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Resilient Quantized Consensus in Multi-Hop Relay Networks

Liwei Yuan, Hideaki Ishii

TL;DR

The paper tackles resilient quantized consensus in directed multi-hop relay networks subject to asynchronous updates and delays by introducing the QMW-MSR algorithm, which combines a multi-hop MSR strategy with a randomized quantizer. It provides necessary and sufficient graph conditions based on $l$-hop robustness (and strict robustness for Byzantine attacks) to guarantee almost-sure convergence to consensus among normal agents despite adversaries. The approach demonstrates improved resilience with fewer relay hops compared to flooding-based methods and extends to both synchronous and asynchronous settings, including scenarios with delays. Numerical examples corroborate the theoretical results, showing practical gains in convergence speed and fault tolerance for various network topologies and attack models.

Abstract

We study resilient quantized consensus in multi-agent systems, where some agents may malfunction. The network consists of agents taking integer-valued states, and the agents' communication is subject to asynchronous updates and time delays. We utilize the quantized weighted mean subsequence reduced algorithm where agents communicate with others through multi-hop relays. We prove necessary and sufficient conditions for our algorithm to achieve the objective under the malicious and Byzantine attack models. Our approach has tighter graph conditions compared to the one-hop algorithm and the flooding-based algorithms for binary consensus. Numerical examples verify the efficacy of our algorithm.

Resilient Quantized Consensus in Multi-Hop Relay Networks

TL;DR

The paper tackles resilient quantized consensus in directed multi-hop relay networks subject to asynchronous updates and delays by introducing the QMW-MSR algorithm, which combines a multi-hop MSR strategy with a randomized quantizer. It provides necessary and sufficient graph conditions based on -hop robustness (and strict robustness for Byzantine attacks) to guarantee almost-sure convergence to consensus among normal agents despite adversaries. The approach demonstrates improved resilience with fewer relay hops compared to flooding-based methods and extends to both synchronous and asynchronous settings, including scenarios with delays. Numerical examples corroborate the theoretical results, showing practical gains in convergence speed and fault tolerance for various network topologies and attack models.

Abstract

We study resilient quantized consensus in multi-agent systems, where some agents may malfunction. The network consists of agents taking integer-valued states, and the agents' communication is subject to asynchronous updates and time delays. We utilize the quantized weighted mean subsequence reduced algorithm where agents communicate with others through multi-hop relays. We prove necessary and sufficient conditions for our algorithm to achieve the objective under the malicious and Byzantine attack models. Our approach has tighter graph conditions compared to the one-hop algorithm and the flooding-based algorithms for binary consensus. Numerical examples verify the efficacy of our algorithm.

Paper Structure

This paper contains 20 sections, 9 theorems, 44 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Quantized Consensus and Update Rule
  4. Attack Models
  5. Resilient Quantized Consensus and Algorithm
  6. Graph Robustness with Multi-Hop Communication
  7. Robustness with Multi-Hop Communication
  8. Strict Robustness with Multi-Hop Communication
  9. The Case of Unbounded Path Length
  10. Classes of Graphs Satisfying Robustness Conditions
  11. Synchronous Network
  12. Asynchronous Network
  13. Asynchronous Updates without Delays
  14. Asynchronous Updates with Delays
  15. Quantized Consensus under the Byzantine Model
  16. ...and 5 more sections

Key Result

Lemma 1

For the following conditions on any directed graph $\mathcal{G} = (\mathcal{V},\mathcal{E})$ under the $f$-total or $f$-local model ($l\geq1$): $(A)$$\mathcal{G}$ is $(2f+1)$-robust with $l$ hops, $(B)$$\mathcal{G}$ is $(f+1)$-strictly robust with $l$ hops, $(C)$$\mathcal{G}$ is $(f + 1,f+1)$-robust

Figures (7)

  • Figure 1: (a) The cycle graph is not (2, 2)-robust with $l\leq 3$ hops but is (2,2)-robust with 4 hops. (b) The graph is 2-strictly robust with 2 hops and hence is (2,2)-robust with 2 hops.
  • Figure 2: Illustration for graphs satisfying our conditions. (a) Complete bipartite graph $\mathcal{K}_{3,3}$ having robustness with $l$ hops. (b) Wheel graph $\mathcal{W}_7$ having strict robustness with $l$ hops.
  • Figure 3: Time responses of the synchronous QMW-MSR algorithm under the 1-total malicious model in the network of Fig. \ref{['graph1']}(a).
  • Figure 4: Time responses of the asynchronous four-hop QMW-MSR algorithm under the 1-total malicious model without delays in the network of Fig. \ref{['graph1']}(a).
  • Figure 5: Time responses of the asynchronous two-hop QMW-MSR algorithm under the 1-total malicious model with delays in the network of Fig. \ref{['graph1']}(b).
  • ...and 2 more figures

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 13 more