Robustness of n interdependent networks with partial support-dependence relationship

We study both analytically and numerically the robustness ofninterdependent networks with partial support-dependence relationship, which reflects real-world networks more realistically. For a starlike network ofnErd˝os-Renyi (ER) networks, we find that the system undergoes from second-order to first-order phase transition as coupling strengthqincreases. Moreover, we notice that the region of the first-order transition becomes larger, while the region of the second-order transition becomes smaller as the number of networksnincreases. However, for a starlike network ofnscale-free (SF) networks, the system undergoes from second-order through hybrid-order to first-order phase transition asqincreases. Furthermore, we also observe that the region of the first-order transition remains constant and appears only forq= 1, however, the region of hybrid-order transition gradually becomes larger and the region of the second-order transition becomes smaller asnincreases. For a looplike network ofnER networks, we find the giant componentp to be independent of the number of networks. Additionally, when the average degree of networks increases, the region of the first-order transition becomes smaller and the region of the second-order transition becomes larger. For the case ofnER networks with partial support-dependence relationship, as average supported degree ˜ ! ,ncoupled networks become independent and only second-order transition is observed, which is similar toq= 0. Copyright c !EPLA, 2013