Optimal Filter Design for Consensus on Random Directed Graphs

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.

Spectral Statistics of Directed Networks With Random Link Model Transpose-Asymmetry

“…One such approach periodically applies a filter to the consensus state every d iterations, where d is the filter degree. Examples that formulate optimization problems for filter design using the spectral asymptotics of large-scale random graphs include [21][22][23][24][25][26].…”

“…In particular, [26] uses Girko's K25 method for directed random network models with transpose-symmetry. It then proposes the following optimization problem for non-timevarying random networks to approximately optimize the convergence rate 1 d ln ρ (p(W ) − J ).…”

“…For example, consider distributed average consensus, the task of iteratively averaging all node data through only local network communications [9], which finds use in several applications [10][11][12][13]. Graph filters applied at each node can accelerate consensus convergence [14][15][16][17][18][19][20], which can benefit from asymptotic spectral information for suitable random networks [21][22][23][24][25][26].…”

“…Graph filters applied at each node can accelerate consensus convergence [14][15][16][17][18][19][20], which can benefit from asymptotic spectral information for suitable random networks [21][22][23][24][25][26].…”

“…The work in this paper accommodates random network models where the link directions have different probabilities, leading to random matrix models that have different distribution from their transposes. In contrast, [26] has a similar purpose but focuses more closely on the filter design aspects and only examines directed networks in which both link directions have identical distribution. Under the less restrictive conditions of this paper, additional analysis is necessary.…”