  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 $\couplingcoeff
k_i^{\eta-1}$. By invoking the mean-field approach, we determine the
synchronization transition point $J_c$, which is zero (finite) when $\eta >
\lambda-2$ ($\eta < \lambda-2$). 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 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.
