Distributed Estimation of Graph Spectrum

TL;DR

This paper presents a novel distributed algorithm for nodes in a graph to collaboratively estimate the spectrum of a matrix associated with the graph, applicable to adjacency and Laplacian matrices, with proven convergence properties.

Contribution

The paper introduces a two-stage distributed algorithm that estimates graph spectrum using linear iterations, Cayley-Hamilton theorem, and Lyapunov or perturbation methods for convergence.

Findings

Algorithm successfully estimates spectrum in simulations

Convergence is exponential if W is cyclic

Approximate solutions with small error using perturbation approach

Abstract

In this paper, we develop a two-stage distributed algorithm that enables nodes in a graph to cooperatively estimate the spectrum of a matrix $W$ associated with the graph, which includes the adjacency and Laplacian matrices as special cases. In the first stage, the algorithm uses a discrete-time linear iteration and the Cayley-Hamilton theorem to convert the problem into one of solving a set of linear equations, where each equation is known to a node. In the second stage, if the nodes happen to know that $W$ is cyclic, the algorithm uses a Lyapunov approach to asymptotically solve the equations with an exponential rate of convergence. If they do not know whether $W$ is cyclic, the algorithm uses a random perturbation approach and a structural controllability result to approximately solve the equations with an error that can be made small. Finally, we provide simulation results that illustrate the algorithm.