Kuramoto model with frequency-degree correlations on complex networks
Physical Review E, (Mar 24, 2013)DOI: 10.1103/PhysRevE.87.032106
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 scale-free networks with the
degree distribution exponent \$2 < < 3\$, the model undergoes a
first-order phase transition, while the transition becomes of the second order
at \$\gamma>3\$. If \$\gamma=3\$, the phase synchronization emerges as a result of
a hybrid phase transition that combines an abrupt emergence of synchronization,
as in first-order phase transitions, and a critical singularity, as in
second-order phase transitions. The critical fluctuations manifest themselves
as avalanches in synchronization process. Comparing our analytical calculations
with numerical simulations for Erd\Hos--Rényi and scale-free networks, we
demonstrate that the annealed network approach is accurate if the 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 first-order phase transition. This shows that
oscillators sitting at hubs in a network may generate a discontinuous
synchronization transition.