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MaxCUCL: Max-Consensus with Deterministic Convergence in Networks with Unreliable Communication

Apostolos I. Rikos, Themistoklis Charalambous, Karl H. Johansson

TL;DR

The paper tackles deterministic max-consensus in networks with unreliable communication by introducing MaxCUCL, a two-phase distributed algorithm that uses narrowband error-free feedback channels for acknowledgments and termination signaling. Phase1 runs a $D$-step max-consensus while marking links, and Phase2 exchanges $\text{dflag}$ to detect convergence, enabling distributed termination when all $\text{dflag}$ become zero. The authors prove deterministic finite-time convergence and provide a probabilistic bound on termination time, with explicit dependence on the network diameter $D$ and packet-drop characteristics. An environmental-monitoring sensor network example demonstrates practical applicability and illustrates how diameter and dropout probabilities affect convergence. The work contributes a first deterministic guarantee for max-consensus under packet drops and a distributed termination mechanism, with insights for robust coordination in resource-constrained networks.

Abstract

In this paper, we present a novel distributed algorithm (herein called MaxCUCL) designed to guarantee that max-consensus is reached in networks characterized by unreliable communication links (i.e., links suffering from packet drops). Our proposed algorithm is the first algorithm that achieves max-consensus in a deterministic manner (i.e., nodes always calculate the maximum of their states regardless of the nature of the probability distribution of the packet drops). Furthermore, it allows nodes to determine whether convergence has been achieved (enabling them to transition to subsequent tasks). The operation of MaxCUCL relies on the deployment of narrowband error-free feedback channels used for acknowledging whether a packet transmission between nodes was successful. We analyze the operation of our algorithm and show that it converges after a finite number of time steps. Finally, we demonstrate our algorithm's effectiveness and practical applicability by applying it to a sensor network deployed for environmental monitoring.

MaxCUCL: Max-Consensus with Deterministic Convergence in Networks with Unreliable Communication

TL;DR

The paper tackles deterministic max-consensus in networks with unreliable communication by introducing MaxCUCL, a two-phase distributed algorithm that uses narrowband error-free feedback channels for acknowledgments and termination signaling. Phase1 runs a -step max-consensus while marking links, and Phase2 exchanges to detect convergence, enabling distributed termination when all become zero. The authors prove deterministic finite-time convergence and provide a probabilistic bound on termination time, with explicit dependence on the network diameter and packet-drop characteristics. An environmental-monitoring sensor network example demonstrates practical applicability and illustrates how diameter and dropout probabilities affect convergence. The work contributes a first deterministic guarantee for max-consensus under packet drops and a distributed termination mechanism, with insights for robust coordination in resource-constrained networks.

Abstract

In this paper, we present a novel distributed algorithm (herein called MaxCUCL) designed to guarantee that max-consensus is reached in networks characterized by unreliable communication links (i.e., links suffering from packet drops). Our proposed algorithm is the first algorithm that achieves max-consensus in a deterministic manner (i.e., nodes always calculate the maximum of their states regardless of the nature of the probability distribution of the packet drops). Furthermore, it allows nodes to determine whether convergence has been achieved (enabling them to transition to subsequent tasks). The operation of MaxCUCL relies on the deployment of narrowband error-free feedback channels used for acknowledging whether a packet transmission between nodes was successful. We analyze the operation of our algorithm and show that it converges after a finite number of time steps. Finally, we demonstrate our algorithm's effectiveness and practical applicability by applying it to a sensor network deployed for environmental monitoring.
Paper Structure (9 sections, 1 theorem, 11 equations, 5 figures)

This paper contains 9 sections, 1 theorem, 11 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. System Model for Packet Dropping Links and Feedback Channels
  4. Node Operation
  5. Problem Formulation
  6. Main Results
  7. Convergence Analysis
  8. Application: Distributed Environmental Monitoring in Sensor Networks
  9. Conclusions and Future Directions

Key Result

Theorem 1

Consider a strongly connected digraph $\mathcal{G}_d = (\mathcal{V}, \mathcal{E})$ with $n = |\mathcal{V}|$ nodes and $m = |\mathcal{E}|$ edges. Each node $v_j \in \mathcal{V}$ has an initial state $x_j[0] \in \mathds{R}$ and Assumptions assum_feedback_channel, str_conn, and diam_known hold. During

Figures (5)

  • Figure 1: Example of digraph for execution of MaxCUCL algorithm.
  • Figure 2: Node states plotted against the number of iterations during execution of MaxCUCL for the digraph shown in Fig. \ref{['prob_example']}.
  • Figure 3: Execution of MaxCUCL over a random digraph of $20$ nodes with diameter $D=4$.
  • Figure 4: Required executions of Phase1 and Phase2 for MaxCUCL to converge, for random digraphs with diameters $D=3, 5, 7$.
  • Figure 5: Required executions of Phase1 and Phase2 for MaxCUCL to converge, for $q_{ji} = 0.9, 0.93, 0.96$, and $0.99$ for each $(v_j, v_i) \in \mathcal{E}$, and diameter $D = 4$.

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Remark 2