Abstract
The dependency among nodes has significant effects on the cascading failures of complex networks. Although the prevention of cascading failures on multilayered networks in which the failures of nodes in one layer affect the functioning of nodes in other layers has been widely investigated, the prevention of catastrophic cascade has rarely been addressed to single-layer networks where nodes are grouped and nodes within the same group are dependent on each other. For such networks, we find that it is already enough to prevent abrupt catastrophic collapses by randomly reinforcing a constant density of nodes. More importantly, we give the analytical solutions to the proportion of needed reinforced nodes for three typical networks, i.e., dependent Erds-Renyi (ER), random regular (RR), and scale-free (SF) networks. Interestingly, the density of reinforced nodes is a constant 0.1756, which holds true for ER networks with group size 2 regardless of average degree, RR, and SF networks with a large average degree. Also, we find the elegant expression of the density with any group size. In addition, we find a hybrid phase transition behavior, which is present in RR and SF networks while absent in ER networks. Our findings might shed some new light on designing more resilient infrastructure networks. Published under license by AIP Publishing.