Accelerating Consensus by Spectral Clustering and Polynomial Filters

It is known that polynomial filtering can accelerate the convergence toward average consensus on an undirected network. In this paper, the gain of a second-order filtering is investigated in more detail. 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 and can favor symmetry breaking. A possible loss of robustness of the polynomial filter is also highlighted.