Figure 2: Structural and physical controllability measures in directed networks.

Structural controllability measure n_D versus directional edge probability P_b for (a) ER random networks and (b) BA scale-free networks of size N = 1000 and three values of the average degree (〈k〉 = 4, 6, and 8). The dash-dotted lines represent the results obtained from the cavity method^4,5, and the squares, triangles, and circles are simulation results from the maximum matching algorithm⁴. (c,d) Measure of physical controllability for ER random and BA scale-free networks of size N = 100, respectively, where is the probability that the condition number of the Gramian matrix is less than some physically reasonable threshold value. Comparing (a) with (c), or (b) with (d), we observe the striking phenomenon that, in the parameter regime where the number of driver nodes is minimized so that the corresponding networks are deemed to be most structurally controllable, they are physically uncontrollable. The phenomenon persists regardless of the network size and type. All nodes are self-loop free. The qualitative behavior is robust against variations in the value of .