Controllability of Large-Scale Networks: The Control Energy Exponents

We characterize the control energy behavior of large-scale linear network systems controlled by a single input. Specifically, we establish conditions under which two widely used control energy metrics, namely the inverse of the minimum eigenvalue and the normalized determinant of the controllability Gramian, grow exponentially fast in the system dimension and derive closed-form expressions for their asymptotic exponential growth rate. We apply our findings to structured and random network architectures.