Finite-time consensus using stochastic matrices with positive diagonals

TL;DR

This paper explores conditions under which finite-time consensus can be achieved in networked systems using stochastic matrices with positive diagonals, highlighting differences between undirected and directed graphs.

Contribution

It establishes that finite-time average consensus is always possible for connected undirected graphs and identifies necessary conditions for directed graphs.

Findings

Finite-time consensus achievable in connected undirected graphs.

Necessary conditions include strong connectivity and even-length cycles in directed graphs.

Finite-time consensus not guaranteed in all directed graph configurations.

Abstract

We discuss the possibility of reaching consensus in finite time using only linear iterations, with the additional restrictions that the update matrices must be stochastic with positive diagonals and consistent with a given graph structure. We show that finite-time average consensus can always be achieved for connected undirected graphs. For directed graphs, we show some necessary conditions for finite-time consensus, including strong connectivity and the presence of a simple cycle of even length.