This paper presents a new method for estimating the eigenvalues of the Laplacian matrix associated with the graph describing the network topology of a multi-agent system. Given an approximate value of the average of the initial condition of the network state and some intermediate values of the network state when performing a Laplacian-based average consensus, the estimation of the Laplacian eigenvalues is obtained by solving the factorization of the averaging matrix. For this purpose, in contrast to the state of the art, we formulate a convex optimization problem that is solved in a distributed way by means of the Alternating Direction Method of Multipliers (ADMM). The main variables in the optimization problem are the coefficients of a polynomial whose roots are precisely the inverse of the distinct nonzero Laplacian eigenvalues. The performance of the proposed method is evaluated by means of simulation results.

In this paper, we present a novel algorithm for estimating eigenvalues of the Laplacian matrix associated with the graph describing the network topology of a multi-agent system or a wireless sensor network. As recently shown, the average consensus matrix can be written as a product of Laplacian based consensus matrices whose stepsizes are given by the inverse of the nonzero Laplacian eigenvalues. Therefore, by solving the factorization of the average consensus matrix, we can infer the Laplacian eigenvalues. We show how solving such a matrix factorization problem in a distributed way. In particular, we formulate the problem as a constrained consensus problem. The proposed algorithm does not require great resources in both computation and storage. This algorithm can also be viewed as a way for decentralizing the design of finite-time average consensus protocol recently proposed in the literature. Eventually, the performance of the proposed algorithm is evaluated by means of simulation results.