  We study the impact of random pinning fields on the emergence of synchrony in
the Kuramoto model on complete graphs and uncorrelated random complex networks.
We consider random fields with uniformly distributed directions and homogeneous
and heterogeneous (Gaussian) field magnitude distribution. In our analysis we
apply the Ott-Antonsen method and the annealed-network approximation to find
the critical behavior of the order parameter. In the case of homogeneous
fields, we find a tricritical point above which a second-order phase transition
gives place to a first-order phase transition when the network is either fully
connected, or scale-free with the degree exponent $\gamma>5$. Interestingly,
for scale-free networks with $2<\gamma \leq 5$, the phase transition is of
second-order at any field magnitude, except for degree distributions with
$\gamma=3$ when the transition is of infinite order at $K_c=0$ independently on
the random fields. Contrarily to the Ising model, even strong Gaussian random
fields do not suppress the second-order phase transition in both complete
graphs and scale-free networks though the fields increase the critical coupling
for $\gamma > 3$. Our simulations support these analytical results.
