  Many real-world networks are coupled together to maintain their normal
functions. Here we study the robustness of multiplex networks with
interdependent and interconnected links under k-core percolation, where a node
fails when it connects to a threshold of less than k neighbors. By deriving the
self-consistency equations, we solve the key quantities of interests such as
the critical threshold and size of the giant component analytically and
validate the theoretical results with numerical simulations. We find a rich
phase transition phenomenon as we tune the inter-layer coupling strength.
Specifically speaking, in the ER-ER multiplex networks, with the increase of
coupling strength, the size of the giant component in each layer first
undergoes a first-order transition and then a second-order transition and
finally a first-order transition. This is due to the nature of inter-layer
links with both connectivity and dependency simultaneously. The system is more
robust if the dependency on the initial robust network is strong and more
vulnerable if the dependency on the initial attacked network is strong. These
effects are even amplified in the cascading process. When applying our model to
the SF-SF multiplex networks, the type of transition changes. The system
undergoes a first-order phase transition first only when the two layers'
mutually coupling is very strong and a second-order transition in other
conditions.
