  The Kuramoto model, which serves as a paradigm for investigating
synchronization phenomenon of oscillatory system, is known to exhibit
second-order, i.e., continuous, phase transitions in the macroscopic order
parameter. Here, we generalize a number of classical results by presenting a
general framework for capturing, analytically, the critical scaling of the
order parameter at the onset of synchronization. Using a self-consistent
approach and constructing a characteristic function, we identify various phase
transitions toward synchrony and establish scaling relations describing the
asymptotic dependence of the order parameter on coupling strength near the
critical point. We find that the geometric properties of the characteristic
function, which depends on the natural frequency distribution, determines the
scaling properties of order parameter above the criticality.
