  In this paper, we investigate synchronization in a small-world network of
coupled nonlinear oscillators. This network is constructed by introducing
random shortcuts in a nearest-neighbors ring. The local stability of the
synchronous state is closely related with the support of the eigenvalue
distribution of the Laplacian matrix of the network. We introduce, for the
first time, analytical expressions for the first three moments of the
eigenvalue distribution of the Laplacian matrix as a function of the
probability of shortcuts and the connectivity of the underlying
nearest-neighbor coupled ring. We apply these expressions to estimate the
spectral support of the Laplacian matrix in order to predict synchronization in
small-world networks. We verify the efficiency of our predictions with
numerical simulations.
