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 lambda. An oscillator of degree k(i) is coupled to its neighboring oscillators with asymmetric and degree-dependent coupling in the form of Jk(i)(eta-1). By invoking the mean-field approach, we find eight different synchronization transition behaviors depending on the values of eta and lambda, and derive the critical exponents associated with the order parameter and the finite-size scaling in each case. The synchronization transition point J(c) is determined as being zero (finite) when eta >lambda-2 (eta <lambda-2). 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.