Critical effects of overlapping of connectivity and dependence links on percolation of networks

In a recent work Parshani et al (2011 Proc. Natl Acad. Sci. USA 108 1007), dependence links have been introduced to the percolation model and used to study the robustness of the networks with such links, which shows that the networks are more vulnerable than the classical networks with only connectivity links. This model usually demonstrates a first order transition, rather than the second order transition found in classical network percolation. In this paper, considering the real situation that the interdependent nodes are usually connected, we study the cascading dynamics of networks when dependence links partially overlap with connectivity links. We find that the percolation transitions are not always sharpened by making nodes interdependent. For a high fraction of overlapping, the network is robust for random failures, and the percolation transition is second order, while for a low fraction of overlapping, the percolation process shows a first order transition. This work demonstrates that the crossover between two types of transitions does not only depend on the density of dependence links but also on the overlapping fraction of connectivity and dependence links. Using generating function techniques, we present exact solutions for the size of the giant component and the critical point, which are in good agreement with the simulations.