Optimal Filter Design for Signal Processing on Random Graphs: Accelerated Consensus

Optimal Filter Design for Consensus on Random Directed Graphs

“…These methods involve solving a system of equations that depends on the random matrix model to find a deterministic equivalent for the empirical spectral distribution of a large-scale matrix. For symmetric matrices, Girko's K1 Equation was applied to network adjacency matrices and consensus iteration matrices in [15,16,24], information that was then used to inform filter design optimization problems for consensus acceleration in [17][18][19]. For the non-symmetric random network models on which this paper focuses, a much more complex method shown, in abridged form, as Theorem 1 (Girko's K25 Equation) is required to perform analysis.…”

“…The parameter τ (small value chosen) fills this role, while the parameter κ (small value chosen) provides a transition region around the equality constraint. Note that some computation could be saved by simply transforming the complement of the region G from Theorem 1, but the above formulation is more directly analogous to that from [17,18]. By introducing ε to bound the maximum filter response magnitude squared and examining the response only at sample points Λ S ⊆ Λ κ,τ , an approximate solution to (19) can be found by solving the following problem.…”

“…The filter should be designed to optimize the convergence rate by minimizing the convergence factor 1 d ln (p (W ) − J ). For constant, random networks, [15][16][17][18] find deterministic approximations for the empirical spectral distribution of consensus iteration matrices for large-scale random, symmetric networks and use this for consensus acceleration filter design, while [19] handles networks with time-varying (switching) network topology. This paper extends previous work connecting spectral asymptotics to graph filter design for consensus acceleration by examining the filter design problem in the context of constant (not time-varying), random networks of large-scale.…”

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

“…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 signal processing: Filter design and spectral statistics

“…Finally, Section V provides concluding remarks. For an extended version of this paper, refer to [16].…”