The Structured Controllability Radius of Symmetric (Brain) Networks

In this paper we propose and analyze a novel notion of controllability of network systems with linear dynamics and symmetric weights. Namely, we quantify the controllability degree of a network with its distance from the set of uncontrollable networks with the same structure, that is, with the minimum Frobenius norm of a structured perturbation rendering the network uncontrollable (structured controllability radius). We derive analytical conditions to compute the structured controllability radius of a network with symmetric weights, and illustrate our results through a number of examples. In particular, we use our theoretical results to study the controllability properties of a set of brain networks reconstructed from diffusion MRI data, and compare them with the controllability properties of a class of random networks. Our results show that brain networks feature a controllability radius that is consistently smaller than the one of random networks with similar weights, indicating that the considered brain networks may not be optimized to favor controllability.