The effect of randomness for dependency map on the robustness of interdependent lattices

The percolation for interdependent networks with identical dependency map follows a second-order phase transition which is exactly the same with percolation on a single network, while percolation for random dependency follows a first-order phase transition. In real networks, the dependency relations between networks are neither identical nor completely random. Thus in this paper, we study the influence of randomness for dependency maps on the robustness of interdependent lattice networks. We introduce approximate entropy(ApEn) as the measure of randomness of the dependency maps. We find that there is critical ApEnc below which the percolation is continuous, but for larger ApEn, it is a first-order transition. With the increment of ApEn, the pc increases until ApEn reaching ApEnc (') and then remains almost constant. The time scale of the system shows rich properties as ApEn increases. Our results uncover that randomness is one of the important factors that lead to cascading failures of spatially interdependent networks.