Cluster synchronization and isolated desynchronization in complex networks with symmetries

Synchronization is of central importance in power distribution, telecommunication, neuronal and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form. Here we present a new framework and develop techniques for the analysis of network dynamics that shows the connection between network symmetries and cluster formation. The connection between symmetries and cluster synchronization is experimentally confirmed in the context of real networks with heterogeneities and noise using an electro-optic network. We experimentally observe and theoretically predict a surprising phenomenon in which some clusters lose synchrony without disturbing the others. Our analysis shows that such behaviour will occur in a wide variety of networks and node dynamics. The results could guide the design of new power grid systems or lead to new understanding of the dynamical behaviour of networks ranging from neural to social.

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

[2]  F. Vallentin.  Symmetry in semidefinite programs , 2007, 0706.4233.

[3]  J. Danckaert,et al.  Synchronization properties of network motifs: influence of coupling delay and symmetry. , 2008, Chaos.

[4]  M. Golubitsky,et al.  The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space , 2002 .

[5]  Francesco Sorrentino,et al.  Synchronization of dynamical hypernetworks: dimensionality reduction through simultaneous block-diagonalization of matrices. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Joos Vandewalle,et al.  Cluster synchronization in oscillatory networks. , 2008, Chaos.

[7]  W. Kinzel,et al.  Synchronization of unidirectional time delay chaotic networks and the greatest common divisor , 2010, 1012.0990.

[8]  Rubén J. Sánchez-García,et al.  Spectral characteristics of network redundancy. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Eckehard Schöll,et al.  Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators. , 2013, Physical review letters.

[10]  Shahin Abdollahy,et al.  PNM smart grid demonstration project from modeling to demonstration , 2012, 2012 IEEE PES Innovative Smart Grid Technologies (ISGT).

[11]  Eckehard Schöll,et al.  Control of synchronization patterns in neural-like Boolean networks. , 2012, Physical review letters.

[12]  Ben D. MacArthur,et al.  Symmetry in complex networks , 2007, Discret. Appl. Math..

[13]  Vito Latora,et al.  Remote synchronization reveals network symmetries and functional modules. , 2012, Physical review letters.

[14]  J. Slotine,et al.  Symmetries, stability, and control in nonlinear systems and networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[16]  Belykh,et al.  Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  K. Judd.  Networked dynamical systems with linear coupling: synchronisation patterns, coherence and other behaviours. , 2013, Chaos.

[18]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[19]  R. Roy,et al.  Experimental observation of chimeras in coupled-map lattices , 2012, Nature Physics.

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

[21]  T. Carroll,et al.  MASTER STABILITY FUNCTIONS FOR SYNCHRONIZED COUPLED SYSTEMS , 1999 .

[22]  M. Tinkham.  Group Theory and Quantum Mechanics , 1964 .

[23]  Carlo R. Laing,et al.  The dynamics of chimera states in heterogeneous Kuramoto networks , 2009 .

[24]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[25]  Johnson,et al.  Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  E. Ott,et al.  Network synchronization of groups. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Liang Huang,et al.  Topological control of synchronous patterns in systems of networked chaotic oscillators , 2013 .

[28]  R. Goodman,et al.  Representations and Invariants of the Classical Groups , 1998 .

[29]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[30]  S. Strogatz,et al.  Chimera states for coupled oscillators. , 2004, Physical review letters.

[31]  Ian Stewart,et al.  Coupled cells with internal symmetry: I. Wreath products , 1996 .

[32]  Jean-Jacques E. Slotine,et al.  Erratum: Symmetries, stability, and control in nonlinear systems and networks [Phys. Rev. E84, 041929 (2011)] , 2012 .

[33]  Eckehard Schöll,et al.  Cluster and group synchronization in delay-coupled networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Elise Bærevar,et al.  Introduction ch , 2007 .

[35]  Seth A. Myers,et al.  Spontaneous synchrony in power-grid networks , 2013, Nature Physics.

[36]  Thilo Gross,et al.  Engineering mesoscale structures with distinct dynamical implications , 2012, New Journal of Physics.

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

[38]  B. Sagan.  The Symmetric Group , 2001 .

[39]  William Stein,et al.  SAGE: Software for Algebra and Geometry Experimentation , 2006 .

[40]  James W. Anderson,et al.  On Automorphism Groups of Networks , 2007, 0705.3215.

[41]  S. Bornholdt,et al.  Handbook of Graphs and Networks , 2012 .

[42]  J. Kurths,et al.  Hierarchical synchronization in complex networks with heterogeneous degrees. , 2006, Chaos.