  Network robustness is a central point in network science, both from a
theoretical and a practical point of view. In this paper, we show that layer
degradation, understood as the continuous or discrete loss of links' weight,
triggers a structural transition revealed by an abrupt change in the algebraic
connectivity of the graph. Unlike traditional single layer networks, multiplex
networks exist in two phases, one in which the system is protected from link
failures in some of its layers and one in which all the system senses the
failure happening in one single layer. We also give the exact critical value of
the weight of the intra-layer links at which the transition occurs for
continuous layer degradation and its relation to the value of the coupling
between layers. This relation allows us to reveal the connection between the
transition observed under layer degradation and the one observed under the
variation of the coupling between layers.
