  We investigate notions of network centrality in terms of the underlying
coupling graph of the network, structure of exogenous uncertainties, and
communication time-delay. Our focus is on time-delay linear consensus networks,
where uncertainty is modeled by structured additive noise on the dynamics of
agents. The centrality measures are defined using the $\mathcal H_2$-norm of
the network. We quantify the centrality measures as functions of time-delay,
the graph Laplacian, and the covariance matrix of the input noise. Several
practically relevant uncertainty structures are considered, where we discuss
two notions of centrality: one w.r.t intensity of the noise and the other one
w.r.t coupling strength between the agents. Furthermore, explicit formulas for
the centrality measures are obtained for all types of uncertainty structures.
Lastly, we rank agents and communication links based on their centrality
indices and highlight the role of time-delay and uncertainty structure in each
scenario. Our counter intuitive grasp is that some of centrality measures are
highly volatile with respect to time-delay.
