  We propose a theoretical framework to study the eigenvalue spectra of the
controllability Gramian of systems with random state matrices, such as
networked systems with a random graph structure. Using random matrix theory, we
provide expressions for the moments of the eigenvalue distribution of the
controllability Gramian. These moments can then be used to derive useful
properties of the eigenvalue distribution of the Gramian (in some cases, even
closed-form expressions for the distribution). We illustrate this framework by
considering system matrices derived from common random graph and matrix
ensembles, such as the Wigner ensemble, the Gaussian Orthogonal Ensemble (GOE),
and random regular graphs. Subsequently, we illustrate how the eigenvalue
distribution of the Gramian can be used to draw conclusions about the energy
required to control random system.
