Sustaining a network by controlling a fraction of nodes

Multi-stability is a widely observed phenomenon in real complex networked systems, such as technological infrastructures, ecological systems, gene regulation, transportation and more. When a system functions normally but there exists also a potential state with abnormal low activity, although the system is at equilibrium it might make a transition into the low activity undesired state due to external disturbances and perturbations. Thus, such a system can be regarded as unsustainable, due to the danger of falling into the potential inactive state. Here we explore, analytically and by simulations, how supporting the activity of a fraction $\rho$ of nodes can turn an unsustainable system to be sustainable by eliminating the inactive potential stable state. We thus unveil a new sustainability phase diagram in the presence of a fraction of controlled nodes $\rho$. This phase diagram could provide guidelines to sustain a network by external intervention and/or by strengthening the connectivity of the network.