Distributed Estimation of the Laplacian Spectrum via Wave Equation and Distributed Optimization

This paper presents a distributed algorithm to estimate all distinct eigenvalues of the Laplacian matrix encoding the unknown topology of a multi-agent system. The agents interact according to the discrete-time wave equation so that their state trajectory persistently oscillates with modes that depend on the eigenvalues of the Laplacian matrix. In this way, the problem of distributed estimation of the eigenvalues of the Laplacian is recast into that of estimating the modes of evolution of the state-trajectory of a linear dynamical system. Unlike previous literature, this paper formulates a distributed optimization problem where, by considering its own state trajectory, each agent estimates all distinct eigenvalues of the Laplacian matrix. The main advantages of the proposed algorithm are the ability of each agent to estimate also eigenvalues corresponding to modes unobservable from its own state trajectory, a much greater numerical stability, and therefore improved scalability to large networks wit h respect to competing approaches, as evidenced by the numerical comparisons.