# Controllability Gramian Spectra of Random Networks

arXiv: https://arxiv.org/abs/1609.04727 · HTML: https://mlsanity.com/paper/1609.04727/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Sep 2016.

## Built on by (most cited first, 1 of 1)

- [Controllability of Coarsely Measured Networked Linear Dynamical Systems (Extended Version)](https://mlsanity.com/paper/2206.10569/) (2022)

## Abstract (arXiv)

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.