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A Linear Push-Pull Average Consensus Algorithm for Delay-Prone Networks

Evagoras Makridis, Themistoklis Charalambous

TL;DR

The paper addresses the discrete-time average consensus problem on directed, delay-prone networks and introduces a linear, surplus-based push-pull algorithm (RPPAC) that tolerates asynchronous updates and heterogeneous bounded delays. RPPAC preserves total mass via a surplus variable and uses push/pull weights to maintain consensus in directed graphs, even when information arrivals are delayed. Convergence analysis employs an augmented digraph with virtual delay buffers and a forward/backward product of time-varying matrices, establishing that asymptotic average consensus is achieved if the base graph is strongly connected and delays are bounded, provided the surplus gain \gamma is sufficiently small. Simulations on a 10-node network demonstrate convergence to the true average (e.g., $\\bar{x}=5.5$ for $x_j(0)=j$) with varying delay bounds, and show how spectral properties and $\\gamma$ tuning influence convergence rate. The work provides a linear, scalable solution for robust average consensus in realistic communication scenarios, and suggests future work on optimal gain selection and extensions to distributed optimization.

Abstract

In this paper, we address the average consensus problem of multi-agent systems for possibly unbalanced and delay-prone networks with directional information flow. We propose a linear distributed algorithm (referred to as RPPAC) that handles asynchronous updates and time-varying heterogeneous information delays. Our proposed distributed algorithm utilizes a surplus-consensus mechanism and information regarding the number of incoming and outgoing links to guarantee state averaging, despite the imbalanced and delayed information flow in directional networks. The convergence of the RPPAC algorithm is examined using key properties of the backward product of time-varying matrices that correspond to different snapshots of the directional augmented network.

A Linear Push-Pull Average Consensus Algorithm for Delay-Prone Networks

TL;DR

The paper addresses the discrete-time average consensus problem on directed, delay-prone networks and introduces a linear, surplus-based push-pull algorithm (RPPAC) that tolerates asynchronous updates and heterogeneous bounded delays. RPPAC preserves total mass via a surplus variable and uses push/pull weights to maintain consensus in directed graphs, even when information arrivals are delayed. Convergence analysis employs an augmented digraph with virtual delay buffers and a forward/backward product of time-varying matrices, establishing that asymptotic average consensus is achieved if the base graph is strongly connected and delays are bounded, provided the surplus gain \gamma is sufficiently small. Simulations on a 10-node network demonstrate convergence to the true average (e.g., for ) with varying delay bounds, and show how spectral properties and tuning influence convergence rate. The work provides a linear, scalable solution for robust average consensus in realistic communication scenarios, and suggests future work on optimal gain selection and extensions to distributed optimization.

Abstract

In this paper, we address the average consensus problem of multi-agent systems for possibly unbalanced and delay-prone networks with directional information flow. We propose a linear distributed algorithm (referred to as RPPAC) that handles asynchronous updates and time-varying heterogeneous information delays. Our proposed distributed algorithm utilizes a surplus-consensus mechanism and information regarding the number of incoming and outgoing links to guarantee state averaging, despite the imbalanced and delayed information flow in directional networks. The convergence of the RPPAC algorithm is examined using key properties of the backward product of time-varying matrices that correspond to different snapshots of the directional augmented network.

Paper Structure

This paper contains 10 sections, 1 theorem, 19 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Network Model
  4. Problem Setup
  5. Average Consensus using Push-Pull Weights (PPAC)
  6. Algorithm Development
  7. Robustified Push-Pull Average Consensus (RPPAC)
  8. Convergence Analysis
  9. Simulation Results
  10. Conclusions and Future Directions

Key Result

Theorem 1

The algorithm in eq:augmented_rppac achieves asymptotic average consensus with the parameter $\gamma>0$ sufficiently small, if and only if the digraph $\mathcal{G}$ is strongly connected, and the transmission delays are bounded, $\tau_{ji}(k) \leq \bar{\tau}_{ji} \leq \bar{\tau}<\infty$ for all $j,i

Figures (7)

  • Figure 1: Delay-prone digraph $\mathcal{G}$ comprised of 10 agents. Self-loops are allowed but not shown for ease of presentation.
  • Figure 2: State variable $x_j(k)$ at each agent .
  • Figure 3: Surplus variable $s_j(k)$ at each agent .
  • Figure 4: Mean square consensus error for different length of delays .
  • Figure 5: Mean spectral gap of $M(k)$ that corresponds to different for different upper bounds on the delays $\bar{\tau}=\{0,2,5\}$ .
  • ...and 2 more figures

Theorems & Definitions (4)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2