On Distributed Average Consensus Algorithms

TL;DR

This work reframes distributed average consensus (AC) as a linearly constrained, exact problem and analyzes finite-time convergence through eigenstep and power-iteration paradigms. It shows that, for undirected graphs, exact AC can be achieved with minimal communication in $2N$ steps via back-substitution on a carefully structured lower-triangular factor, or via graph-filtering approaches that realize the same projection with ${\cal O}(N^2)$ additions; these approaches avoid problematic online normalizations and improve robustness compared to traditional eigenstep methods. For directed graphs, the authors extend the eigenstep framework with appropriate normalization, and discuss stability and error-accumulation concerns inherent to unsupervised gradient-like updates. The paper also connects AC to diffusion/adaptive networks, demonstrating how exact or accelerated AC can be integrated with LMS-type learning while keeping computational demands low. Overall, the results offer practical, low-complexity strategies for exact AC in large-scale distributed systems, with clear implications for diffusion, sensing, and networked signal processing.

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.

Figures (8)

• Figure 1: Cooperative network wit $N=20$ nodes, arbitrarily numbered.
• Figure 2: An undirected network with $N=53$ nodes, containing a linear strech sprouting from node 3 and exhibiting a few short branched paths.
• Figure 3: Convergence of several methods to the exact solution for the undirected network of Fig. \ref{['fig.net_asymm_20']} (with target vector normalized to $\mathbbm{1}$).
• Figure 4: Convergence of the eigenstep method for ($i$) $\boldsymbol{A}_o={ \boldsymbol{\cal L}}$, ($ii$) $\boldsymbol{A}_o=\boldsymbol{A}_c$, and ($iii$) power iteration with $\boldsymbol{A}_i=\boldsymbol{A}_c$, for the directed graph of Appendix A. In the former case, the curve lies flat until sudden, exact convergence.
• Figure 5: Directed graph constructed from the network of Fig. \ref{['fig.netill']}.