  We study the effect of random scattering in quantum walks on a finite graph
and compare it with the effect of repeated measurements. To this end, a
constructive approach is employed by introducing a localized and a delocalized
basis for the underlying Hilbert space. This enables us to design Hamiltonians
whose eigenvectors are either localized or delocalized. By presenting some
specific examples we demonstrate that the localization of eigenvectors
restricts the transition probabilities on the graph and leads to dark states in
the monitored evolution. We conclude that repeated measurements as well as
random scattering provide efficient tools for controlling quantum walks.
