Evolution of Complex Networks via Edge Snapping

In this paper, we present a novel adaptive strategy for consensus and synchronization of complex networks. The strategy is inspired by bistable phenomena that are observed in a variety of mechanical systems. The novelty is that the adaptation involves the topology of the network itself rather than its coupling gains. In particular, we model the evolution of each coupling gain as a second order dynamical system that is subject to the action of a double-well potential. Through a new mechanism, termed as edge snapping, an unweighted network topology emerges at steady state. We assess the stability properties of the proposed scheme through analytical methods and numerical investigations. We conduct an extensive numerical study of the topological properties of the emerging network to elucidate the correlation between the initial conditions of the nodes' dynamics and the network structure.

[1]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[2]  Silvano Donati,et al.  Synchronization of chaotic injected-laser systems and its application to optical cryptography , 1996 .

[3]  Xiwei Chen,et al.  Network Synchronization with an Adaptive Coupling Strength , 2006 .

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

[5]  Changsong Zhou,et al.  Dynamical weights and enhanced synchronization in adaptive complex networks. , 2006, Physical review letters.

[6]  Benjamin Van Roy,et al.  Distributed Optimization in Adaptive Networks , 2003, NIPS.

[7]  Seunghoon Park,et al.  Pre-shaped buckled-beam actuators: Theory and experiments , 2008 .

[8]  G. Grassi,et al.  Synchronizing hyperchaotic systems by observer design , 1999 .

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

[10]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[11]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

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

[13]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[14]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[15]  George J. Pappas,et al.  Flocking in Fixed and Switching Networks , 2007, IEEE Transactions on Automatic Control.

[16]  Andrea Maggiolo-Schettini,et al.  Dynamic Graphs , 1996, MFCS.

[17]  M. Hasler,et al.  Synchronization in complex networks with blinking interactions , 2005, Proceedings. 2005 International Conference Physics and Control, 2005..

[18]  P. Holmes,et al.  A nonlinear oscillator with a strange attractor , 1979, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[19]  Pu Zhou,et al.  Synchronization and coherent combining of two pulsed fiber lasers , 2008 .

[20]  M. Porfiri,et al.  Global pulse synchronization of chaotic oscillators through fast-switching: theory and experiments , 2009 .

[21]  M. Imboden,et al.  Electrostatically actuated silicon-based nanomechanical switch at room temperature , 2008, 0903.2491.

[22]  Dragoslav D. Šiljak,et al.  Large-Scale Dynamic Systems: Stability and Structure , 1978 .

[23]  M. Porfiri,et al.  Node-to-node pinning control of complex networks. , 2009, Chaos.

[24]  Dimos V. Dimarogonas,et al.  On the Rendezvous Problem for Multiple Nonholonomic Agents , 2007, IEEE Transactions on Automatic Control.

[25]  K. Aihara,et al.  Synchronization of coupled nonidentical genetic oscillators , 2006, Physical biology.

[26]  M. Hasler,et al.  Blinking model and synchronization in small-world networks with a time-varying coupling , 2004 .

[27]  R. L. Badzey,et al.  Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance , 2005, Nature.

[28]  I. Couzin,et al.  Effective leadership and decision-making in animal groups on the move , 2005, Nature.

[29]  Maurizio Porfiri,et al.  Consensus Seeking Over Random Weighted Directed Graphs , 2007, IEEE Transactions on Automatic Control.

[30]  Randal W. Beard,et al.  Distributed Consensus in Multi-vehicle Cooperative Control - Theory and Applications , 2007, Communications and Control Engineering.

[31]  Chai Wah Wu.  Localization of effective pinning control in complex networks of dynamical systems , 2008, 2008 IEEE International Symposium on Circuits and Systems.

[32]  M. Porfiri,et al.  Vibrations of narrow microbeams predeformed by an electric field , 2008 .

[33]  Stefano Vidoli,et al.  Distributed piezoelectric actuation of a bistable buckled beam , 2007 .

[34]  Jian Zhao,et al.  Post-buckling and Snap-Through Behavior of Inclined Slender Beams , 2008 .

[35]  Maurizio Porfiri,et al.  Criteria for global pinning-controllability of complex networks , 2008, Autom..

[36]  Fei Ge,et al.  An Adaptive Energy Efficient Topology for Wireless Sensor Networks , 2008, 2008 Second International Conference on Future Generation Communication and Networking.

[37]  Reza Olfati-Saber,et al.  Flocking for multi-agent dynamic systems: algorithms and theory , 2006, IEEE Transactions on Automatic Control.

[38]  P. Benfey,et al.  Evolution, Interactions, and Biological Networks , 2007, PLoS biology.

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

[40]  F. Garofalo,et al.  Synchronization of complex networks through local adaptive coupling. , 2008, Chaos.

[41]  J. Anthonisse.  The rush in a directed graph , 1971 .

[42]  Maurizio Porfiri,et al.  Synchronization in Random Weighted Directed Networks , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[43]  Guo-Ping Jiang,et al.  A State-Observer-Based Approach for Synchronization in Complex Dynamical Networks , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[44]  P. McClintock.  Synchronization:a universal concept in nonlinear science , 2003 .

[45]  Wei Wu,et al.  Cluster Synchronization of Linearly Coupled Complex Networks Under Pinning Control , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[46]  R. L. Badzey,et al.  Signal Processing and Control in Nonlinear Nanomechanical Systems , 2006 .

[47]  Jie Lin,et al.  The multi-agent rendezvous problem , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[48]  E. Seneta,et al.  Towards consensus: some convergence theorems on repeated averaging , 1977, Journal of Applied Probability.

[49]  Jian-Ming Chen,et al.  Research and Improvement of Adaptive Topology Algorithm LEACH for Wireless Sensor Network , 2008, 2008 4th International Conference on Wireless Communications, Networking and Mobile Computing.

[50]  E. Ott,et al.  Adaptive synchronization of dynamics on evolving complex networks. , 2008, Physical review letters.

[51]  Tomasz Kapitaniak,et al.  Dynamics of an array of mutually coupled semiconductor lasers. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  Maurizio Porfiri,et al.  Master-Slave Global Stochastic Synchronization of Chaotic Oscillators , 2008, SIAM J. Appl. Dyn. Syst..

[53]  Duncan J. Watts,et al.  The Structure and Dynamics of Networks: (Princeton Studies in Complexity) , 2006 .

[54]  J. Fewell.  Social Insect Networks , 2003, Science.

[55]  Mikhail I. Rabinovich,et al.  Self-regularization of chaos in neural systems: experimental and theoretical results , 1997 .

[56]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[57]  M. Degroot.  Reaching a Consensus , 1974 .

[58]  Licheng Jiao,et al.  Robust adaptive global synchronization of complex dynamical networks by adjusting time-varying coupling strength , 2008 .

[59]  Santo Banerjee.  Synchronization of time-delayed systems with chaotic modulation and cryptography , 2009 .

[60]  Mario di Bernardo,et al.  How to Synchronize Biological Clocks , 2009, J. Comput. Biol..

[61]  Romesh C. Batra,et al.  Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems , 2009 .

[62]  R.W. Johnstone,et al.  A pulse width modulation controlled bistable microelectromechanical system , 2005, Canadian Conference on Electrical and Computer Engineering, 2005..

[63]  Davide Spinello,et al.  Vibrations and pull-in instabilities of microelectromechanical von Kármán elliptic plates incorporating the Casimir force , 2008 .

[64]  Guanrong Chen,et al.  Global synchronization and asymptotic stability of complex dynamical networks , 2006, IEEE Transactions on Circuits and Systems II: Express Briefs.

[65]  Tongxi Yu,et al.  Mechanics of adhesion in MEMS—a review , 2003 .

[66]  Dragoslav D. Šiljak,et al.  On connective stability and instability of competitive equilibrium , 1974 .

[67]  Ying-Cheng Lai,et al.  Controlling bistability in microelectromechanical resonators. , 2008, Chaos.

[68]  Tianping Chen,et al.  Synchronization in general complex delayed dynamical networks , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[69]  Leon O. Chua,et al.  Cryptography based on chaotic systems , 1997 .

[70]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[71]  Deborah Estrin,et al.  ASCENT: adaptive self-configuring sensor networks topologies , 2004, IEEE Transactions on Mobile Computing.

[72]  A S Perelson,et al.  Modeling adaptive biological systems. , 1989, Bio Systems.

[73]  Leonard M. Freeman,et al.  A set of measures of centrality based upon betweenness , 1977 .

[74]  M. Porfiri,et al.  Review of modeling electrostatically actuated microelectromechanical systems , 2007 .

[75]  L Chen,et al.  Synchronization with on-off coupling: Role of time scales in network dynamics. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[76]  Sonia Martínez,et al.  Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions , 2006, IEEE Transactions on Automatic Control.

[77]  Daizhan Cheng,et al.  Characterizing the synchronizability of small-world dynamical networks , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[78]  Dragoslav D. Siljak,et al.  A learning scheme for dynamic neural networks: equilibrium manifold and connective stability , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[79]  Stephen P. Boyd,et al.  Distributed average consensus with least-mean-square deviation , 2007, J. Parallel Distributed Comput..

[80]  Mario di Bernardo,et al.  Synchronization of Complex Networks via Hybrid Adaptive Coupling and Evolving Topologies , 2009 .

[81]  Liang Chen,et al.  Synchronization of the Time-Varying Discrete Biological Networks , 2007, 2007 IEEE International Symposium on Circuits and Systems.

[82]  Mario di Bernardo,et al.  Novel decentralized adaptive strategies for the synchronization of complex networks , 2009, Autom..

[83]  Xingming Guo,et al.  Reversible mechanical bistability of single-walled carbon nanotubes under axial strain , 2006 .

[84]  R. Vitushinsky,et al.  Bistable Thin-Film Shape Memory Actuators for Applications in Tactile Displays , 2009, Journal of Microelectromechanical Systems.

[85]  Xiang Li,et al.  Pinning a complex dynamical network to its equilibrium , 2004, IEEE Trans. Circuits Syst. I Regul. Pap..