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Asymptotic Subspace Consensus in Dynamic Networks

Matthias Függer, Thomas Nowak

TL;DR

It is shown that a large class of algorithms used for asymptotic consensus gracefully degrades to asymptotic subspace consensus in distributed systems with weaker assumptions on the communication network.

Abstract

We introduce the problem of asymptotic subspace consensus, which requires the outputs of processes to converge onto a common subspace while remaining inside the convex hull of initial vectors.This is a relaxation of asymptotic consensus in which outputs have to converge to a single point, i.e., a zero-dimensional affine subspace. We give a complete characterization of the solvability of asymptotic subspace consensus in oblivious message adversaries. In particular, we show that a large class of algorithms used for asymptotic consensus gracefully degrades to asymptotic subspace consensus in distributed systems with weaker assumptions on the communication network. We also present bounds on the rate by which a lower-than-initial dimension is reached.

Asymptotic Subspace Consensus in Dynamic Networks

TL;DR

It is shown that a large class of algorithms used for asymptotic consensus gracefully degrades to asymptotic subspace consensus in distributed systems with weaker assumptions on the communication network.

Abstract

We introduce the problem of asymptotic subspace consensus, which requires the outputs of processes to converge onto a common subspace while remaining inside the convex hull of initial vectors.This is a relaxation of asymptotic consensus in which outputs have to converge to a single point, i.e., a zero-dimensional affine subspace. We give a complete characterization of the solvability of asymptotic subspace consensus in oblivious message adversaries. In particular, we show that a large class of algorithms used for asymptotic consensus gracefully degrades to asymptotic subspace consensus in distributed systems with weaker assumptions on the communication network. We also present bounds on the rate by which a lower-than-initial dimension is reached.
Paper Structure (3 sections, 1 theorem, 1 figure)

This paper contains 3 sections, 1 theorem, 1 figure.

Table of Contents

  1. Introduction
  2. Computational Model \leanurl{DefComputationalModel.lean}
  3. Asymptotic Subspace Consensus

Key Result

Theorem 1

Any product of at least $\frac{\pi^2+6}{6}n + 1$ communication graphs that are $k$-rooted is $k$-broadcastable.

Figures (1)

  • Figure 1: Averaging algorithm running in dynamic networks.a Distributed system with $5$ nodes, executing an averaging algorithm in $\mathbb{R}^3$. Nodes start with initial values in $\mathbb{R}^3$ and average values received within the round's communication graph ($2$-rooted graphs shown). b--d Execution of the equal neighbor algorithm in $\mathbb{R}^3$ for $10$ rounds. Initial values (blue) and round-10 values (black) shown. b In a $1$-rooted oblivious message adversary, convergence onto a single point (black). c In a $2$-rooted oblivious message adversary, convergence onto a line (black) d In a $3$-rooted oblivious message adversary, convergence onto a plane (black). Animated (Movies \suppurl{Movie_S1_execution_1_root_animation.mp4}, \suppurl{Movie_S2_execution_2_roots_animation.mp4}, \suppurl{Movie_S3_execution_3_roots_animation.mp4}) and interactive (Documents \suppurl{Document_S1_execution_1_root_execution.html}, \suppurl{Document_S2_execution_2_roots_execution.html}, \suppurl{Document_S3_execution_3_roots_execution.html}) versions of b--d are provided as supplementary material. e Simulation of averaging algorithm within an oblivious message adversary with $2$-broadcastable communication graphs (broadcasting set marked in red). Initial values ($t=0$) and outputs until $t=4$ are shown (black). The linear subspace spanned by the processes in the current round's broadcasting set is shown (dashed red). Its polar coordinates ($\varphi$ and $\theta$) are seen to converge. An animated execution oscillating on a $1$-dimensional subspace is provided as Movie \suppurl{Movie_S4_execution_oscillating_on_1d_animation.mp4} and Document \suppurl{Document_S4_execution_oscillating_on_1d_execution.html}.

Theorems & Definitions (1)

  • Theorem 1: el2023asymptotically