Figure 2

Controllability characteristics. The variation of \(n_{\mathrm{D}}\) and \(m_{\mathrm{D}}\) in (a, d) ER networks, (b, e) EX networks, and (c, f) SF networks as the function of the coupling density P, the average degree \(\langle k \rangle \) and the power exponent \(\gamma \), respectively. The results of (g) \(n_{\mathrm{D}}\) and (j) \(m_{\mathrm{D}}\) in real networks with \(P=0\), \(P=0.2\), \(P=0.4\), and \(P=1\), respectively. Numbers refer to the network indices in Table 1. The results of (h) \(N_{\mathrm{D}}^\mathrm{rand-degree}\) and (k) \(M_{\mathrm{D}}^\mathrm{rand-degree}\) obtained from the degree-preserving randomized version of the real networks, compared with the real results, respectively. The theoretical predictions of (i) \(N_{{\mathrm{D}}}^{\mathrm{analytic}}\) and (l) \(M_{\mathrm{D}}^{\mathrm{analytic}}\) compared with the real results, respectively. All the numerical results are obtained by averaging over 100 independent network realizations.