Synchronization transition of heterogeneously coupled oscillators on scale-free networks.

We investigate the synchronization transition of the modified Kuramoto model where the oscillators form a scale-free network with degree exponent lambda . An oscillator of degree k_{i} is coupled to its neighboring oscillators with asymmetric and degree-dependent coupling in the form of Jk_{i};{eta-1} . By invoking the mean-field approach, we find eight different synchronization transition behaviors depending on the values of eta and lambda , and derive the critical exponents associated with the order parameter and the finite-size scaling in each case. The synchronization transition point J_{c} is determined as being zero (finite) when eta>lambda-2 (eta<lambda-2) . The synchronization transition is also studied from the perspective of cluster formation of synchronized vertices. The cluster-size distribution and the largest cluster size as a function of the system size are derived for each case using the generating function technique. Our analytic results are confirmed by numerical simulations.

[1]  Changsong Zhou,et al.  Universality in the synchronization of weighted random networks. , 2006, Physical review letters.

[2]  B Kahng,et al.  Sandpile on scale-free networks. , 2003, Physical review letters.

[3]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[4]  Michael E. Fisher,et al.  Critical Exponents for Long-Range Interactions , 1972 .

[5]  B. Kahng,et al.  Sandpile avalanche dynamics on scale-free networks , 2004 .

[6]  K. Goh,et al.  Universal behavior of load distribution in scale-free networks. , 2001, Physical review letters.

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

[8]  Sergey N. Dorogovtsev,et al.  Ising Model on Networks with an Arbitrary Distribution of Connections , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Deok-Sun Lee.  Synchronization transition in scale-free networks: clusters of synchrony. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[11]  G. Church,et al.  Analysis of optimality in natural and perturbed metabolic networks , 2002 .

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

[13]  S. Boccaletti,et al.  Synchronization is enhanced in weighted complex networks. , 2005, Physical review letters.

[14]  B. Kahng,et al.  Evolution of scale-free random graphs: Potts model formulation , 2004 .

[15]  B Kahng,et al.  Spin-glass phase transition on scale-free networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Deok-Sun Lee,et al.  Branching Process Approach to Avalanche Dynamics on Complex Networks , 2004 .

[17]  J. Kurths,et al.  Network synchronization, diffusion, and the paradox of heterogeneity. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  A. Barabasi,et al.  Global organization of metabolic fluxes in the bacterium Escherichia coli , 2004, Nature.

[19]  Jacek Klinowski,et al.  Values of the Legendre chi and Hurwitz zeta functions at rational arguments , 1999, Math. Comput..

[20]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[21]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[22]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[23]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[24]  E. Ott,et al.  Onset of synchronization in large networks of coupled oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Beom Jun Kim,et al.  Synchronization on small-world networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[27]  T. Ichinomiya.  Frequency synchronization in a random oscillator network. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[29]  Yoshiki Kuramoto,et al.  Cooperative Dynamics of Oscillator Community : A Study Based on Lattice of Rings , 1984 .

[30]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.