi , xi

)

1 ∑

=

2

k,h=1

[

∑

ω (Q i )

(kh)

Fi

(

(k)

(h)

xi , xi

)

( )] (k) + Fi(hk) x(h) . i , xi

(h,k)∈E(CQi )

Qi ∈Qi

In this case, the assumption (A3) can be replaced by (A3’) Along each directed cycle C (i) of weighted digraph (G , Ai ), 1 ≤ i ≤ n, there is

[

∑

(

(kh)

(k)

)

(h)

(hk)

xi (t − τk ), xi (t − τh ) + Fi

Fi

(

(h)

)]

(k)

xi (t − τh ), xi (t − τk )

≤ 0.

(20)

(h,k)∈E(C (i)) (kh)

Moreover, if (G , Ai ) is balanced, Fi (kh)

Fi

(

(k)

is much easier to get since it just needs to satisfy

)

(h)

(

(k)

)

(h)

(

(h)

)

(k)

xi (t − τk ), xi (t − τh ) ≤ Pkh xi (t − τk ), xi (t − τh ) − Phk xi (t − τh ), xi (t − τk ) .

In this case, it holds that

∑

[

(kh)

Fi

(

(k)

(h)

)

(k)

(h)

)

(hk)

xi (t − τk ), xi (t − τh ) + Fi

(

(h)

(k)

xi (t − τh ), xi (t − τk )

)]

(h,k)∈E(C (i))

∑

≤

[(

(

(

(h)

(k)

Pkh xi (t − τk ), xi (t − τh ) − Phk xi (t − τh ), xi (t − τk )

))

(h,k)∈E(C (i))

) ( ))] ( ( (k) (h) (k) + Phk x(h) i (t − τh ), xi (t − τk ) − Pkh xi (t − τk ), xi (t − τh ) =0, for each directed cycle C (i) of the weighted digraph (G , Ai ). With these statements above, an important corollary is showed below. Corollary 2. Suppose that for any 1 ≤ i ≤ n, (G , Ai ) is balanced. Then the conclusion of Theorem 1 and Corollary 1 holds if assumption (A3) is replaced by (A3’). 3.2. Coefficients-type theorem Exponential stability criteria in Theorem 1 and Corollary 1 are given in the form of Lyapunov functions. Indeed, we need to check whether we can find some Lyapunov functions satisfying those conditions in Theorem 1 and Corollary 1. This is the task of this subsection. Theorem 2. The origin of (3) is exponentially stable in mean square provided that the following conditions hold for any 1 ≤ k, h ≤ l, 1 ≤ i ≤ n. (k)

(

(k)

(C1) There are constants ηi , δi

(k)

and βi

) j

,

( ) ξi(k) , 1 ≤ j ≤ n such that j

n ( n ( ) ) ∑ ∑ 2 2 (k) (k) (xi )T fi (x(k) (t), x(k) (t − τk ), t) ≤ βi(k) |x(k) ξi(k) |x(k) j (t)| + j (t − τk )| , j

j=1 2

2

j

j=1 2

|gi(k) (x(k) (t), x(k) (t − τk ), t)| ≤ ηi(k) |x(k) (t)| + δi(k) |x(k) (t − τk )| . (kh)

(C2) There exists a constant ai

such that

⏐ ⏐ ⏐ ( )⏐ ⏐ ( )⏐ ⏐ (kh) (h) ⏐ ⏐ (kh) (h) ⏐ (kh) ⏐ (h) ⏐ xi ⏐ ∨ ⏐Ni xi ⏐ ≤ ai ⏐xi ⏐ , ⏐Hi (kh)

and digraph (G , Ai ), Ai = (ai

)l×l is strongly connected and balanced.

(C3) It holds that

γi(k) > εi(k) , ) ) ∑l ∑n (( (k) ) ∑ (( ) ∑ (k) (kh) where γi = − h=1 ai − 2 j=1 βj + ηj(k) cj(k) /ci(k) , εi(k) = 2 nj=1 ξj(k) + δj(k) cj(k) /ci(k) + lh=1 a(kh) i (hk)

(2lai

i

i

+ 1) and ci(k) is the cofactor of the kth diagonal element of Laplacian matrix of (G , Ai ).

12

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827 (k)

(k) 2

(k)

(k)

(k)

(kh)

Proof. For any 1 ≤ k ≤ l, 1 ≤ i ≤ n, define a function Vi (xi , t) = |xi | . For brevity, we write fi , gi , Hi (k) (k) (kh) (h) (kh) (h) instead of fi (x(k) (t), x(k) (t − τk ), t), gi (x(k) (t), x(k) (t − τk ), t), Hi (xi (t − τh )), Ni (xi (t − τh )), respectively. Making use of (5) and (C1), we can get that (k)

, Ni(kh)

(k)

LVi (xi , t)

=2

(

(k) xi

[

)T

fi

(k)

+

l ∑

] (kh) Hi

h=1

⏐ ⏐2 l ⏐ ⏐ ⏐ (k) ∑ (kh) ⏐ + ⏐gi + Ni ⏐ ⏐ ⏐ h=1

n ( n ( l ⏐ ⏐ ) ) ∑ ∑ ∑ 2 2 ⏐ (k) (kh) ⏐ (h) ≤2 βi(k) |x(k) ξi(k) |x(k) ai ⏐xi (t − τh )⏐ j | +2 j (t − τk )| + 2|xi | j

j=1

j

j=1

h=1 l

2

2

+ 2ηi(k) |x(k) (t)| + 2δi(k) |x(k) (t − τk )| + 2l

∑

(kh) 2

2

(h)

) |xi (t − τh )| .

(ai

h=1

Therein, (k)

2|xi |

l ∑

l l ⏐2 ⏐ ⏐ ∑ ⏐ ∑ 2 ⏐ ⏐ ⏐ (h) (kh) ⏐ (h) (kh) (k) (t) | + ai ⏐xi (t − τh )⏐ . | x ≤ a (t − τ ) x ⏐ i h ⏐ i i

(kh)

ai

h=1

h=1

h=1

Hence, n ∑

(k)

(k)

(k)

ci LVi (xi (t), t)

i=1

≤

n ∑

⎡ n ( l ) ∑ ∑ 2 2 2 2 (k) (k) (kh) (k) (k) βi(k) |x(k) ai |xi (t)| + 2ηi |x(k) (t)| + 2δi |x(k) (t − τk )| + 2 ci ⎣ j (t)| j

j=1

h=1

i=1

⎤ n ( l [( ) ) ] ∑ ∑ 2 2 (k) (k) (kh) 2 (kh) (h) +2 ξi |xj (t − τk )| + 2l(ai ) + ai |xi (t − τh )| ⎦ j

j=1 n

=

h=1

l

∑

(k)

ci

i=1

∑

(kh)

ai

2

|xi(k) (t)| + 2

∑

h=1

+2

n l ∑ ∑

n ∑

2

(k) (k)

ci ηi |x(k) (t)| + 2

i=1

n n ∑ ∑

2

(k) (k)

ci δi |x(k) (t − τk )|

i=1

n n ( ) ( ) ∑ ∑ 2 2 (k) (k) (t) | + 2 ci ξi(k) |x(k) ci βi(k) |x(k) j j (t − τk )| j

i=1 j=1

+

n

(k)

ci

[(

(kh)

(kh) 2

) + ai

2l(ai

j

i=1 j=1

)

2

]

2

]

|xi(h) (t − τh )|

i=1 h=1

=

n ∑

(k)

ci

i=1

l ∑

2

|x(k) i (t)| + I + II + III + IV

h=1 n l ∑ ∑

+

(kh)

ai

(21) (k) ci

[(

(kh) 2l(ai )2

+

(kh) ai

)

(h) xi (t

|

− τh )|

.

i=1 h=1

In what follows, we make a calculation for I , II , III and IV . I=2

n n ∑ ∑

(k) (k) ci i

2 (k) xj (t)

η |

| =

i=1 j=1

n n ∑ ∑

(k) (k) j cj

2η

i=1 j=1

2 (k) xi (t)

|

| =

n ∑ i=1

( ∑n (k) ci

2

j=1

ηj(k) cj(k)

(k) ci

)

2

|x(k) i (t)| .

(22)

Similarly, it is trivial to get that II =

n ∑ i=1

( ∑n (k) ci

2

j=1

δj(k) cj(k)

(k) ci

)

2

|x(k) i (t − τk )| ,

⎛ ∑ ( )⎞ (k) n 2 c βj(k) ∑ (k) ⎜ j=1 j 2 i ⎟ (k) III = ci ⎝ ⎠ |xi (t)| , (k)

(23)

n

i=1

ci

(24)

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

⎛ ∑ ( )⎞ (k) n n 2 j=1 cj ξj(k) ∑ 2 (k) ⎜ i ⎟ (k) IV = ci ⎝ ⎠ |xi (t − τk )| . (k)

13

(25)

ci

i=1

Plugging (22)–(25) into (21) gives n ∑

(k)

(k)

(k)

ci LVi (xi (t), t)

i=1

≤

n ∑

⎛

l ∑ 2 (k) ⎜ (kh) ci ⎝ ai +

i=1

+

(k)

n ∑

⎛ 2 (k) ⎜

ci

(k) (k) j=1 δj cj (k) ci

n l ∑ ∑

(k) (kh)

ci ai

[(

(kh)

2lai

( )⎞ βj(k) 2 i ⎟ (k) ⎠ |xi (t)| (k)

(k) j=1 cj

∑n

+

ci

∑n

⎝

2

ci

h=1

i=1

+

(k) (k) j=1 ηj cj

∑n

2

(

(k) j=1 cj

∑n

+

ξj(k)

) i

(k) ci

+

l ∑

⎞ (kh)

ai

(

(hk)

2lai

2 ⎟ + 1 ⎠ |x(k) i (t − τk )|

)

h=1

(26)

) ( ) ] 2 2 + 1 |xi(h) (t − τh )| − 2la(hk) + 1 |x(k) i i (t − τk )|

i=1 h=1

≜−

n ∑

(k)

(k)

(k)

(k)

ci γi Vi (xi (t), t) +

(k)

(k) (k)

(k)

ci εi Vi (xi (t − τk ), t − τk )

i=1

i=1 n

+

n ∑

l

∑

(k)

ci

i=1

∑

(kh) (kh) (k) Fi (xi (t

ai

h=1 (k)

By condition (C3), γi (kh) (G , (ai )l×l ), there is

[

∑

− τk ), xi(h) (t − τh )).

(kh)

Fi

(k)

> εi(k) for 1 ≤ k ≤ l, 1 ≤ i ≤ n. Clearly, along each directed cycle C (i) of balanced digraph (h)

(hk)

(xi (t − τk ), xi (t − τh )) + Fi

(h)

(k)

]

(xi (t − τh ), xi (t − τk )) = 0,

(h,k)∈E(C (i)) (k)

(h)

for all xi , xi

∈ Rmi . Till here, all the conditions in Corollary 2 have been checked, so this conclusion holds of course. □ (k)

(k) 2

Remark 4. In Theorem 2, we find a set of Lyapunov functions {Vi = |xi | , 1 ≤ k ≤ l, 1 ≤ i ≤ n} satisfying conditions (A1), (B2) and (A3′ ). Meanwhile, we have obtained another kind of exponential stability criterion for the trivial solution of SDCNMW (3) based on Corollary 2. The stability criterion in Theorem 2 is shown with coefficients of SDCNMW (3), called as coefficients-type theorem, which can be checked easily in the real applications. (kh)

Remark 5. We know from (C2) that there are n kinds of weights. Hence, there exist n digraphs (G , Ai ), Ai = (ai )l×l , 1 ≤ i ≤ n. Clearly, the graph-theoretic method appeared in the existing literature [22,36] is not applicable any more since what they used in their paper is based on a single digraph. So, a novel systematic multi-graph theoretic method is employed to study the stability of stochastic delayed complex networks with multiple weights. 4. Application to stochastic oscillators network In this section, we will apply the main results in Section 3 to discuss the stability for stochastic oscillators network with two weights. 4.1. Stability analysis In physics, mechanics and engineering, one of the commonest examples is as follow: x¨ (t) + θ x˙ (t) + x(t) = 0, t ≥ 0,

(27)

where θ ≥ 0 is the damping coefficient. Let y(t) = x˙ (t) + ωx(t), ω > 0. Then (27) can be rewritten as x˙ (t) = y(t) − ωx(t), y˙ (t) = (ω − θ )y + ω(θ − ω)x(t) − x(t). Some coupled oscillators on networks have been studied as models of self-excited systems [22,43].

(28)

14

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

By using (28) as the vertex system and taking the delay and noise into consideration, we get stochastic delayed oscillators network with weights A and B, A = (akh )l×l and B = (bkh )l×l :

[ dxk (t) = yk (t) − ωxk (t) +

l ∑

] akh xh (t − τh ) dt + gk (xk (t), xk (t − τk ), t)dW (t), 1 ≤ k ≤ l,

h=1

[ dyk (t) = (ω − θk ) yk (t) + ω (θk − ω) xk (t) − xk (t) +

l ∑

(29)

] bkh yh (t − τh ) dt + hk (yk (t), yk (t − τk ), t)dW (t).

h=1

Without loss of generality, assume that akh ≥ 0, bkh ≥ 0. Otherwise, we use |akh | and |bkh | instead of akh and bkh , respectively. Assume that gk (0, 0, t) = hk (0, 0, t) = 0. It is straightforward to see that (29) has a trivial solution Υ ∗ = (x1 , y1 , . . . , xl , yl )T = 0 corresponding to the initial condition (x1 (0), y1 (0), . . . , xl (0), yl (0))T = 0. We first split the stochastic delayed oscillators network with weights A and B into two single weighted complex networks (G , A) and (G , B). Denote dk and ek as the cofactor of the kth diagonal element of the Laplacian matrix of (G , A) and (G , B), respectively. ( ) Denote x(k) (t) =

(k)

(k)

x1 (t), x2 (t)

T

= (xk (t), yk (t))T . Compared to (3), we can easily get that f1(k) (x(k) (t), x(k) (t −

τk ), t) = yk (t) − ωxk (t), f2(k) (x(k) (t), x(k) (t − τk ), t) = (ω − θk ) yk (t) + ω (θk − ω) xk (t) − xk (t), g1(k) (x(k) (t), x(k) (t − (kh) (h) (k) τk ), t) = gk (xk (t), xk (t − τk ), t), g2(k) (x(k) (t) ( , x (t − τ)k ), t) = h(k (yk (t), yk (t )− τk ), t), H1 (x1 )(t − τh ) = akh xh (t − τh ), (kh) (h) (kh) (h) (kh) (h) x2 (t − τh ) = 0. For simplicity, we always write them x1 (t − τh ) = N2 H2 (x2 (t − τh )) = bkh yh (t − τh ), N1 (k)

(k)

(k)

(kh)

(k)

as f1 , f2 , g1 , g2 , H1 noise.

, H2(kh) , N1(kh) , N2(kh) , respectively. Let us now make an assumption for the intensity of white

(D1) There are constants αk , α¯ k , ρk and ρ¯ k such that

|gk (xk , uk )|2 ≤ αk |xk |2 + α¯ k |uk |2 ; |hk (yk , vk )|2 ≤ ρk |yk |2 + ρ¯ k |vk |2 . (k)

(k)

∑l

For brevity, denote γ1 = ω − 2αk − h=1 akh − (|ω(θk − ω) − 1|+ 2ρk )ek /dk ; γ2 = 2θk − 2ω − 2ρk −|ω(θk − ω) − 1|− ∑ ∑ (k) ¯ k + 2ρ¯ k ek /dk + lh=1 akh (2lahk + 1); ε2(k) = 2ρ¯ k + 2α¯ k dk /ek + lh=1 bkh (2lbhk + 1). h=1 bkh − (1 + 2ωαk )dk /(ω ek ); ε1 = 2α

∑l

(k)

Theorem 3. Let digraphs (G , A) and (G , B) be strongly connected. Suppose that (D1) is satisfied and γi i = 1, 2, 1 ≤ k ≤ l. Then the trivial solution of (29) is exponentially stable in mean square.

> εi(k) for

Proof. In the following, we just check the conditions of Theorem 2. First, choosing ε such that ε 2 = ω and then we can obtain that (k)

xk f 1

(k)

yk f2

1

y2k

1 1 2 − ωx2k = − ωx2k + y ; 2ε 2 2 2ω k = (ω − θk )y2k + (ω(θk − ω) − 1)xk yk

= xk yk − ωx2k ≤

≤

2

|ω(θk − ω) − 1| 2

ε 2 x2k +

x2k

( +

|ω(θk − ω) − 1| 2

2

2

2

2

2

2

)

− θk + ω y2k ;

|g1(k) | ≤ αk |x(k) (t)| + α¯ k |x(k) (t − τk )| ; |g2(k) | ≤ ρk |x(k) (t)| + ρ¯ k |x(k) (t − τk )| ; |H1(kh) | ≤ akh |xh (t − τh )|; |H2(kh) | ≤ bkh |yh (t − τh )|. (k)

(k)

(k)

(k)

Additionally, γ1 > ε1 , γ2 > ε2 for 1 ≤ k ≤ l. Till here, all the conditions of Theorem 2 has been successfully verified. This implies immediately that the origin of (29) is exponentially stable in mean square. □ Remark 6. In general, the stability of stochastic linear system is not too complicated. However, the dimension of coupled systems is always very large, which makes its stability analysis is not easy any more. Additionally, time delays, multiple weights and stochastic perturbations are taken into account in (29). Hence, it becomes much more difficult to do the stability analysis of (29). In this case, Theorem 3 presents a new insight to study the stability of this kind of systems. Furthermore, we can obtain the similar stability results if the damping coefficient is time-varying.

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

15

Fig. 3. The topology map for (29) in four oscillators with two weights and its split.

4.2. Numerical simulations

For better understanding of oscillators network, we provide numerical simulations for coupled oscillators on digraph (G , A) and (G , B), where two weighted matrices A and B are as follows:

⎛

0 ⎜ 0.01 A=⎝ 0 0

0 0 0.01 0

0 0 0 0.01

0.01 0 ⎟ , 0 ⎠ 0

⎞

⎛

0 ⎜ 0.01 B=⎝ 0 0

0.02 0 0.02 0

0 0 0 0.02

⎞

0 0.01 ⎟ . 0 ⎠ 0

In this case, the topology map for (29) with two weights and its split are shown in Fig. 3. Let ω = 0.5, θk = 2.5 and τk = 0.1 (1 ≤ k ≤ 4). The noise terms of (29) are gk (xk , uk , t) = 0.01xk + 0.01uk sin t , hk (yk , vk , t) = 0.01yk + 0.01vk cos t .

∑4

∑4

Clearly, (0, 0, 0, 0, 0, 0, 0, 0)T is the trivial solution of (29) in our setting. h=1 |akh | = 0.01 and h=1 |bkh | = 0.02, αk = α¯ k = ρk = ρ¯ k = 0.0002, dk = 0.013 (1 ≤ k ≤ 4) and ek = 4 × 10−6 (k = 1, 3, 4), e2 = 0.023 . Then we can calculate that

γ1(1) = 0.488, ε1(1) = 0.012, γ2(1) = 3.4795, ε2(1) = 0.0221, γ1(2) = 0.4864, ε1(2) = 0.0136, γ2(2) = 3.72955, ε2(2) = 0.02205, γ1(3) = 0.488, ε1(3) = 0.012, γ2(3) = 3.4795, ε2(3) = 0.0205, γ1(4) = 0.488, ε1(4) = 0.012, γ2(4) = 3.4795, ε2(1) = 0.0205. (k)

(k)

Hence, γi > εi for i = 1, 2, 1 ≤ k ≤ 4. It follows by Theorem 3 that the trivial solution of (29) with l = 4 is exponential stable in mean square. Numerical simulation is provided in Fig. 4, in which the initial values are x1 (t) = −0.5 sin t , y1 (t) = −0.5 cos t − 0.25 sin t , x2 (t) = 0.3 cos t , y2 (t) = −0.3 sin t + 0.15 cos t , x3 (t) = −0.2 cos t , y3 (t) = 0.2 sin t − 0.1 cos t , x4 (t) = 0.5 sin t , y4 (t) = 0.5 cos t + 0.25 sin t.

16

C. Zhang and B.-S. Han / Physica A 538 (2020) 122827

Fig. 4. The second moment of solution to system (29).

5. Conclusions and discussions In this paper, we first split SDCNMW into multiple single weighted stochastic delayed complex networks. Then, Razumikhin-type technique and graph-theoretic method are effectively melted into classic Lyapunov method to investigate the exponential stability of SDCNMW. The developed systematic method can be used to study the stability of other stochastic complex networks with multi-weights. In [11,12], the method of network split is used to effectively investigate the complex networks with multi-weights. But the effect of noise perturbations to complex networks is not considered therein. In [36,37], Razumikhin-type technique, graph-theoretic method and classic Lyapunov method have been combined to study the stability of stochastic functional differential equations on single weighted network. Complex networks with multi-weights are important as mentioned in Introduction, but only few papers have studied it due to the difficult of mathematics [28,44]. It should be pointed out that [28] aimed to investigate the stability of coupled reaction–diffusion systems on networks without stochastic perturbations and [44] only considered stochastic stability for complex networks with proportional delay. Compared to relative work in the literature, multiple weights, time delays and stochastic perturbations are all taken into account in complex networks in this paper. Moreover, the method of network split, Razumikhin technique and systematic multi-digraph theoretic method are successfully combined to study the exponential stability of our model. Therefore, model considered in this paper is more general and research method is novel. In our future work, we aims to study the stability and synchronization of multiple weighted stochastic complex networks with Lévy noise or Markov switching or semi-Markov switching [45–47]. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 11601445, 11801470) and the Fundamental Research Funds for the Central Universities, PR China (No. 2682018CX60). References [1] Q. Zhu, J. Cao, R. Rakkiyappan, Exponential input-to-state stability of stochastic Cohen–Grossberg neural networks with mixed delays, Nonlinear Dynam. 79 (2015) 1085–1098. [2] P. Wang, X. Wang, H. Su, Stability analysis for complex-valued stochastic delayed networks with Markovian switching and impulsive effects, Commun. Nonlinear Sci. Numer. Simul. 73 (2019) 35–51. [3] K. Shi, S. Zhong, H. Zhu, X. Liu, Y. Zeng, New delay-dependent stability criteria for neutral-type neural networks with mixed random time-varying delays, Neurocomputing 168 (2015) 896–907. [4] Y. Zhao, J. Kurths, L. Duan, Input-to-state stability analysis for memristive BAM neural networks with variable time delays, Phys. Lett. A 383 (2019) 1143–1150. [5] K. Shi, J. Wang, Y. Tang, S. Zhong, Reliable asynchronous sampled-data filtering of T–S fuzzy uncertain delayed neural networks with stochastic switched topologies, Fuzzy Set. Syst. (2019) http://dx.doi.org/10.1016/j.fss.2018.11.017. [6] Z. Wang, X. Liu, Exponential stability of impulsive complex-valued neural networks with time delay, Math. Comput. Simulation 156 (2019) 143–157. [7] Y. Muroya, Y. Enatsu, T. Kuniya, Global stability of extended multi-group sir epidemic models with patches through migration and cross patch infection, Acta Math. Sci. 33 (2013) 341–361.

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