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) approximately k(-gamma) at large k, we found that the finite-size exponent nu takes on the value 5/2 when gamma>5, the same as in the globally coupled Kuramoto model. For highly heterogeneous networks (3<gamma<5), nu and the order parameter exponent beta depend on gamma. 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.

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