Vulnerability of continuous-time network synchronization processes: A minimum energy perspective

We characterize the vulnerability of a linear network synchronization process to intrusion by an adversary that can actuate a single network component. Specifically, we model the intruder as seeking to move the state of the synchronization process to an undesirable value or set (which may or may not be known to system operators) via a local actuation. We evaluate the network vulnerability in terms of the whether or not the intruder can achieve its goal, and also the minimum actuation energy (or expected minimum energy, if the goal is unknown) required of the adversary to achieve the goal. We formalize that the required energy is related to the inverse of the controllability Gramian for the process, and statistics defined thereof (e.g., its trace and determinant). We then obtain explicit formulas for the Gramian inverse and its associated statistics. These explicit formulae yield interesting structural and graph-theoretic characterizations of the energy-based vulnerability measures.