噪聲擾動(dòng)下時(shí)滯復(fù)雜網(wǎng)絡(luò)動(dòng)力學(xué)參數(shù)及拓?fù)浣Y(jié)構(gòu)辨識(shí)

噪聲擾動(dòng)下時(shí)滯復(fù)雜網(wǎng)絡(luò)動(dòng)力學(xué)參數(shù)及拓?fù)浣Y(jié)構(gòu)辨識(shí)

衛(wèi)亭1，楊曉麗1，孫中奎2

(1.陜西師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,西安710062；2.西北工業(yè)大學(xué)應(yīng)用數(shù)學(xué)系，西安710072)

摘要：針對(duì)隨機(jī)噪聲及時(shí)間滯后普遍存在于耦合網(wǎng)絡(luò)，且其結(jié)構(gòu)往往未知或部分未知問(wèn)題，基于網(wǎng)絡(luò)間隨機(jī)廣義投影滯后同步原理，通過(guò)合理設(shè)計(jì)控制器與自適應(yīng)更新規(guī)則，構(gòu)建辨識(shí)網(wǎng)絡(luò)模型未知?jiǎng)恿W(xué)參數(shù)及拓?fù)浣Y(jié)構(gòu)的識(shí)別方案；結(jié)合隨機(jī)時(shí)滯微分方程LaSalle型不變性原理，從數(shù)學(xué)上嚴(yán)格證明識(shí)別方案的準(zhǔn)確性。通過(guò)具體網(wǎng)絡(luò)模型，借助計(jì)算仿真驗(yàn)證識(shí)別方案的有效性。數(shù)值模擬結(jié)果表明，網(wǎng)絡(luò)未知?jiǎng)恿W(xué)參數(shù)及拓?fù)浣Y(jié)構(gòu)不但能準(zhǔn)確辨識(shí)，且識(shí)別方案不依賴(lài)耦合時(shí)滯、更新增益及網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu)等選取。

關(guān)鍵詞：網(wǎng)絡(luò)結(jié)構(gòu)識(shí)別；網(wǎng)絡(luò)同步；耦合時(shí)滯；隨機(jī)噪聲

中圖分類(lèi)號(hào):O322文獻(xiàn)標(biāo)志碼：A

基金項(xiàng)目：國(guó)家自然科學(xué)基金資助項(xiàng)目(11372081)

收稿日期：2014-09-22修改稿收到日期：2014-11-06

基金項(xiàng)目：國(guó)家自然科學(xué)基金重大研究計(jì)劃重點(diǎn)支持項(xiàng)目“古建木構(gòu)的狀態(tài)評(píng)估、安全極限與性能保持”(51338001)；北京交通大學(xué)人才基金項(xiàng)目(2014RC011)

收稿日期：2014-07-29修改稿收到日期：2014-10-17

Identification of system parameters and network topology of delay-coupled complex networks under circumstance noise

WEITing1,YANGXiao-Li1,SUNZhong-kui2(1. Shanxi Normal University, Xi’an 710062, China; 2. Northwestern Polytechnical University, Xi’an 710072, China)

Abstract:Noting that the random noise and time delay are prevalent in complex networks and the topology of a network is often unknown or partially unknown, based on the principle of random generalized projective lag synchronization, an approach was proposed to estimate the system parameters and topological structure of delay-coupled complex networks under circumstance noise. By constructing an appropriate controller and adaptive updating rules, the unknown network parameters and topological structure of the concerned networks were identified simultaneously. The accuracy of the method was rigorously proved by the LaSalle-type theorem for stochastic differential delay equations．An example of network with chaotic oscillator was provided to illustrate the method. The numerical results indicate that the unknown network parameters and topological structure can be accurately identified, and yet the proposed method is robust against the time delay, the update gain and the network topology.

Key words:topology identification; network synchronization; coupled delay; random noise

在網(wǎng)絡(luò)結(jié)構(gòu)已知條件下，有關(guān)其統(tǒng)計(jì)特征(如平均路徑長(zhǎng)度、度分布、聚類(lèi)系數(shù))的動(dòng)力學(xué)行為及控制、網(wǎng)絡(luò)結(jié)構(gòu)對(duì)動(dòng)力學(xué)影響等獲得廣泛研究[1-7]。然而，由于各種因素的不確定性，諸多(如蛋白質(zhì)相互作用、生物神經(jīng)、電力等)網(wǎng)絡(luò)動(dòng)力學(xué)參數(shù)或拓?fù)浣Y(jié)構(gòu)往往未知或部分未知，因此，辨識(shí)網(wǎng)絡(luò)模型的動(dòng)力學(xué)參數(shù)及拓?fù)浣Y(jié)構(gòu)在復(fù)雜動(dòng)力網(wǎng)絡(luò)研究中具有重要的理論意義與應(yīng)用價(jià)值。

Yu等[8]首次提出利用網(wǎng)絡(luò)動(dòng)力學(xué)演化信息，通過(guò)構(gòu)造含控制器的新網(wǎng)絡(luò)，并基于兩網(wǎng)絡(luò)間完全同步追蹤原網(wǎng)絡(luò)的拓?fù)浣Y(jié)構(gòu)。該思想在具有相同[9-11]、不同節(jié)點(diǎn)動(dòng)力學(xué)[12-13]網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu)及動(dòng)力學(xué)參數(shù)識(shí)別中廣泛研究。文獻(xiàn)[14]采用攝動(dòng)法反演網(wǎng)絡(luò)模型的連接矩陣，由于信息傳遞速度的有限性、交通堵塞及腦神經(jīng)系統(tǒng)與其它網(wǎng)絡(luò)系統(tǒng)信息傳播路徑長(zhǎng)短不同，網(wǎng)絡(luò)節(jié)點(diǎn)之間進(jìn)行信息傳輸時(shí)存在時(shí)間滯后。因此，對(duì)網(wǎng)絡(luò)結(jié)構(gòu)識(shí)別研究需拓展到含耦合時(shí)滯的復(fù)雜網(wǎng)絡(luò)情形?；诰W(wǎng)絡(luò)間完全同步，文獻(xiàn)[15-21]通過(guò)設(shè)計(jì)自適應(yīng)反饋控制器，分別研究含常數(shù)時(shí)滯或時(shí)變時(shí)滯、節(jié)點(diǎn)動(dòng)力學(xué)相同或不同、加權(quán)網(wǎng)絡(luò)或等權(quán)復(fù)雜網(wǎng)絡(luò)的動(dòng)力學(xué)參數(shù)或拓?fù)浣Y(jié)構(gòu)識(shí)別。網(wǎng)絡(luò)間其它同步類(lèi)型如投影同步[22]、超前同步[23]及滯后同步[24]在時(shí)滯耦合網(wǎng)絡(luò)結(jié)構(gòu)識(shí)別中也發(fā)揮重要作用。而基于最優(yōu)化法[25]、穩(wěn)態(tài)控制法[26]也用于時(shí)滯復(fù)雜網(wǎng)絡(luò)的拓?fù)浣Y(jié)構(gòu)識(shí)別研究。因現(xiàn)實(shí)網(wǎng)絡(luò)系統(tǒng)會(huì)受隨機(jī)噪聲影響，如何識(shí)別復(fù)雜網(wǎng)絡(luò)的拓?fù)浣Y(jié)構(gòu)及動(dòng)力學(xué)參數(shù)仍為具有挑戰(zhàn)性的前沿課題。對(duì)含噪聲的網(wǎng)絡(luò)模型, Ren等[27]通過(guò)定義網(wǎng)絡(luò)節(jié)點(diǎn)動(dòng)力學(xué)信息間相關(guān)性,推導(dǎo)信息相關(guān)矩陣與決定網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu)的Laplacian矩陣關(guān)系。吳曉群等[28]通過(guò)設(shè)計(jì)自適應(yīng)反饋控制器，基于網(wǎng)絡(luò)間隨機(jī)同步研究節(jié)點(diǎn)動(dòng)力學(xué)含隨機(jī)噪聲的復(fù)雜動(dòng)力網(wǎng)絡(luò)模型拓?fù)浣Y(jié)構(gòu)識(shí)別。文獻(xiàn)[29-33]采用基于最優(yōu)化法、格林因果檢驗(yàn)法、ROC曲線分析法及反復(fù)性理論法研究噪聲擾動(dòng)下復(fù)雜網(wǎng)絡(luò)的拓?fù)浣Y(jié)構(gòu)識(shí)別。

考慮隨機(jī)噪聲及耦合時(shí)滯普遍存在于復(fù)雜網(wǎng)絡(luò)，尤其利用網(wǎng)絡(luò)間同步原理反演網(wǎng)絡(luò)結(jié)構(gòu)時(shí)，耦合網(wǎng)絡(luò)亦會(huì)受時(shí)間滯后、隨機(jī)噪聲影響。針對(duì)噪聲擾動(dòng)下含耦合時(shí)滯的復(fù)雜網(wǎng)絡(luò)模型，采用反復(fù)性理論法[34]、時(shí)滯反饋控制法[35]及信息論法[36]研究網(wǎng)絡(luò)模型的拓?fù)浣Y(jié)構(gòu)識(shí)別。而基于同步法辨識(shí)噪聲擾動(dòng)下時(shí)滯復(fù)雜網(wǎng)絡(luò)模型的動(dòng)力學(xué)參數(shù)及網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu)研究較少。本文構(gòu)建含耦合時(shí)滯及噪聲擾動(dòng)的驅(qū)動(dòng)-響應(yīng)網(wǎng)絡(luò)模型，通過(guò)合理設(shè)計(jì)控制器與自適應(yīng)更新規(guī)則，基于網(wǎng)絡(luò)間隨機(jī)廣義投影滯后同步辨識(shí)網(wǎng)絡(luò)模型的動(dòng)力學(xué)參數(shù)與拓?fù)浣Y(jié)構(gòu)，并數(shù)值仿真驗(yàn)證理論推理的有效性。

1網(wǎng)絡(luò)模型及預(yù)備知識(shí)

1.1網(wǎng)絡(luò)模型

考慮含N個(gè)節(jié)點(diǎn)的一般復(fù)雜動(dòng)力網(wǎng)絡(luò)，其動(dòng)力學(xué)方程為

(1)

式中：xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn為第i個(gè)節(jié)點(diǎn)狀態(tài)變量；f∈Rn×1，F(xiàn)∈Rn×m1為光滑向量函數(shù)及矩陣函數(shù)；ξ∈Rm1為未知或不確定的動(dòng)力學(xué)參數(shù)； B=(bij)N×N為耦合矩陣，表示未知或不確定的網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu)，bij定義為：若從節(jié)點(diǎn)i到節(jié)點(diǎn)j有一個(gè)連接，則bij≠0，否則，bij=0；?！蔙n×n為決定變量間相互關(guān)系的內(nèi)部耦合矩陣；τ(t)為網(wǎng)絡(luò)內(nèi)部節(jié)點(diǎn)間時(shí)變耦合時(shí)滯。

將方程(1)作為驅(qū)動(dòng)網(wǎng)絡(luò)，構(gòu)建響應(yīng)網(wǎng)絡(luò)為

dyi(t)=[g(yi(t))+G(yi(t))θ+

σi(yi(t)-γxi(t-δ),yi(t-τ(t))-

γxi(t-τ(t)-δ),t)dW(t)，(i=1,2,…,N)

(2)

式中：yi(t)=(yi1(t),yi2(t),…,yin(t))T∈Rn為響應(yīng)網(wǎng)絡(luò)第i個(gè)節(jié)點(diǎn)狀態(tài)變量；g∈Rn×1，G∈Rn×m2為光滑向量函數(shù)及矩陣函數(shù)，其中驅(qū)動(dòng)網(wǎng)絡(luò)與響應(yīng)網(wǎng)絡(luò)可具有不同節(jié)點(diǎn)動(dòng)力學(xué)；θ∈Rm2為未知或不確定的動(dòng)力學(xué)參數(shù)； D=(dij)N×N為耦合矩陣，表示對(duì)耦合矩陣B的估計(jì)；Ui(t)∈Rn為控制器；δ為網(wǎng)絡(luò)間耦合時(shí)滯；σi:Rn×Rn×R+→Rn×m為噪聲強(qiáng)度函數(shù)；W(t)=(w1(t),w2(t),…,wm(t))T∈Rm為定義在完備概率空間(Ω,H,P)的m維布朗運(yùn)動(dòng)，σidW(t)用于刻畫(huà)響應(yīng)網(wǎng)絡(luò)與驅(qū)動(dòng)網(wǎng)絡(luò)的耦合過(guò)程中，會(huì)受外界環(huán)境浮動(dòng)、耦合強(qiáng)度設(shè)計(jì)不精確性等不確定因素影響。本文設(shè)m=1。

1.2預(yù)備知識(shí)

引理對(duì)任意向量x，y∈Rn及正定矩陣Q∈Rn×n，有2xTy≤xTQx+yTQ-1y成立。

2網(wǎng)絡(luò)動(dòng)力學(xué)參數(shù)及拓?fù)浣Y(jié)構(gòu)識(shí)別方案

通過(guò)設(shè)計(jì)適當(dāng)?shù)目刂破髋c自適應(yīng)更新規(guī)則,基于網(wǎng)絡(luò)間隨機(jī)廣義投影滯后同步,研究網(wǎng)絡(luò)模型的動(dòng)力學(xué)參數(shù)與拓?fù)浣Y(jié)構(gòu)識(shí)別。

定理對(duì)驅(qū)動(dòng)-響應(yīng)網(wǎng)絡(luò)模型式(1)、(2),在假設(shè)1-3下，所用控制器及自適應(yīng)更新規(guī)則為

(3)

(4)

(5)

(6)

(7)

dei(t)=dyi(t)-γdxi(t-δ)=

ki(t)ei(t)]dt+σi(ei(t),ei(t-τ(t)),t)dW(t)

(8)

建立V函數(shù)為

式中：V∈C1,2(R+;G)，G=Rm1+m2+N2+N+nN ；k*為可確定的足夠大正常數(shù)。

(9)

由假設(shè)1知

peTi(t)ei(t)

式(9)進(jìn)一步變?yōu)?/p>

令e(t)=(eT1(t),eT2(t),…,eTN(t))T∈RnN，A=B?Γ，將lV寫(xiě)成緊積形式為

peT(t)e(t)+qeT(t-τ(t))e(t-τ(t))

(10)

由引理知

將式(10)改寫(xiě)為

qeT(t-τ(t))e(t-τ(t))?-ω1(x)+ω2(y)

據(jù)隨機(jī)時(shí)滯微分方程LaSalle型不變性原理[37-38]，可得

由假設(shè)3、更新規(guī)則(7)、狀態(tài)誤差系統(tǒng)(8)，得

0,ki(t)=const,ei(t)=0}

至此，在控制器、更新規(guī)則作用及幾乎必然漸近穩(wěn)定性意義下，網(wǎng)絡(luò)模型未知?jiǎng)恿W(xué)參數(shù)與拓?fù)浣Y(jié)構(gòu)能得到正確識(shí)別、反饋強(qiáng)度能自適應(yīng)調(diào)整到常數(shù)、驅(qū)動(dòng)網(wǎng)絡(luò)與響應(yīng)網(wǎng)絡(luò)間亦能實(shí)現(xiàn)隨機(jī)廣義投影滯后同步。

3數(shù)值仿真

對(duì)具體網(wǎng)絡(luò)系統(tǒng)進(jìn)行數(shù)值仿真，驗(yàn)證推論推理的有效性。設(shè)驅(qū)動(dòng)網(wǎng)絡(luò)局部動(dòng)力學(xué)為四維超混沌系統(tǒng)[39],節(jié)點(diǎn)數(shù)N=6,構(gòu)造含未知?jiǎng)恿W(xué)參數(shù)及未知拓?fù)浣Y(jié)構(gòu)的驅(qū)動(dòng)網(wǎng)絡(luò)為

dxi(t)=[f(xi(t))+F(xi(t))ξ+

式中：

構(gòu)造含未知?jiǎng)恿W(xué)參數(shù)的響應(yīng)網(wǎng)絡(luò)[40]為

dyi(t)=[g(yi(t))+G(yi(t))θ+

σi(yi(t)-γxi(t-δ),yi(t-τ(t))-

γxi(t-τ(t)-δ),t)dW(t)，(i=1,2,…,6)

式中：

選噪聲強(qiáng)度函數(shù)為

σi(ei(t),ei(t-τ(t)),t)=(ei1(t)+ei1(t-τ(t)),

ei2(t)+ei2(t-τ(t)),ei3(t)+ei3(t-τ(t)),

ei4(t)+ei4(t-τ(t)))T

易驗(yàn)證該函數(shù)滿(mǎn)足假設(shè)1,即

eTi(t-τ(t))ei(t-τ(t))

數(shù)值仿真時(shí)，為使網(wǎng)絡(luò)節(jié)點(diǎn)動(dòng)力學(xué)呈混沌行為，驗(yàn)證所提識(shí)別方案的有效性，選動(dòng)力學(xué)參數(shù)為

ξ=(a1,b1,c1,k1,g1)T=(35,3,35,-8,-10)T

θ=(a2,b2,c2,k2)T=(-1.0,0.25,0.5,0.05)T

選決定驅(qū)動(dòng)網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu)的耦合矩陣為

為進(jìn)一步刻畫(huà)驅(qū)動(dòng)網(wǎng)絡(luò)與響應(yīng)網(wǎng)絡(luò)間的同步動(dòng)力學(xué)，引入兩網(wǎng)絡(luò)間同步總誤差為

式中：wk∈Ω；h為樣本軌道。

取內(nèi)部耦合矩陣Γ=diag(1,0,0,0)、網(wǎng)絡(luò)內(nèi)部與網(wǎng)絡(luò)之間的耦合時(shí)滯分別為τ(t)=0.02與δ=0.03、比例因子γ=2.0、更新增益λi=30.0，樣本軌道h=10。易驗(yàn)證構(gòu)造的驅(qū)動(dòng)-響應(yīng)網(wǎng)絡(luò)模型滿(mǎn)足假設(shè)3。

圖1　響應(yīng)網(wǎng)絡(luò)耦合矩陣d ij(t)的演化曲線 Fig.1 The evolution of the topological structure d ij(t) of response network

圖2　網(wǎng)絡(luò)未知?jiǎng)恿W(xué)參數(shù)的估計(jì)值 ξ(t)與θ(t)的演化曲線 Fig.2 The evolution of the unknown parameter’s estimation ξ(t) and θ(t)

圖3　反饋強(qiáng)度k i(1≤i≤6)的演化曲線 Fig.3 The evolution of the feedback strength k i(1≤i≤6)

圖4　不同更新增益λ i下網(wǎng)絡(luò)同步總誤差Δ(t)演化曲線 Fig.4 The evolution of the total synchronization error Δ(t) for different λ i

圖5　不同耦合時(shí)滯δ下網(wǎng)絡(luò)同步總誤差Δ(t)演化曲線 Fig.5 The evolution of the total synchronization errorΔ(t) for different δ

圖6　響應(yīng)網(wǎng)絡(luò)耦合矩陣d ij(t)的演化曲線 Fig.6 The evolution of the topological structure d ij(t) of response network

由6圖計(jì)算結(jié)果看出，隨時(shí)間演化，dij(t)仍能分別收斂到預(yù)設(shè)的bij，即驅(qū)動(dòng)網(wǎng)絡(luò)耦合矩陣B通過(guò)響應(yīng)網(wǎng)絡(luò)耦合矩陣D得到正確識(shí)別。

4結(jié)論

(1)針對(duì)噪聲擾動(dòng)下含耦合時(shí)滯的驅(qū)動(dòng)-響應(yīng)網(wǎng)絡(luò)模型，基于網(wǎng)絡(luò)間隨機(jī)廣義投影滯后同步原理，提出辨識(shí)網(wǎng)絡(luò)未知?jiǎng)恿W(xué)參數(shù)及拓?fù)浣Y(jié)構(gòu)的研究方案。

(2)通過(guò)設(shè)計(jì)合理的控制器及自適應(yīng)更新規(guī)則，利用隨機(jī)時(shí)滯微分方程的LaSalle型不變性原理，嚴(yán)格證明識(shí)別方案不僅能使網(wǎng)絡(luò)未知?jiǎng)恿W(xué)參數(shù)及拓?fù)浣Y(jié)構(gòu)得到正確識(shí)別，亦能使驅(qū)動(dòng)網(wǎng)絡(luò)、響應(yīng)網(wǎng)絡(luò)在幾乎必然漸近穩(wěn)定性意義下實(shí)現(xiàn)隨機(jī)廣義投影滯后同步。

(3)通過(guò)具體網(wǎng)絡(luò)模型，利用計(jì)算機(jī)仿真驗(yàn)證理論推理的有效性，且數(shù)值模擬結(jié)果也表明識(shí)別方案對(duì)耦合時(shí)滯、更新增益、拓?fù)浣Y(jié)構(gòu)等的選取具有魯棒性。

參考文獻(xiàn)

[1]Arenas A, Díaz-Guilera A, Kurths J, et al. Synchronization in complex networks[J]. Physics Reports, 2008, 469:93-153.

[2]Kocarev L, Amato P. Synchronization in power-law networks [J].Chaos, 2005, 15:24101.

[3]Newman M E J. The structure and function of complex networks[J]. SIAM Review, 2003, 45(2):167-256.

[4]Watts D J, Strogatz S H. Collective dynamics of small-world networks[J].Nature, 1998, 6684 (393):440-442.

[5]Guimera R, Diaz-Guilera A, Vega-Redondo F, et al. Optimal network topologies for local search with congestion[J]. Physical Review Letters, 2002, 89(24):248701.

[6]Wang Xiao-nan, Lu Ying, Jiang Min-xi, et al. Attraction of spiral waves by localized inhomogeneities with small-world connections in excitable media[J].Physical Review E, 2004, 69:056223.

[7]Roxin X, Riecke H, Solla S A. Self-sustained activity in a small-world network of excitable neurons[J].Physical Review Letters, 2004, 92(19):198101.

[8]Yu D C, Righero M, Kocarev L. Estimating topology of networks[J]. Physical Review Letters, 2006, 97:188701.

[9]Zhou Jin, Lu Jun-an. Topology identification of weighted complex dynamical networks[J]. Physica A,2007,386:481-491.

[10]Xu Yu-hua, Zhou Wu-neng, Fang Jian-an, et al. Structure identification and adaptive synchronization of uncertain general complex dynamical networks[J].Physics Letters A, 2009, 374:272-278.

[11]Fan Chun-xia, Wan You-hong, Jiang Guo-ping. Topology identification for a class of complex dynamical networks using output variables[J]. Chinese Physics B, 2012, 21:020510.

[12]Xu Yu-hua, Zhou Wu-neng, Fang Jian-an, et al. Topology identification and adaptive synchronization of uncertain complex networks with adaptive double scaling functions[J]. Commum. Nonlinear Sci. Numer. Simulat., 2011, 16:3337-3343.

[13]Sun Zhi-yong, Si Gang-quan. Comment on“topology identification and adaptive synchronization of uncertain complex networks with adaptive double scaling functions”[J]. Commum. Nonlinear Sci. Numer. Simulat.,2012, 17: 3461-3463.

[14]Timme M. Revealing network connectivity from response dynamics[J].Physical Review Letters, 2007, 98:224101.

[15]Zhao Jun-chan, Li Qin, Lu Jun-an, et al. Topology identification of complex dynamical networks[J].Chaos, 2010, 20:23119.

[16]Wu Xiao-qun. Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay[J]. Physica A, 2008, 387: 997-1008.

[17]Guo Wan-li, Chen Shi-hua, Sun Wen. Topology identification of the complex networks with non-delayed and delayed coupling [J].Physics Letters A, 2009, 373:3724-3729.

[18]Xu Jiang, Zheng Song, Cai Guo-liang. Topology identification of weighted complex dynamical networks with non-delayed and time-varying delayed coupling[J].Chinese Journal of Physics, 2010, 48(4):482-492.

[19]Liu Hui, Lu Jun-an, Lü Jin-hu, et al. Structure identification of uncertain general complex dynamical networks with time delay[J].Automatica, 2009, 45:1799-1807.

[20]Xu Yu-hua, Zhou Wu-neng, Fang Jian-an. Topology identification of the modified complex dynamical networks with non-delayed and delayed coupling[J]. Nonlinear Dyn., 2012, 68:195-205.

[21]Wu Xiang-jun, Lu Hong-tao. Outer synchronization of uncertain general complex delayed networks with adaptive coupling[J]. Neurocomputing, 2012, 82:157-166.

[22]Zheng Song, Bi Qin-sheng, Cai Guo-liang. Adaptive projective synchronization in complex networks with time-varying coupling delay[J].Physics Letters A, 2009, 373:1553-1559.

[23]Che Yan-qiu, Li Rui-xue, Han Chun-xiao, et al. Topology identification of uncertain nonlinearly coupled complex networks with delays based on anticipatory synchronization [J].Chaos, 2013, 23:13127.

[24]Che Yan-qiu, Li Rui-xue, Han Chun-xiao, et al. Adaptive lag synchronization based topology identification scheme of uncertain general complex dynamical networks[J].Eur. Phys. J. B, 2012, 85:265.

[25]Tang Sheng-xue, Chen Li, He Yi-gang. Optimization-based topology identification of complex networks[J].Chinese Physics B, 2011, 20:110502.

[26]Yu Dong-chuan. Estimating the topology of complex dynamical networks by steady state control：Generality and Limitation[J]. Automatica, 2010, 46:2035-2040.

[27]Ren Jie, Wang Wen-xu, Li Bao-wen, et al. Noise bridges dynamical correlation and topology in complex oscillator networks[J]. Physical Review Letters, 2010, 104:58701.

[28]吳曉群,趙雪漪,呂金虎．節(jié)點(diǎn)動(dòng)力學(xué)含隨機(jī)噪聲的復(fù)雜動(dòng)力網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu)識(shí)別[C].第30屆中國(guó)控制會(huì)議, 2011.

[29]He Tao, Lu Xi-liang, Wu Xiao-qun, et al. Optimization-based structure identification of dynamical networks[J]. Physica A, 2013, 392:1038-1049.

[30]Wu Xiao-qun, Zhou Chang-song, Chen Guan-rong, et al. Detecting the topologies of complex networks with stochastic perturbations[J]. Chaos, 2011, 21:43129.

[31]Wu Xiao-qun, Wang Wei-han, Zheng Wei-xing. Inferring topologies of complex networks with hidden variables[J]. Physical Review E, 2012, 86:46106.

[32]Chen Juan, Lu Jun-an, Zhou Jin. Topology identification of complex networks from noisy time series using ROC curve analysis[J]. Nonlinear Dyn., 2014, 75:761-768.

[33]Nawrath J, Romano M C, Thiel M, et al. Distinguishing direct from indirect interactions in oscillatory networks with multiple time scales[J]. Physical Review Letters, 2010, 104:38701.

[34]Marwan N, Romano M C, Thiel M, et al. Recurrence plots for the analysis of complex systems[J]. Physics Reports, 2007, 438:237-329.

[35]Yu D C, Parlitz U. Inferring network connectivity by delayed feedback control[J]. Plos One, 2011,6(9):e24333.

[36]Pomped B, Runge J. Momentary information transfer as a coupling measure of time series[J].Physical Review E, 2011, 83:51122.

[37]Mao Xue-rong. Stochastic versions of the LaSalle theorem[J]. Journal of Differential Equations, 1999, 153:175-195.

[38]Mao Xue-rong. A note on the LaSalle-type theorems for stochastic differential delay equations[J]. Journal of Mathematical Analysis and Applications, 2002, 268:125-142.

[39]劉明華．一類(lèi)新超混沌系統(tǒng)的產(chǎn)生、同步及其應(yīng)用[D]．廣州：華南理工大學(xué)，2011.

[40]劉秉正,彭建華．非線性動(dòng)力學(xué)[M]．北京：高等教育出版社，2007:504.

[41]陸君安,呂金虎,劉慧,等．復(fù)雜動(dòng)力網(wǎng)絡(luò)結(jié)構(gòu)識(shí)別的某些新進(jìn)展[J]．復(fù)雜系統(tǒng)與復(fù)雜性科學(xué),2010,7(2/3):63-69.

LU Jun-an, Lü Jin-hu, LIU Hui, et al. New progress on structure identification of complex dynamical networks[J]. Complex Systems and Complexity Science, 2010,7(2/3): 63-69.

第一作者鄒廣平男，博士，教授，博士生導(dǎo)師，1963年生

第一作者高延安男，博士生，1986年生

通信作者楊慶山男，博士，教授，1968年生