Synchronization transition of heterogeneously coupled oscillators on scalefree networks

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Synchronization transition of heterogeneously
coupled oscillators on scale-free networks
E. Oh1, D.-S. Lee2, B. Kahng1,3 and D.
Kim1 1School of Physics and Astronomy, Seoul
National University, Seoul, Korea 2Theoretische
Physik Universitat des Saarlandes, Saarbrucken,
Germany 3Center for Nonlinear Studies, Los Alamos
National Laboratory, Los Alamos, NM
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1. Synchronization in nature
Christiaan Huygens (16291695)

• Syn (common, together) chronos (time)
• shared or common time

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1. Synchronization in nature

• Pacemaker cells in the heart
• Circadian pacemaker cells in the brain
• Metabolic activity in yeast cell suspensions
• Crickets that chirp in unison
• Neural activity in the brain
• Oscillating chemical reactions
• Semiconductor lasers
• Josephson junction arrays
• Menstrual synchrony

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2. Models Kuramoto model
Y. Kuramoto, Chemical Oscillations, Waves, and
Turbulence (Springer-Verlag, 1984)

• Order parameter for the synchronization r

• Critical coupling strength

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2. Models Kuramoto model
Classical views on Network
Fully connected network
Regular lattice
Degree Distribution (Degree number of neighbors
connecting to a node)
Homogeneity, Isotropy
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2. Models Kuramoto model
Power law degree distribution

• Hubs and Scale-free networks ( Albert, Jeong,
  Barabási 1999)

Metabolic network Jeong et al. 2000
WWW Albert et al. 1999
Scale-free networks
Heterogeneity
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2. Models Kuramoto model
How to describe ?
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2. Models Kuramoto model
Modified model
element of an adjacency matrix It has
1(0) if i and j are connected (disconnected)
degree of i-th node
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3. Mean-field Solutions
Local order parameter Fokker-Planck equation
local order parameter
Continuum equation
probability distribution that the phase of an
oscillator with the frequency and the
degree is equal to .
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3. Mean-field Solutions
Local order parameter in continuum limit
Continuum version
Average over all possible states
Static model
Conditional probability of a given node with
degree k to be linked to a node with degree k
Gaussian distributionwith zero mean and unit
variance
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3. Mean-field Solutions
Singular term
Thermodynamic limit
The generalized harmonic number
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3. Mean-field Solutions

• Critical coupling strength

I
II
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4. Scaling behaviors
Finite scaling
Increasing J
Here, we consider nodes whose phases satisfy
as synchronized nodes (filled squares) and
connect them. Then a giant clustered component
emerges in a synchronized state.
Sc size of the largest giant cluster
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4. Scaling behaviors
Cluster size distribution
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4. Scaling behaviors
Generating function
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4. Scaling behaviors
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4. Scaling behaviors
Numerical Results
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4. Scaling behaviors
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5. Summary

• We have investigated the nature of the
  synchronizationtransition by a modified Kuramoto
  model with the asymmetric and degree dependent
  weighted couplingstrength.
• The critical exponents associated with the order
  parameterand the finite-size scaling are
  determined in terms of thetwo tunable
  parameters.
• The parameter space of (?, ?) is divided into
  eight different domains.