Fig. 2: Theory. The forced system has a new phase diagram.
From: Sustaining a network by controlling a fraction of nodes
a The relation, Eq. (8), between the system state (〈x〉) and its connectivity (β) provides the phase diagrams of both free (ρ = 0) and controlled (ρ > 0) systems for any dynamics. Here we demonstrate this for three distinct systems. b, c Cellular dynamics. b A free system diagram shows a suppressed function at x0 for small β, and a bi-stable regime for large β, where x0 and x1 both exist and are stable. Thus the system is unsustainable and has a risk to fail into a nonfunctional state. c In contrast, a forced system, controlled by holding a fraction ρ of nodes with high-value Δ, shows a new phase diagram, having an s-shape curve, with a new region for large β in which only x1 appears (blue). Thus, a system in the blue region is now safe and not only active, but also sustainable. d, e Brain dynamics. d A free system exhibits three regimes: inactive for sparse topology, active for dense topology, and bi-stable in between. e Forcing the system with certain Δ and ρ pushes the twist of the s-shape to a lower β-value, and as such, makes the blue regime becoming sustainable rather than bi-stable. f, g Spin dynamics. f A free system has a zero inactive stable state for sparse connectivity, while for dense connectivity, there are two symmetric active states. g Controlled system shows a new phase diagram including a sustainable regime where only the positive stable state appears (blue shade). This dynamics does not fall into the formula in (a), however, a similar analysis can be done, see Supplementary Note 4 Section 4.3.
