Finite-size scaling of synchronized oscillation on complex networks

The onset of synchronization in a system of random frequency oscillators coupled through a random network is investigated. Using a mean-field approximation, we characterize sample-to-sample fluctuations for networks of finite size, and derive the corresponding scaling properties in the critical region. For scale-free networks with the degree distribution P (k) ∼ k-γ at large k, we found that the finite-size exponent ν̄ takes on the value 5/2 when γ>5, the same as in the globally coupled Kuramoto model. For highly heterogeneous networks (3<γ<5), ν̄ and the order parameter exponent β depend on γ. The analytical expressions for these exponents obtained from the mean-field theory are shown to be in excellent agreement with data from extensive numerical simulations.