Critical exponents in coupled phase-oscillator models on small-world networks

A coupled phase-oscillator model consists of phase-oscillators, each
of which has the natural frequency obeying a probability distribution
and couples with other oscillators through a given periodic coupling
function. These models are widely studied since they describe the
synchronization transition, which emerges between the nonsynchronized
state and partially synchronized states, and which is characterized by
the critical exponents. Among them, we focus on the critical exponent
defined by coupling strength dependence of the order parameter. The
synchronization transition is not limited in the all-to-all
interaction, whose number of links is of O(N^2) with N oscillators,
and occurs in small-world networks whose links are of O(N). In the
all-to-all interaction, values of the critical exponent depend on the
natural frequency distribution and the coupling function. A natural
question is, therefore, in small-world networks, whether the
dependency remains irrespective of the order of links. To answer this
question, in this thesis, we numerically compute the critical exponent
on small-world networks with coupling functions up to the second
harmonics and with unimodal and symmetric natural frequency
distributions by using the finite-size scaling method. Our numerical
results suggest that the considered systems are collapsed into two
universality classes, while the all-to-all interaction is known to
give an infinite number of universality classes.