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.

我们根据网络的基础耦合图，外部不确定性的结构和通信时间延迟来研究网络中心性的概念。我们的重点是时滞线性共识网络，其中不确定性是通过关于代理动力学的结构化加性噪声建模的。集中度度量使用${\mathcal{H}}_2$-网络规范。我们将中心性度量量化为时间延迟，图拉普拉斯算子和输入噪声的协方差矩阵的函数。考虑了几种与实际相关的不确定性结构，在这里我们讨论了两个中心性概念：一种是噪声的强度，另一种是代理之间的耦合强度。此外，针对所有类型的不确定性结构获得了集中度度量的明确公式。最后，我们根据代理人和通信链接的中心度对它们进行排名，并突出时间延迟和不确定性结构在每种情况下的作用。我们的违反直觉的理解是，某些中心性措施在时间延迟方面非常不稳定。