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Synchronization transition on random scale-free networks : Analytic approach

Synchronization transition on random scale-free networks : Analytic approach. Seoul National University, Seoul, Korea. Eulsik Oh. Contents. Synchronization Networks Model on scale-free networks Analytic Results Numerical Results Summary. http://cnrl.snu.ac.kr. Large J. Small J.

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Synchronization transition on random scale-free networks : Analytic approach

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  1. Synchronization transition on random scale-free networks : Analytic approach Seoul National University, Seoul, Korea Eulsik Oh

  2. Contents • Synchronization • Networks • Model on scale-free networks • Analytic Results • Numerical Results • Summary http://cnrl.snu.ac.kr

  3. Large J Small J 1. Synchronization • Oscillators (nodes) • Connections (network) • Communications (coupling strength : J) Constituents http://cnrl.snu.ac.kr

  4. 1. Synchronization flashing fireflies pacemaker cells in the heart neurons in the brain Josephson junction arrays Chemical reactions http://cnrl.snu.ac.kr

  5. : natural frequency of ith node selected from the distribution function 1. Synchronization Mathematical model (Kuramoto, 1975) The states of oscillators are described by phases The individual property is quantified by the natural frequency. : phase of ith node http://cnrl.snu.ac.kr

  6. 1. Synchronization Ordering http://cnrl.snu.ac.kr

  7. Regular lattice Fully connected network 2. Networks Previous Research Degree Distribution (Degree : number of neighbors connect to a node) Homogeneity, Isotropy http://cnrl.snu.ac.kr

  8. Scale-free networks 2. Networks In nature Actor www (Albert et al., 1999) Heterogeneity http://cnrl.snu.ac.kr

  9. 2. Networks How to describe ? http://cnrl.snu.ac.kr

  10. : degree of i-th node 3. Model on scale-free networks Modified model : element of a adjacency matrix It has 1(0) if i and j are connected (disconnected) http://cnrl.snu.ac.kr

  11. 3. Model on scale-free networks Static model (Goh et al., 2001) • Each vertex is indexed by an integer i • They are assigned its weight • Two different vertices (i, j) are selected with the their weights and connected 1 2 Scale-free network Uncorrelated http://cnrl.snu.ac.kr

  12. 4. Analytic Results Local order parameter & Fokker-Planck equation ri: local order parameter Fokker-Planck equation :probability distribution that the phase of an oscillator with the frequency and the degree is equal to . http://cnrl.snu.ac.kr

  13. 4. Analytic Results 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 http://cnrl.snu.ac.kr

  14. 4. Analytic Results Singular term Thermodynamic limit http://cnrl.snu.ac.kr

  15. , where for 4. Analytic Results Results Region A Region B http://cnrl.snu.ac.kr

  16. 4. Analytic Results Order Parameter http://cnrl.snu.ac.kr

  17. 4. Analytic Results Region A Region B Region C http://cnrl.snu.ac.kr

  18. 4. Analytic Results Finite scaling Increasing J Here, we consider nodes whose phases satisfy as synchronizednodes (filled squares) and connect them. Then a giant clustered componentemerges in synchronized state. Sc: size of the largest giant cluster http://cnrl.snu.ac.kr

  19. 4. Analytic Results Regions http://cnrl.snu.ac.kr

  20. Largest cluster size 4. Analytic Results Cluster size distribution http://cnrl.snu.ac.kr

  21. 4. Analytic Results Region I, IIand III Region VI, IV http://cnrl.snu.ac.kr

  22. 4. Analytic Results Region V http://cnrl.snu.ac.kr

  23. 5. Numerical Results Region I, II, and III http://cnrl.snu.ac.kr

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