  Entrainment by a pacemaker, representing an element with a higher frequency,
is numerically investigated for several classes of random networks which
consist of identical phase oscillators. We find that the entrainment frequency
window of a network decreases exponentially with its depth, defined as the mean
forward distance of the elements from the pacemaker. Effectively, only shallow
networks can thus exhibit frequency-locking to the pacemaker. The exponential
dependence is also derived analytically as an approximation for large random
asymmetric networks.
