Consensus State Gram Matrix Estimation for Stochastic Switching Networks from Spectral Distribution Moments

TL;DR

This paper introduces a method to estimate the Gram matrix of error in distributed consensus over stochastic switching networks using spectral distribution moments, enhancing filter design for faster convergence.

Contribution

It develops a novel approach to approximate the error Gram matrix in random networks by leveraging spectral moments and Monte Carlo simulation, aiding in filter optimization.

Findings

The spectral distribution moments can be estimated via Monte Carlo methods.

The proposed approach improves consensus error estimation in stochastic networks.

Simulation results validate the effectiveness of the spectral moment-based approximation.

Abstract

Reaching distributed average consensus quickly and accurately over a network through iterative dynamics represents an important task in numerous distributed applications. Suitably designed filters applied to the state values can significantly improve the convergence rate. For constant networks, these filters can be viewed in terms of graph signal processing as polynomials in a single matrix, the consensus iteration matrix, with filter response evaluated at its eigenvalues. For random, time-varying networks, filter design becomes more complicated, involving eigendecompositions of sums and products of random, time-varying iteration matrices. This paper focuses on deriving an estimate for the Gram matrix of error in the state vectors over a filtering window for large-scale, stationary, switching random networks. The result depends on the moments of the empirical spectral distribution, which can be estimated through Monte-Carlo simulation. This work then defines a quadratic objective function to minimize the expected consensus estimate error norm. Simulation results provide support for the approximation.