Controlling Complex Networks: How Much Energy Is Needed?

The outstanding problem of controlling complex networks is relevant to many areas of science and engineering, and has the potential to generate technological breakthroughs as well. We address the physically important issue of the energy required for achieving control by deriving and validating scaling laws for the lower and upper energy bounds. These bounds represent a reasonable estimate of the energy cost associated with control, and provide a step forward from the current research on controllability toward ultimate control of complex networked dynamical systems.

Figure 1

Lower bound of the energy cost $E_{\min }\equiv 1/{\eta }_{\max }$ versus the control time $T_f$. All networks are weighted BA scale-free networks except one weighted ER random network in (a), with the same size $N=500$ and $⟨s⟩=20$. The $s_c$ denotes the strength of directly controlled node. In (a) $a=-150$ which makes $\mathbf{A}$ PD. In (b) $a=-50$ thus $\mathbf{A}$ is not PD. The dashed line in the semilog plot in (b) has a slope $2{\lambda }_N$. The symbols represent the same quantities calculated numerically and the solid lines represent the results from the estimation ${\eta }_{\max }\approx \mathrm{Tr}[\mathbf{H}]$.

Figure 2

Upper bound of the control energy cost $E_{\max }\equiv 1/{\eta }_{\min }$ for a weighted BA network with 20 nodes. In (a), $a=2$ thus $\mathbf{A}$ is ND. In (b), $a=-5$. In (c), $a=-20$ so that $\mathbf{A}$ is PD. In (a)–(c), (●) represent the upper bound $E_{\max }$ while (◀) represent the corresponding lower bound $E_{\min }$ (included for comparison). In (d) the decaying behavior of $E_{\max }$ is shown for different $s_c$ and $a$ values. The dashed line has a slope $-36$. In (e) the exponential decay of $E_{\max }$ for large $T_f$ is plotted for different values of $a$. The slopes of dashed lines are $2{\lambda }_1$, respectively. In (f) the constant values of the energy cost in (c) are shown as a function of $|a+s_c|$. The slope of the dashed line is 2.

Figure 3

(a) The ratio ${\epsilon }_{\mathrm{BA}}/{\epsilon }_{\mathrm{ER}}$ for different network size $N$. In order to eliminate the effects of nodal dynamics and strength, we fix the values of $a_{ii}$ and $⟨s⟩$. The results include the ratio for optimal driver node (◀) and the ratio of averaging over different driver nodes (●). The error bars are caused by different generations of network topology and link weights. (b) $E_{\min }$ (◀, left) and $E_{\max }$ (●, right) versus $n_c$, the number of directly controlled nodes. The dot pointed by the arrow corresponds to the node with largest degree in the network.