Accelerating Consensus by Spectral Clustering and Polynomial Filters
For researchers in distributed computing and multi-agent systems, this work provides a method to improve consensus speed, though it is an incremental extension of known filtering techniques.
The paper investigates how second-order polynomial filtering can accelerate consensus in undirected networks, identifying graphs where finite-time consensus is possible and proposing a preconditioner to optimize convergence speed by clustering eigenvalues. It also highlights a potential robustness issue with the polynomial filter.
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.