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1、精品论文节点动力学参数影响复杂网络结构识别汤龙坤1,2 , 陆君安1, 吕金虎3, 吴晓群11 武汉大学数学与统计学院，武汉4300722 华侨大学数学科学学院，泉州3620213 中科院数学与系统科学研究院，北京 100190 摘要：本文以统一混沌系统为节点动力学，研究了两种固定内联矩阵和不同耦合强度下，动力 学参数的改变对复杂网络结构识别的影响。研究结果表明，在耦合强度足够小或足够大的情况 下，节点动力学参数的变动不影响网络拓扑识别的效果。足够小的耦合强度可以使得网络拓扑 结构完全识别，而足够大的耦合强度容易使得拓扑识别失败，完全不能识别。而对于某些耦合 强度，随着节点动力学参数的增大，网
2、络拓扑结构出现了完全不能识别到部分可识别再到完全 识别的过程。这些结果表明：（1）对于混沌振子为节点动力学的网络，小的耦合强度有利于拓 扑识别；（2）基于自适应同步技术的网络结构识别依赖于节点动力学，即使同一动力学不同的 参数对识别效率也会产生很大的影响；（3）完全内同步阻碍结构识别，更一般的，投影同步也 会阻碍网络拓扑识别。这些结果加深了我们对结构识别条件的深入理解和探索。 关键词：复杂网络; 结构识别; 同步；节点动力学参数；混沌吸引子中图分类号： O231.5; O241.8; O193.Impact of node dynamics parameters on topology ide
3、ntication of complex dynamical networksTANG Longkun1,2 , LU Jun-an1, L Jinhu3, WU Xiaoqun11 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China2 School of Mathematical Science, Huaqiao University, Quanzhou 362021, PR China3 Institute of Systems Science, Academy of Mathemat
4、ics and Systems Science, ChineseAcademy of Sciences, Beijing 100190, PR ChinaAbstract: This paper aims at investigating the topology identication problem of complex dynamical networks with varying node dynamics parameters and xed inner coupling matrices. In particular, by employing the unied chaotic
5、 system as node dynamics, this work further explores the inuence of continuously changing node dynamics parameters on基金项目： The Doctoral Fund of Ministry of Education of China (200804861072), the National Natural Science Foun- dation of China (11172215, 61174028)作 者 简 介：Tang Longkun （1977-）， male， ma
6、ster， major research direction： Complex networks.E-mail:tomlk. Lu Jun-an (1944-), male, professor, major research direction: Complex networks, nonlinear systems. Email:jalu. L Jinhu (1974-), male, professor, major research direction: Complex networks, complex systems. Email: jhlu. Correspondence aut
7、hor：Wu Xiaoqun（1978-），female，associate professor，major research direction： Complex networks, time series. Email: xqwu.- 10 -topology identication of complex dynamical networks with dierent coupling strengths. Results show that for suciently small or large coupling strengths, the performance of topol
8、ogy identication is not aected by the change of node parameters. Specically, for enough small coupling strengths, the topological structure can be completely identied regardless of the change of node parameters, while for suciently large coupling strengths, none of the connectivity (presence and abs
9、ence of connections) can be successfully identied. Furthermore, for some certain coupling strengths, with the increase of node dynamics parameters, the topology identication varies from completely unidentiable, to partially identiable, then to completely identiable. Therefore, the synchronization-ba
10、sed topology identication depends on node dynamics. Even for the same node dynamical model, dierent parameters can have a signicant impact on identication results. Furthermore, for networks consisting of chaotic oscillators as node dynamics, small coupling strengths are conducive to topology identic
11、ation. A broader conclusion is that projective synchronization, rather than just complete synchronization, is an obstacle to the network topology identication. The ndings in this paper will add to our understanding of conditions for identifying topologies of complex networks.Key words:Complex networ
12、k ; Topology identication ; Synchronization ; Node dynamics parameter ; Chaotic oscillator0 IntroductionThe study of complex dynamical networks has penetrated into various disciplines, from cell biology to ecology, from computer sciences to transportation, and from social sciences to humanities. Und
13、erstanding the interaction between a networks structure and its dynamics is a crucial issue in the research of complex networks. In recent years, there have been extensive investigations 1, 2, 3 studying the eect of network global structural properties (such as degree distribution, average shortest
14、path, clustering coecient, and so on) on network dynamics, under the premise of known topologies. However, in practice, the topology structure of a network is usually unknown or uncertain. For instance, the interaction between genes in gene regulatory networks is unclear, and the structure of the ne
15、urons in biological neural networks is also uncertain. Therefore, it is of great important to predict the connectivity between nodes via observed dynamical information from the networks considered.Since the synchronization-based method was presented by Yu et al. in 2006 4 for recover- ing network to
16、pologies, this topic has attracted wide interest. So far, many methods for topolo- gy identication have been developed 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, including the synchronization-based technique 4, 5, 6, 7, 8, 9, the perturbation method 10, the steady-state contro
17、l approach 11, and those on basis of time series 21, 22, 23, 24, 25.Among the various approaches, that based on adaptive synchronization 4, 5, 6, 7, 8, 9 is of high concern.If the study of synchronization and control of complex dynamical networks with known topological structures is a direct problem
18、, then topology identication of a network is an inverse one. Solving an inverse problem tends to be more challenging than a direct one. Generally speaking, the solution of an inverse problem is not unique, thus some additional constraints are needed to establish the correct result. Therefore, the ad
19、ditional constraints are especially critical to whether the network topology can be successfully identied. It is especially important to understand in depth and properly employ these conditions. In 8, Chen et al. pointed out that complete synchronization is an obstacle to topology identication based
20、 on adaptive synchronization. A natural question arises, then, whether or not non-complete synchronization leads to correct topology identication? In 18, Liu et al. addressed that the coupling strength and node dierence in an unknown complex network also aect the performance of topology identication
21、. Recently, our work 26 has studied the shifting of network synchronized states with the change of node dynamics parameters. As a result, a question arises as to how the varying node dynamics parameters aect topology identication of a network?The rest of this paper is arranged as follows. The method
22、 of topology identication based on adaptive synchronization and related theories are briey introduced in Section 2. Some fac- tors aecting topology identication, such as complete synchronization, the coupling strength, and node dierence are presented in Section 3. The inuence of successively varying
23、 node dynamics parameters on transitions of synchronous states and topology identication are in-vestigated in Section 4. Conclusions are nally drawn in Section 5.1Topology identication based on adaptivesynchronizationConsider a complex dynamical network consisting of N coupled oscillators:Nx i = Fi
24、(xi ) c Lij Hj (xj ), i = 1, 2, , N,j=1(1)where xi = (xi1 , xi2 , , xin )T Rn is the state vector of the ith oscillator. The networks topological structure is represented by the coupling matrix L = (Lij ) RNN , in which Lij = 0 if there is a connection from oscillator j to oscillator i, and Lij = 0
25、otherwise. It is noticed that the coupling matrix can be asymmetrical or non-dissipative. The parameter c 0 is the coupling strength, and Hj () : Rn Rn is the inner coupling function.To recover the topology of network (1), one can take (1) as the drive network and constructa response network as foll
26、ows:Nx i = Fi (xi ) c L ij Hj (xj ) + ui , i = 1, 2, , N, (2)j=1where xi = (xi1 , xi2 , , xin )T Rn is the state vector of the ith oscillator of the response network, ui Rn (i = 1, 2, , N ) is the adaptive controller to be designed, and L ij is an estimator of Lij (i, j = 1, 2, , N ). Denote xi = xi
27、 xi , and L ij = L ij Lij , then the error system isNx i =Fi (xi ) Fi (xi ) c L ij Hj (xj )j=1(3)N+ c Lij Hj (xj ) + ui , i = 1, 2, , Nj=1j=1i=1Assumption 1: It is assumed that the vector group Hj (xj ) Nis linearly independent on the synchronization manifold xi = xi N .Assumption 2: For node dynami
28、cs i (), it is assumed that there exists a non-negative constant K1 such thatFi (xi ) Fi (yi ) K1 xi yi ,i = 1, 2, , N. (4)Assumption 3: For the coupling function Hj (), it is assumed that there exists a non- negative constant K2 satisfyingHj (xj ) Hj (yj ) K2 xj yj ,j = 1, 2, , N. (5) According to
29、the synchronization-based identication technique, adaptive controllers aredesigned so that the constructed response network can synchronize the drive network, mean- while the unknown topological structure can be recovered.Theorem 1: Suppose that Assumptions 1 to 3 hold. Then the unknown conguration
30、matrix L of complex dynamical network (1) can be accurately estimated by L using the followingresponse network and adaptive controllersNx i = Fi (xi ) c j=1 L ij Hj (xj ) + ui(6)ui = ki xi ,k i = di xi , = xT H (x )2 Lijijjwhere i, j = 1, 2, , N . Namely, L ij Lij (i, j = 1, 2, , N ) as t .In fact,
31、from the adaptive controllers and the error system (3), one getsNN Hj (xj )L ij Lij = Hj (xj )Lij = 0.(7)j=1j=1According to Theorem 1, Lij Lij (i, j = 1, 2, , N ) as t .Remark 1: It should be noticed that the synchronization manifold in Assumption 1 means the outer synchronization between the drive
32、and response networks 27, 28, that is, xi xi (t ) for i = 1, 2, ., N .j=1Remark 2: If Hj (xj ) Nare not linearly independent on the synchronization manifoldNxi = xi i=1 , although L ij (i, j = 1, 2, ., N ) will asymptotically tends to a constant satisfying(7), it may not be the true value of Lij . T
33、his results in pseudo-identication of the network topology.2 Some factors aecting topology identication2.1 Complete synchronizationConsider the same coupling function for each oscillator, then the drive network (1) becomesNx i = Fi (xi ) c Lij H (xj ), i = 1, 2, , N. (8)j=1Under similar assumptions,
34、 by employing the following response network and controllersNx i = Fi (xi ) c j=1 L ij H (xj ) + ui ,(9)ui = ki xi ,k i = di xi , = xT H (x ),2 Liji jthe unknown topological structure L can be successfully estimated by L .Remark 3: Here, complete synchronization refers to the inner synchronization o
35、f all the oscillators in the drive network, namely xi = s, i = 1, 2, , N .Remark 4: Generally speaking, complete synchronization of the drive network makesthe topology unidentiable, and partial synchronization implies a part of the topology being unidentiable.As a matter of fact, when all the nodes
36、in the drive network reach a synchronized statexi = s, i = 1, 2, , N , the linear independence condition in Assumption 1 no longer holds. Thus, complete synchronization in a network is an obstacle to topology identication. Intuitively, when the drive network arrives at complete synchronization, the
37、oscillators can not be distinguished from one another, which leads to failure of identication.On the other hand, if a part of the drive network reaches synchronization, that is, x1 =x2 = = xm = s, i = 1, 2, , m, m N , then (7) can be rewritten asmH (s) Lij +j=1Nj=m+1H (xj )L ij = 0(10)ij ij ij j=1 i
38、jFurthermore, one can only get L= L L = 0 (j = m+ 1, , N ) and m L = 0 if H (s)and H (xj ) (j = m + 1, , N ) are linearly independent on the synchronization manifold. In other words, partial complete synchronization will lead to a part of topology being unidentiable 8.2.2 Node dierenceHere, node die
39、rence indicates the dierence of node dynamics, including dierent node dynamical models and dierent dynamical parameters in identical models. The bigger the node dierence, the shorter the time needed for identication, and the higher the identication eciency. This has been veried by Liu 18. Therein, t
40、he inner coupling matrix was set as an identity matrix, and the coupling strength remains unchanged. The L system with dynamical parameters (36, i2 , 3) was chosen as node dynamics, and i2 for the ith node was set to be i2 = 20 + i and i2 = 20 + 0.2i, separately, to describe node dierence. Results s
41、how that the identication time for i2 = 20 + i is remarkably shorter than that for i2 = 20 + 0.2i, implying that bigger node dierence leads to shorter identication time.2.3 Coupling strengthConsider identical node dynamics and the same inner coupling function H () : Rn Rn , then the network consisti
42、ng of N coupled oscillators can be described byNx i = F (xi ) c Lij H (xj ), i = 1, 2, , N, (11)j=1where L = (Lij ) RNN is the topology conguration to be determined. Under similar assumptions, by using the following response network and adaptive controllers:Nx i = Fi (xi ) c j=1 L ij H (xj ) + ui ,(
43、12)ui = ki xi ,k i = di xi , = xT H (x ),2 Liji jthe conguration matrix L can be successfully estimated by L .The master stability function method 29 is employed here to study the impact of the coupling strength on topology identication. The matrix L is required to be dissipative. For simplicity, L
44、is supposed to be symmetric and irreducible, which has only one zero characteristic root and the other roots are all positive, namely 0 = 1 2 N .According to 26, the synchronized regions of complex networks fall into four types: thebound, the unbound, multiple regions and empty set.i) The bound SR = (1 , 2 ). That is, when 1 /2 c 1 /2 . Generally, when the coupling strength is larger than a certain threshold, any network can achieve complete synchronization. In this case, topology identication fails.iii) Multiple disconnected regions SR = (1 , 2 ) (3 , 4 ) (2k1 , 2k
