Notions of centrality in consensus protocols with structured uncertainties

We introduce new insights into the network centrality based not only on the network topology but also on the network dynamics. The focus of this paper is on the class of uncertain linear consensus networks in continuous time, where the network uncertainty is modeled by structured additive Gaussian white noise input on the update dynamics of each agent. The performance of the network is measured by the expected dispersion of its states in steady-state. This measure is equal to the square of the H2-norm of the network, and it quantifies the extent by which its state is away from the consensus state in steady-state. We show that this performance measure can be explicitly expressed as a function of the Laplacian matrix of the network and the covariance matrix of the noise input. We investigate several structures for the noise input and provide engineering insights on how each uncertainty structure can be relevant in real-world settings. Then, a new centrality index is defined to assess the influence of each agent or link on the network performance. For each noise structure, the value of the centrality index is calculated explicitly, and it is shown that how it depends on the network topology as well as the noise structure. Our results assert that agents or links can be ranked according to this centrality index and their rank can drastically change from the lowest to the highest, or vice versa, depending on the noise structure.

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