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Asynchronous Approximate Byzantine Consensus: A Multi-hop Relay Method and Tight Graph Conditions

Liwei Yuan, Hideaki Ishii

TL;DR

The multi-hop weighted mean subsequence reduced (MW-MSR) algorithm is developed, which describes a tight graph condition for this algorithm to achieve Byzantine consensus, expressed in the novel notion of strictly robust graphs.

Abstract

We study a multi-agent resilient consensus problem, where some agents are of the Byzantine type and try to prevent the normal ones from reaching consensus. In our setting, normal agents communicate with each other asynchronously over multi-hop relay channels with delays. To solve this asynchronous Byzantine consensus problem, we develop the multi-hop weighted mean subsequence reduced (MW-MSR) algorithm. The main contribution is that we characterize a tight graph condition for our algorithm to achieve Byzantine consensus, which is expressed in the novel notion of strictly robust graphs. We show that the multi-hop communication is effective for enhancing the network's resilience against Byzantine agents. As a result, we also obtain novel conditions for resilient consensus under the malicious attack model, which are tighter than those known in the literature. Furthermore, the proposed algorithm can be viewed as a generalization of the conventional flooding-based algorithms, with less computational complexity. Lastly, we provide numerical examples to show the effectiveness of the proposed algorithm.

Asynchronous Approximate Byzantine Consensus: A Multi-hop Relay Method and Tight Graph Conditions

TL;DR

The multi-hop weighted mean subsequence reduced (MW-MSR) algorithm is developed, which describes a tight graph condition for this algorithm to achieve Byzantine consensus, expressed in the novel notion of strictly robust graphs.

Abstract

We study a multi-agent resilient consensus problem, where some agents are of the Byzantine type and try to prevent the normal ones from reaching consensus. In our setting, normal agents communicate with each other asynchronously over multi-hop relay channels with delays. To solve this asynchronous Byzantine consensus problem, we develop the multi-hop weighted mean subsequence reduced (MW-MSR) algorithm. The main contribution is that we characterize a tight graph condition for our algorithm to achieve Byzantine consensus, which is expressed in the novel notion of strictly robust graphs. We show that the multi-hop communication is effective for enhancing the network's resilience against Byzantine agents. As a result, we also obtain novel conditions for resilient consensus under the malicious attack model, which are tighter than those known in the literature. Furthermore, the proposed algorithm can be viewed as a generalization of the conventional flooding-based algorithms, with less computational complexity. Lastly, we provide numerical examples to show the effectiveness of the proposed algorithm.
Paper Structure (19 sections, 16 equations, 8 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 16 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Setting
  3. Network Model under Multi-hop Communication
  4. Update Rule and Threat Models
  5. Resilient Asymptotic Consensus and Algorithm
  6. Strictly Robust Graphs with Multi-hop Communication
  7. The Notion of $(r,s)$-Robust Graphs with $l$ Hops
  8. The Notion of $r$-Strictly Robust Graphs with $l$ Hops
  9. Synchronous Byzantine Consensus
  10. Discussions on Different Graph Conditions
  11. Asynchronous Byzantine Consensus
  12. Consensus Analysis
  13. Comparison with Conventional Methods
  14. Advantages in Threat Models and Graph Conditions
  15. Advantages in Computational Complexity
  16. ...and 4 more sections

Figures (8)

  • Figure 1: The flow of determining $\overline{\mathcal{R}}_i[k]$.
  • Figure 2: (a) Node $i$ has two independent paths originating from the outside of $\mathcal{V}_1$ and do not go through the nodes in the set $\mathcal{F}=\{j\}$. (b) Node $i$ has only one independent path sharing the same property.
  • Figure 3: Both undirected graphs are not $2$-strictly robust with $1$ hop but are $2$-strictly robust with $2$ hops under the 1-local model.
  • Figure 4: (a) $3$-robust. (b) $2$-strictly robust. (c) $(2,2)$-robust.
  • Figure 5: Time responses of the synchronous one-hop W-MSR algorithm in the 7-node network of Fig. \ref{['1lcoal']}(a).
  • ...and 3 more figures