Note that random noise and time delay are prevalent in complex networks. Moreover, the topology of a network is often unknown or partially unknown．Therefore，based on the random generalized projective lag synchronization, we suggest an approach to estimate the system parameters and topological structure of delay-coupled complex networks under circumstance noise. By constructing an appropriate controller and adaptive update laws, the unknown network parameters and topological structure of the concerned networks can be identified simultaneously. The accuracy of the method is rigorously proved by the LaSalle-type theorem for stochastic differential delay equations．An example with networks of chaotic oscillator is provided to illustrate our method. The numerical results indicate that not only the unknown network parameters and topological structure can be accurately identify，but also the proposed method is robust against the time delay，the update gain and the network topology．
卫亭1 杨晓丽1 孙中奎2. 噪声扰动下时滞复杂网络的动力学参数及网络拓扑结构辨识[J]. 振动与冲击, 2015, 34(22): 138-143.
Wei Ting1 Yang Xiao-Li1 Sun Zhong-Kui2. Identification of system parameters and network topology in delay-coupled complex networks under circumstance noise. JOURNAL OF VIBRATION AND SHOCK, 2015, 34(22): 138-143.
[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, 15:024101.
[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, 393(6684):440-442.
[5]Guimera R, Diaz-Guilera A, Vega-Redondo F, ea al. Optimal network topologies for local search with congestion[J].Physical Review Letters, 2002, 89(24):248701.
[6]Wang Xiaonan, Lu Ying, Jiang Minxi, 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 Dongchuan, 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 compl-
ex dynamical networks[J]. Physica A, 2007, 386:481-491.
[10]Xu Yuhua, Zhou Wuneng, 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 Chunxia, Wan Youhong, Jiang Guoping. Topology identification for a class of complex dynamical networks using output variables[J]. Chinese Physics B, 2012, 21:020510.
[12]Xu Yuhua, Zhou Wuneng, 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 Zhiyong, Si Gangquan. 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 Junchan, Li Qin, Lu Jun-an, et al. Topology identification of complex dynamical networks[J].Chaos, 2010, 20:023119.
[16]Wu Xiaoqun. Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay[J]. Physica A, 2008, 387:997-1008.
[17]Guo Wanli, Chen Shihua, 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 Guoliang. 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 Jinhu, et al. Structure identification of uncertain general complex dynamical networks with time delay[J].Automatica, 2009, 45:1799-1807.
[20]Xu Yuhua, Zhou Wuneng, 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 Xiangjun, Lu Hongtao. Outer synchronization of uncertain general complex delayed networks with adaptive coupling[J]. Neurocomputing, 2012, 82:157-166.
[22]Zheng Song, Bi Qinsheng, Cai Guoliang. Adaptive projective synchronization in complex networks with time-varying coupling delay[J].Physics Letters A, 2009, 373:1553-1559.
[23]Che Yanqiu, Li Ruixue, Han Chunxiao, et al. Topology identification of uncertain nonlinearly coupled complex networks with delays based on anticipatory synchronization[J].Chaos, 2013, 23:013127.
[24]Che Yanqiu, Li Ruixue, Han Chunxiao, 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 Shengxue, Chen Li, He Yigang. Optimization-based topology identification of complex networks[J].Chinese Physics B, 2011, 20:110502.
[26]Yu Dongchuan. Estimating the topology of complex dynamical networks by steady state control：Generality and Limitation[J]. Automatica, 2010, 46:2035-2040.
[27]Ren Jie, Wang Wenxu, Li Baowen, et al. Noise bridges dynamical correlation and topology in complex oscillator networks[J]. Physical Review Letters, 2010, 104:058701.
[28]吴晓群，赵雪漪，吕金虎．节点动力学含随机噪声的复杂动力网络拓扑结构识别[J].Proceedings of the 30th Chinese Control Conference, 2011, Yantai, China.
Wu Xiaoqun, Zhao Xueyi, Lü Jinhu. Topology identification of dynamical networks with stochastic perturbation[J]. Proceedings of the 30th Chinese Control Conference, 2011, Yantai, China.
[29]He Tao, Lu Xiliang, Wu Xiaoqun, et al. Optimization-based structure identification of dynamical networks[J]. Physica A, 2013, 392:1038-1049.
[30]Wu Xiaoqun, Zhou Changsong, Chen Guanrong, et al. Detecting the topologies of complex networks with stochastic perturbations[J]. Chaos, 2011, 21:043129.
[31]Wu Xiaoqun, Wang Weihan, Zheng Weixing. Inferring topologies of complex networks with hidden variables[J]. Physical Review E, 2012, 86:046106.
[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]Jokob Nawrath, M.Carmen Romano, Marco Thiel, et al. Distinguishing Direct from Indirect Interactions in Oscillatory Networks with Multiple Time Scales[J]. Physical Review Letters, 2010, 104:038701.
[34]Norbert Marwan, M.Carmen Romano, Marco Thiel, et al. Recurrence plots for the analysis of complex systems[J]. Physics Reports, 2007, 438:237-329.
[35]Yu Dongchuan, Parlitz U. Inferring network connectivity by delayed feedback control[J]. Plos One, 2011,6(9):e24333.
[36]Bernd Pomped, Jakob Runge. Momentary information transfer as a coupling measure of time series[J].Physical Review E, 2011, 83:051122.
[37]Mao Xuerong. Stochastic versions of the LaSalle Theorem[J].Journal of Differential Equations, 1999, 153:175-195.
[38]Mao Xuerong. A note on the LaSalle-Type Theorems for stochastic differential delay equations[J]. Journal of Mathematical Analysis and Applications, 2002, 268:125-142.
[39]刘明华．一类新超混沌系统的产生、同步及其应用[D]．广州：华南理工大学，2011．
Liu Minghua.The generation, synchronization and application for a class of novel hyperchaotic systems[D].
[40]刘秉正，彭建华．非线性动力学[M]．北京：高等教育出版社，2007重印，pp504．
Liu Bingzheng, Peng Jianhua. Nonlinear dynamics[M] . Beijing: Higher Education Press, 2007 reprints, pp504.
[41]陆君安，吕金虎，刘 慧等．复杂动力网络结构识别的某些新进展[J]．复杂系统与复杂性科学，2010,7(2-3):63-69.
Lu Jun-an, Lü Jinhu, 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.
