Control energy scaling for target control of complex networks

To control complex networks in practice, recent work has focused on the control energy required to drive the associated system from an initial state to any final state within finite time when it is fully controllable. However, beyond the prohibitively high cost for controlling all nodes of a network, it is usually adequate to control some target nodes of most natural and technological networks. In other words, what we usually need is to implement target control frequently. Yet, understanding the control energy for target control remains an outstanding challenge so far. Here we theoretically present an efficient method to calculate the minimum control energy required for implementing target control of complex networks, which bypasses the sophisticated calculation of the traditional Gramian matrix of the original system. Surprisingly, we uncover that the scaling behavior is only determined by the controllable part of the network. Furthermore, for the upper and lower bounds of the minimum control energy, we systematically derive the exact scaling behavior in terms of the control time. In addition, for controlling temporal networks composed of a sequence of uncontrollable snapshots, we demonstrate that our method offers a more efficient and effective way for analyzing the associated control energy. Our theoretical results are all verified numerically, which paves the way for implementing realistic target control over much broader applications.