Centrality Measures in Linear Consensus Networks With Structured Network Uncertainties

Abstract

We propose new insights into the network centrality based not only on the network graph, but also on a more structured model of network uncertainties. 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 $\mathcal {H}_2$ -norm of the network, and it quantifies the extent by which its state deviates 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 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 in order 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 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. This fact hints at emergence of fundamental tradeoffs on network centrality in the presence of multiple concurrent network uncertainties with different structures.