Optimal Filter Design for Consensus on Random Directed Graphs

TL;DR

This paper extends spectral analysis and optimal filter design methods from undirected to directed random networks, using Girko's theorems to approximate eigenvalue distributions for improved consensus acceleration.

Contribution

It introduces a novel approach for designing filters in large-scale directed random networks by applying Girko's spectral theorems to handle complex eigenvalues.

Findings

Effective filter design for directed networks demonstrated through simulations

Spectral approximation methods enable better consensus acceleration

Limitations of the approach are discussed

Abstract

Optimal design of consensus acceleration graph filters relates closely to the eigenvalues of the consensus iteration matrix. This task is complicated by random networks with uncertain iteration matrix eigenvalues. Filter design methods based on the spectral asymptotics of consensus iteration matrices for large-scale, random undirected networks have been previously developed both for constant and for time-varying network topologies. This work builds upon these results by extending analysis to large-scale, constant, random directed networks. The proposed approach uses theorems by Girko that analytically produce deterministic approximations of the empirical spectral distribution for suitable non-Hermitian random matrices. The approximate empirical spectral distribution defines filtering regions in the proposed filter optimization problem, which must be modified to accommodate complex-valued eigenvalues. Presented numerical simulations demonstrate good results. Additionally, limitations of the proposed method are discussed.