On Distributed Average Consensus Algorithms

TL;DR

This paper introduces new iterative algorithms for distributed average consensus that are computationally efficient, requiring only linear complexity in the number of agents, and are applicable to both directed and undirected graphs.

Contribution

It proposes novel exact solutions for average consensus with reduced complexity, especially using the eigenstep method for finite-time convergence in directed graphs.

Findings

Achieves exact consensus with ${ m O}(N)$ additions.

Outperforms existing schemes requiring ${ m O}(KN^2)$ multiplications.

Provides algorithms applicable to directed and undirected graphs.

Abstract

Average consensus (AC) strategies play a key role in every system that employs cooperation by means of distributed computations. To promote consensus, an $N$-agent network can repeatedly combine certain node estimates until their mean value is reached. Such algorithms are typically formulated as (global) recursive matrix-vector products of size $N$, where consensus is attained either asymptotically or in finite time. We revisit some existing approaches in these directions and propose new iterative and exact solutions to the problem. Considering directed graphs, this is carried out by interplaying with standalone conterparts, while underpinned by the so-called eigenstep method of finite-time convergence. In particular, we focus on reducing complexity so as to require, overall, as little as ${\cal O}(N)$ additions to achieve the solution exactly. For undirected graphs, the latter compares favorably to existing schemes that require, in total, ${\cal O}(KN^2)$ multiplications to deliver the AC, where $K$ refers to the number of distinct eivenvalues of the underlying graph Laplacian matrix.