  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.
