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 and/or directed) network of interacting dynamical agents from dynamical observations. These observations are discrete, temporal measurements about the evolution of the outputs of a subset of agents (potentially one) during a finite time horizon; notably, we do not require knowledge of which agents are contributing to our measurements. We propose an efficient algorithm to exactly recover the (potentially complex) eigenvalues corresponding to network modes that 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, twice the number of agents in the network, but smaller in practice. The proposed technique can be applied to networks of multiagent systems with arbitrary dynamics in both continuous- and discrete-time. Finally, we illustrate our results with numerical simulations.

💡 Research Summary

The paper addresses the problem of estimating the eigenvalues of an arbitrary network of interacting dynamical agents—potentially weighted, directed, with self‑loops and multi‑edges—using only a finite sequence of scalar output measurements. The authors assume that the observer has access to the output of a single agent or a weighted linear combination of a subset of agents, without knowing which agents contribute to the measurement or the initial state of the network. The central contribution is an efficient algorithm that exactly recovers all eigenvalues that are observable from the given output, with a worst‑case sample requirement of at most twice the number of agents (2n) and typically far fewer samples in practice.

The methodology begins with a discrete‑time linear system x