Figure 1. Condition number C_W versus control precision e_X for random and scale-free networks.

Network size is N = 100 for (a–d) and 200 for (e–h), average degree is 〈k〉 = 6 for ER random networks [(a), (b), (e) and (f)] and 8 for BA scale-free networks [(c), (d), (g) and (h)]. Directional link probability between any pair of nodes is P_b = 0.1. Panels (a), (c), (e) and (g) show the scaling relation between the condition number C_W and the control precision e_X. Panels (b), (d), (f) and (h) show the fraction R_CW of the networks with a certain C_W number. The scaling relation holds within some C_W-e_X region with boundaries specified as the black dashed lines. The e_X values are not physically meaningful outside the boundaries that are defined according to the precision limit of computation. The thresholds of C_W and e_X used in the computations are 10¹² and 10⁻⁴, respectively, which are indicated as the blue dashed lines. The threshold values are chosen to lie within the physical boundaries so that the calculations for all C_W values are meaningful.