  We study a zero-temperature phase transition in the random field Ising model
on scale-free networks with the degree exponent $\gamma$. Using an analytic
mean-field theory, we find that the spins are always in the ordered phase for
$\gamma<3$. On the other hand, the spins undergo a phase transition from an
ordered phase to a disordered phase as the dispersion of the random fields
increases for $\gamma > 3$. The phase transition may be either continuous or
discontinuous depending on the shape of the random field distribution. We
derive the condition for the nature of the phase transition. Numerical
simulations are performed to confirm the results.
