Synchronization transition of heterogeneously coupled oscillators on scale-free networks

We investigate the synchronization transition of the modified Kuramoto model where the oscillators form a scale-free network with degree exponent $λ$. An oscillator of degree $k_i$ is coupled to its neighboring oscillators with asymmetric and degree-dependent coupling in the form of $\couplingcoeff k_i^{η-1}$. By invoking the mean-field approach, we determine the synchronization transition point $J_c$, which is zero (finite) when $η> λ-2$ ($η< λ-2$). We find eight different synchronization transition behaviors depending on the values of $η$ and $λ$, and derive the critical exponents associated with the order parameter and the finite-size scaling in each case. The synchronization transition is also studied from the perspective of cluster formation of synchronized vertices. The cluster-size distribution and the largest cluster size as a function of the system size are derived for each case using the generating function technique. Our analytic results are confirmed by numerical simulations.

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