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We present a graph-theoretic approach to analyzing the robustness ofleader-follower consensus dynamics to disturbances and time delays. Robustnessto disturbances is captured via the system $\mathcal{H}_2$ and$\mathcal{H}_{\infty}$ norms and robustness to time delay is defined as themaximum allowable delay for the system to remain asymptotically stable. Ouranalysis is built on understanding certain spectral properties of the groundedLaplacian matrix that play a key role in such dynamics. Specifically, we givegraph-theoretic bounds on the extreme eigenvalues of the grounded Laplacianmatrix which quantify the impact of disturbances and time-delays on theleader-follower dynamics. We then provide tight characterizations of theserobustness metrics in Erdos-Renyi random graphs and random regular graphs.Finally, we view robustness to disturbances and time delay as networkcentrality metrics, and provide conditions under which a leader in a networkoptimizes each robustness objective. Furthermore, we propose a sufficientcondition under which a single leader optimizes both robustness objectivessimultaneously.

Robustness of Leader-Follower Networked Dynamical Systems