  We present a graph-theoretic approach to analyzing the robustness of
leader-follower consensus dynamics to disturbances and time delays. Robustness
to disturbances is captured via the system $\mathcal{H}_2$ and
$\mathcal{H}_{\infty}$ 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 which quantify the impact of disturbances and time-delays on the
leader-follower dynamics. We then provide tight characterizations of these
robustness metrics in Erdos-Renyi random graphs and random 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.
