  We study networks of noisy phase oscillators whose nodes are characterized by
a random degree counting the number of its connections. Both these degrees and
the natural frequencies of the oscillators are distributed according to a given
probability density. Replacing the randomly connected network by an all-to-all
coupled network with weighted edges, allows us to formulate the dynamics of a
single oscillator coupled to the mean field and to derive the corresponding
Fokker-Planck equation. From the latter we calculate the critical coupling
strength for the onset of synchronization as a function of the noise intensity,
the frequency distribution and the first two moments of the degree
distribution. Our approach is applied to a dense small-world network model, for
which we calculate the degree distribution. Numerical simulations prove the
validity of the made replacement. We also test the applicability to more
sparsely connected networks and formulate homogeneity and absence of
correlations in the degree distribution as limiting factors of our approach.
