Kuramoto model with frequencydegree correlations on complex networks
Abstract
We study the Kuramoto model on complex networks, in which natural frequencies of phase oscillators and the vertex degrees are correlated. Using the annealed network approximation and numerical simulations, we explore a special case in which the natural frequencies of the oscillators and the vertex degrees are linearly coupled. We find that in uncorrelated scalefree networks with the degree distribution exponent 2<γ<3, the model undergoes a firstorder phase transition, while the transition becomes second order at γ>3. If γ=3, the phase synchronization emerges as a result of a hybrid phase transition that combines an abrupt emergence of synchronization, as in firstorder phase transitions, and a critical singularity, as in secondorder phase transitions. The critical fluctuations manifest themselves as avalanches in the synchronization process. Comparing our analytical calculations with numerical simulations for ErdősRényi and scalefree networks, we demonstrate that the annealed network approach is accurate if the mean degree and size of the network are sufficiently large. We also study analytically and numerically the Kuramoto model on star graphs and find that if the natural frequency of the central oscillator is sufficiently large in comparison to the average frequency of its neighbors, then synchronization emerges as a result of a firstorder phase transition. This shows that oscillators sitting at hubs in a network may generate a discontinuous synchronization transition.
 Publication:

Physical Review E
 Pub Date:
 March 2013
 DOI:
 10.1103/PhysRevE.87.032106
 arXiv:
 arXiv:1211.5690
 Bibcode:
 2013PhRvE..87c2106C
 Keywords:

 05.70.Fh;
 05.45.Xt;
 64.60.aq;
 Phase transitions: general studies;
 Synchronization;
 coupled oscillators;
 Networks;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Soft Condensed Matter;
 Mathematics  Statistics Theory
 EPrint:
 11 pages