  To appropriately select control nodes of a large-scale network system, we
propose two control centralities called volumetric and average energy
controllability scores. The scores are the unique solutions to convex
optimization problems formulated using the controllability Gramian. The
uniqueness is proven for stable cases and for unstable cases that include
multi-agent systems. We show that the scores can be efficiently calculated by
using a proposed algorithm based on the projected gradient method onto the
standard simplex. Numerical experiments demonstrate that the proposed algorithm
is more efficient than an existing interior point method, and the proposed
scores can correctly capture the importance of each state node on
controllability, outperforming existing control centralities.
