  The robustness of complex networks with dependencies has been studied in
recent years. However, previous studies focused on the robustness of networks
composed of dependency links without network topology. In this study, we will
analyze the percolation properties of a realistic network model where
dependency links follow certain network topology. We perform the theoretical
analysis and numerical simulations to show the critical effects of topology of
dependency links on robustness of complex networks. For Erd\"os-R\'enyi (ER)
connectivity network, we find that the system with dependency of RR topology is
more vulnerable than system with dependency of ER topology. And RR-RR (i.e.
random-regular (RR) network with dependency of RR topology) disintegrates in an
abrupt transition. In particular, we find that the system of RR-ER shows
different types of phase transitions. For system of different combinations, the
type of percolation depends on the interaction between connectivity topology
and dependency topology.
