  Controlling complex networks is of paramount importance in science and
engineering. Despite recent efforts to improve controllability and synchronous
strength, little attention has been paid to the speed of pinning
synchronizability (rate of convergence in pinning control) and the
corresponding pinning node selection. To address this issue, we propose a
hypothesis to restrict the control cost, then build a linear matrix inequality
related to the speed of pinning controllability. By solving the inequality, we
obtain both the speed of pinning controllability and optimal control strength
(feedback gains in pinning control) for all nodes. Interestingly, some
low-degree nodes are able to achieve large feedback gains, which suggests that
they have high influence on controlling system. In addition, when choosing
nodes with high feedback gains as pinning nodes, the controlling speed of real
systems is remarkably enhanced compared to that of traditional large-degree and
large-betweenness selections. Thus, the proposed approach provides a novel way
to investigate the speed of pinning controllability and can evoke other
effective heuristic pinning node selections for large-scale systems.
