  Scientists have been considering the Kuramoto model to understand the
mechanism behind the appearance of collective behaviour, such as frequency
synchronisation (FS) as a paradigm, in real-world networks with a finite number
of oscillators. A major current challenge is to obtain an analytical solution
for the phase-angles. Here, we provide an approximate analytical solution for
this problem by deriving a master solution for the finite-size Kuramoto model,
without imposing any restriction on the distribution of the natural frequencies
of the oscillators. The master solution embodies all particular solutions of
the finite-size Kuramoto model for any frequency distribution and coupling
strength larger than the critical one. Furthermore, we present a criterion to
determine the stability of the FS solution. This allows one to analytically
infer the relationship between the physical parameters and the stable behaviour
of networks.
