Consensus-based distributed estimation of Laplacian eigenvalues of undirected graphs

Abstract

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.

Distributed Estimation of Graph Laplacian Eigenvalues by the Alternating Direction of Multipliers Method

“…( Tran and Kibangou (2013)). Let λ 2 , · · · , λ D+1 ̸ = 0 be the D distinct nonzero eigenvalues of the graph Laplacian matrix L, then, up to permutation, the sequence {α i } i=1,··· ,D , with α i = unique sequence, which allows getting the minimal factorization of the averaging matrix as…”

“…Instead of solving the non-convex optimization problem (7) as in Tran and Kibangou (2013), we first reformulate the problem into a convex one and then solve it in a distributed way.…”

“…The later assumption means that several iterations of the average consensus protocol have been run prior to the distributed estimation of the Laplacian eigenvalues. Based on these assumptions, in Tran and Kibangou (2013), the authors proposed to solve the following problem in a distributed way:…”

“…However, this method is only applicable to networks having nodes with sufficient storage and computation capabilities. In Tran and Kibangou (2013), the authors have proposed a method for estimating the distinct nonzero Laplacian eigenvalues by solving the factorization of the averaging matrix as a product of Laplacian-based consensus matrices. Precisely, an equality constrained consensus problem, which turns to be a non-convex optimization problem, was formulated.…”

Distributed Design of Finite-time Average Consensus Protocols

“…In fact, for self-configuration of Laplacian based finite-time average consensus protocols, distributed estimation of Laplacian eigenvalues is required. Such a task can be carried out by means of distributed or decentralized algorithms such as those proposed in Tran and Kibangou (2013), Sahai et al (2012), and Franceschelli et al (2009).…”