Accelerating Consensus by Spectral Clustering and Polynomial Filters

TL;DR

This paper explores how second-order polynomial filters can speed up consensus processes on networks by clustering eigenvalues, proposing a preconditioner to optimize convergence speed while discussing robustness trade-offs.

Contribution

It introduces a novel preconditioner for spectral clustering of eigenvalues to enhance polynomial filter-based consensus acceleration.

Findings

Finite-time consensus on certain graphs

Preconditioner improves convergence speed

Trade-off with robustness of polynomial filters

Abstract

It is known that polynomial filtering can accelerate the convergence towards average consensus on an undirected network. In this paper the gain of a second-order filtering is investigated. A set of graphs is determined for which consensus can be attained in finite time, and a preconditioner is proposed to adapt the undirected weights of any given graph to achieve fastest convergence with the polynomial filter. The corresponding cost function differs from the traditional spectral gap, as it favors grouping the eigenvalues in two clusters. A possible loss of robustness of the polynomial filter is also highlighted.