Graph diameter, eigenvalues, and minimum-time consensus

TL;DR

This paper investigates the minimum number of linear iterations needed for average consensus on undirected graphs, providing counterexamples to a longstanding conjecture and identifying classes of graphs where the conjecture holds.

Contribution

It presents the first counterexample to the definitive consensus conjecture and establishes algebraic conditions under which the conjecture is valid, especially for distance-regular graphs.

Findings

Counterexample to the definitive consensus conjecture.

Improved lower bounds for linear consensus protocols.

Distance-regular graphs satisfy the conjecture.

Abstract

We consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed, undirected graph. We are motivated by the task of deriving lower bounds for consensus protocols and by the so-called "definitive consensus conjecture" which states that for an undirected connected graph G with diameter D there exist D matrices whose nonzero-pattern complies with the edges in G and whose product equals the all-ones matrix. Our first result is a counterexample to the definitive consensus conjecture, which is the first improvement of the diameter lower bound for linear consensus protocols. We then provide some algebraic conditions under which this conjecture holds, which we use to establish that all distance-regular graphs satisfy the definitive consensus conjecture.