A dynamical approach to efficient eigenvalue estimation in general multiagent networks

We propose a method to efficiently estimate the eigenvalues of any arbitrary (potentially weighted or directed) network of interacting dynamical agents in the presence of control inputs from dynamical observations. These observations are discrete, temporal measurements of the evolution of the aggregated outputs from a subset of agents (potentially one) during a finite time horizon. Notably, we do not require knowledge of which agents contribute to our measurements. We propose an efficient algorithm to exactly recover the (potentially complex) eigenvalues corresponding to network modes which are observable from the output measurements. The length of the sequence of measurements required by our method to generate a full reconstruction of the observable eigenvalue spectrum is at most three times the number of agents in the network, but in practice fewer are required, dependent on the number of observable network modes. The proposed technique can be applied to networks of non-autonomous multiagent systems with arbitrary dynamics in both continuous- and discrete-time. Finally, we illustrate our results with numerical simulations.