Finite-size scaling of synchronized oscillation on complex networks

Abstract

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 $P(k)\sim k^{-γ}$ at large $k$, we found that the finite size exponent $\barν$ takes on the value 5/2 when $γ>5$, the same as in the globally coupled Kuramoto model. For highly heterogeneous networks ($3<γ<5$), $\barν$ and the order parameter exponent $β$ depend on $γ$. The analytic expressions for these exponents obtained from the mean field theory are shown to be in excellent agreement with data from extensive numerical simulations.