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.