Upper bound of the minimum energy cost for controlling complex networks

For controlling complex systems ultimately, recent years have witnessed a surge of interest in the controllability of complex networks. Controllability, a system's basic attribute, represents whether we could control the system from any initial to any final states with appropriate external inputs within a finite time. However, in order to implement control in realistic systems, it is far from enough by solely detecting the so-called controllability. In other words, the minimum energy cost, with which to actuate the evolution of system states, must be systematically investigated to accomplish the control of various complex systems. Here we show the results on the scaling behavior of control energy for controlling complex networks. Specifically, we focus on the upper bound of the minimum control energy over all possible control directions within a certain control distance between the initial and final states. Numerical validations on all of our theoretical results are also provided. Our results pave the way to implement realistic control over various complex networks with the minimum energy cost.