Redundant interdependencies boost the robustness of multilayer networks

In the standard model of percolation on multilayer networks, a node is functioning only if its copies in all layers are simultaneously functioning. According to this model, a multilayer network becomes more and more fragile as the number of layers increases. In this respect, the addition of a new layer of interdependent nodes to a preexisting multilayer network will never improve its robustness. Whereas such a model seems appropriate to understand the effect of interdependencies in the simplest scenario of a network composed of only two layers, it may seem not reasonable for real systems where multiple network layers interact one with the other. It seems in fact unrealistic that a real system, such a living organism, evolved, through the development of multiple layers of interactions, towards a fragile structure. In this paper, we introduce a model of percolation where the condition that makes a node functional is that the node is functioning in at least two of the layers of the network. The model reduces to the standard percolation model for multilayer networks when the number of layers equals two. For larger number of layers, however, the model describes a scenario where the addition of new layers boosts the robustness of the system by creating redundant interdependencies among layers. We prove this fact thanks to the development of a message-passing theory able to characterize the model in both synthetic and real-world multilayer graphs.