  Multilayer networked systems are ubiquitous in nature and engineering, and
the robustness of these systems against failures is of great interest. A main
line of theoretical pursuit has been percolation induced cascading failures,
where interdependence between network layers is conveniently and tacitly
assumed to be symmetric. In the real world, interdependent interactions are
generally asymmetric. To uncover and quantify the impact of asymmetry in
interdependence on network robustness, we focus on percolation dynamics in
double-layer systems and implement the following failure mechanism: once a node
in a network layer fails, the damage it can cause depends not only on its
position in the layer but also on the position of its counterpart neighbor in
the other layer. We find that the characteristics of the percolation transition
depend on the degree of asymmetry, where the striking phenomenon of a switch in
the nature of the phase transition from first- to second-order arises. We
derive a theory to calculate the percolation transition points in both network
layers, as well as the transition switching point, with strong numerical
support from synthetic and empirical networks. Not only does our work shed
light upon the factors that determine the robustness of multilayer networks
against cascading failures, but it also provides a scenario by which the system
can be designed or controlled to reach a desirable level of resilience.
