On pinning synchronisability of complex networks with arbitrary topological structure

This article aims to investigate the pinning synchronisability of complex networks under arbitrary topological structures with a focus on the case with directed graph topology. More specifically, we explore the necessary and sufficient conditions on choosing the pinned nodes to guarantee the pinning synchronisability of complex networks. It is found that the pinning synchronisability of complex networks relies totally on the way to pin the nodes as long as the coupling strength is large enough. Furthermore, it is proven that the entire network can synchronise exponentially fast. The least synchronisation speed can be specified as well. Finally, based on the theoretical findings proposed in this article, we provide the specific procedures for finding a least number of pinned nodes to realise the pinning synchronisability of complex networks.

[1]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[2]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[3]  Guanrong Chen,et al.  Pinning control of scale-free dynamical networks , 2002 .

[4]  Daniel W. C. Ho,et al.  Pinning Stabilization of Linearly Coupled Stochastic Neural Networks via Minimum Number of Controllers , 2009, IEEE Transactions on Neural Networks.

[5]  Youxian Sun,et al.  Adaptive synchronization of weighted complex dynamical networks through pinning , 2008 .

[6]  B. Bollobás.  The evolution of random graphs , 1984 .

[7]  Zhi Liu,et al.  A Probabilistic Wavelet System for Stochastic and Incomplete Data-Based Modeling , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[8]  F. Garofalo,et al.  Controllability of complex networks via pinning. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Xiao Fan Wang,et al.  Synchronization in scale-free dynamical networks: robustness and fragility , 2001, cond-mat/0105014.

[10]  Zidong Wang,et al.  A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances , 2008 .

[11]  Tianping Chen,et al.  New approach to synchronization analysis of linearly coupled ordinary differential systems , 2006 .

[12]  Timothy W. McLain,et al.  Coordination Variables and Consensus Building in Multiple Vehicle Systems , 2004 .

[13]  Tianping Chen,et al.  Synchronisation in complex networks of coupled systems with directed topologies , 2009, Int. J. Syst. Sci..

[14]  Wenwu Yu,et al.  On pinning synchronization of complex dynamical networks , 2009, Autom..

[15]  Zidong Wang,et al.  Global synchronization for delayed complex networks with randomly occurring nonlinearities and multiple stochastic disturbances , 2009 .

[16]  Yeung Sam Hung,et al.  Distributed H∞-consensus filtering in sensor networks with multiple missing measurements: The finite-horizon case , 2010, Autom..

[17]  M. Cross,et al.  Pinning control of spatiotemporal chaos , 1997, chao-dyn/9705001.

[18]  Zidong Wang,et al.  Exponential synchronization of complex networks with Markovian jump and mixed delays , 2008 .

[19]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[20]  C. Wu.  Synchronization in networks of nonlinear dynamical systems coupled via a directed graph , 2005 .

[21]  Xiang Li,et al.  Global stabilization of complex networks with digraph topologies via a local pinning algorithm , 2010, Autom..

[22]  Tianping Chen,et al.  Pinning Complex Networks by a Single Controller , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[23]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[24]  Jürgen Kurths,et al.  Consensus over directed static networks with arbitrary finite communication delays. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Junan Lu,et al.  Pinning adaptive synchronization of a general complex dynamical network , 2008, Autom..

[26]  Ji Xiang,et al.  On the V-stability of complex dynamical networks , 2007, Autom..

[27]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[28]  Huijun Gao,et al.  Discrete-time complex networks: a new synchronisation stability criterion , 2009, Int. J. Syst. Sci..

[29]  Zidong Wang,et al.  Synchronization and State Estimation for Discrete-Time Complex Networks With Distributed Delays , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[30]  Ligang Wu,et al.  Exponential stabilization of switched stochastic dynamical networks , 2009 .

[31]  D. Ho,et al.  Local and global synchronization in general complex dynamical networks with delay coupling , 2008 .

[32]  Zidong Wang,et al.  Pinning control of fractional-order weighted complex networks. , 2009, Chaos.

[33]  Yeung Sam Hung,et al.  Distributed $H_{\infty}$ Filtering for Polynomial Nonlinear Stochastic Systems in Sensor Networks , 2011, IEEE Transactions on Industrial Electronics.

[34]  Guanrong Chen,et al.  Pinning a complex dynamical network to its equilibrium , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[35]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[36]  Zidong Wang,et al.  Global Synchronization for Discrete-Time Stochastic Complex Networks With Randomly Occurred Nonlinearities and Mixed Time Delays , 2010, IEEE Transactions on Neural Networks.