  In this paper, we investigate the linear controllability framework for
complex networks from a physical point of view. There are three main results.
(1) If one applies control signals as determined from the structural
controllability theory, there is a high probability that the control energy
will diverge. Especially, if a network is deemed controllable using a single
driving signal, then most likely the energy will diverge. (2) The energy
required for control exhibits a power-law scaling behavior. (3) Applying
additional control signals at proper nodes in the network can reduce and
optimize the energy cost. We identify the fundamental structures embedded in
the network, the longest control chains, which determine the control energy and
give rise to the power-scaling behavior. (To our knowledge, this was not
reported in any previous work on control of complex networks.) In addition, the
issue of control precision is addressed. These results are supported by
extensive simulations from model and real networks, physical reasoning, and
mathematical analyses.
  Notes on the submission history of this work: This work started in late 2012.
The phenomena of power-law energy scaling and energy divergence with a single
controller were discovered in 2013. Strategies to reduce and optimize control
energy was articulated and tested in 2013. The senior co-author (YCL) gave
talks about these results at several conferences, including the NETSCI 2014
Satellite entitled "Controlling Complex Networks" on June 2, 2014. The paper
was submitted to PNAS in September 2014 and was turned down. It was revised and
submitted to PRX in early 2015 and was rejected. After that it was revised and
submitted to Nature Communications in May 2015 and again was turned down.
