We study the percolation in coupled networks with both inner-dependency and
inter-dependency links, where the inner- and inter-dependency links represent
the dependencies between nodes in the same or different networks, respectively.
We find that when most of dependency links are inner- or inter-ones, the
coupled networks system is fragile and makes a discontinuous percolation
transition. However, when the numbers of two types of dependency links are
close to each other, the system is robust and makes a continuous percolation
transition. This indicates that the high density of dependency links could not
always lead to a discontinuous percolation transition as the previous studies.
More interestingly, although the robustness of the system can be optimized by
adjusting the ratio of the two types of dependency links, there exists a
critical average degree of the networks for coupled random networks, below
which the crossover of the two types of percolation transitions disappears, and
the system will always demonstrate a discontinuous percolation transition. We
also develop an approach to analyze this model, which is agreement with the
simulation results well.Comment: 9 pages, 4 figure
