Figure 5: Optimal repairing strategies.
From: Multiple tipping points and optimal repairing in interacting networks
The optimal repairing procedure (least expensive in terms of the number of individual node repairs) depends on the initial condition of the collapsed system. The total cost of repair is 
and the problem of optimal repairing translates into finding the minimal Manhattan distance from the point (in the phase diagram) where the collapsed system is initially situated (Si) to the nearest border of the green region where it becomes fully functional. For a system having the initial condition within the red section (for example, point S1), there are two solutions: it is equally optimal to reach any of the two triple points R1 and R2 by decreasing 
and 
. For the systems starting in the yellow regions, it is optimal to reach only one triple point, R1, for the sector containing point S2, or R2 for the sector containing point S3. Starting in the dark blue regions it is optimal to decrease 
only, that is, repairing only network B. Similarly, in the light blue regions it is optimal to decrease 
only. Triple points play a crucial role when both networks are initially significantly damaged (red and yellow regions).
