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Estimating modes of a complex dynamical network from impulse response data: Structural and graph-theoretic characterizations

Abstract:

We examine the role played by a linear dynamical network's topology in inference of its eigenvalues from noisy impulse-response data. Specifically, for a canonical linear-time-invariant network dynamics, we relate the Cramer-Rao bounds on eigenvalue estimator performance (from impulse-response data) to structural properties of the transfer function and in turn, to the network's topological structure. We begin by reviewing and enhancing algebraic characterizations of such eigenvalue estimates, which are based on pole-residue and pole-zero representations of the network's dynamics. We use these results to characterize mode estimation in networks with slow-coherency structures, finding that stimulus and observation in each strongly connected network subgraph is needed for high-fidelity estimation. We also obtain spectral and graphical characterizations of estimator performance for other graph classes (e.g., trees) and for the general case. These characterizations are used to determine the role of measurement and actuation locations in estimation performance. Finally, application of our results in dynamical-network security is illustrated through a simple example, and a concrete procedure for network mode estimation that draws on our structural results is introduced to conclude the article.