  For interdependent networks with identity dependency map, percolation is
exactly the same with that on a single network and follows a second-order phase
transition, while for random dependency, percolation 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 $ApEn_c$
below which the percolation is continuous, but for larger $ApEn$, it is a
first-order transition. With the increment of $ApEn$, the $p_c$ increases until
$ApEn$ reaching ${ApEn}_c'$ 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.
