Robustness of Leader–Follower Networked Dynamical Systems

We present a graph-theoretic approach to analyze the robustness of leader-follower consensus dynamics to disturbances and time delays. Robustness to disturbances is captured via the system H₂ and H_∞ norms, and robustness to time delay is defined as the maximum-allowable delay for the system to remain asymptotically stable. Our analysis is built on understanding certain spectral properties of the grounded Laplacian matrix that play a key role in such dynamics. Specifically, we give graph-theoretic bounds on the extreme eigenvalues of the grounded Laplacian matrix that quantify the impact of disturbances and time delays on the leader-follower dynamics. We then provide tight characterizations of these robustness metrics in Erdös-Rényi random graphs and random d-regular graphs. Finally, we view robustness to disturbances and time delay as network centrality metrics, and provide conditions under which a leader in a network optimizes each robustness objective. Furthermore, we propose a sufficient condition under which a single leader optimizes both robustness objectives simultaneously.