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Average consensus with resilience and privacy guarantees without losing accuracy

Guilherme Ramos, Daniel Silvestre, André M. H. Teixeira, Sérgio Pequito

TL;DR

The paper tackles private and resilient average consensus in discrete-time networks with up to $f$ faulty agents, aiming to preserve initial-state privacy while achieving exact average consensus among non-faulty nodes. It introduces a noise-scheduling privacy mechanism integrated with an augmented-state design and a block-structured weight design to maintain accurate averaging across subnetworks, without requiring left-eigenvector computations. The main contributions include a modular framework (weight design, finite-window noise injection, and state-exclusion-based selection) with formal guarantees of soundness and privacy under resilience, plus a polynomial-time complexity bound in $n$ and $f$, validated by two illustrative examples. This work has practical implications for secure and robust distributed coordination in networks where faults and eavesdropping risk privacy leakage, offering a scalable alternative to differential privacy while preserving accuracy.

Abstract

This paper addresses the challenge of achieving private and resilient average consensus among a group of discrete-time networked agents without compromising accuracy. State-of-the-art solutions to attain privacy and resilient consensus entail an explicit trade-off between the two with an implicit compromise on accuracy. In contrast, in the present work, we propose a methodology that avoids trade-offs between privacy, resilience, and accuracy. We design a methodology that, under certain conditions, enables non-faulty agents, i.e., agents complying with the established protocol, to reach average consensus in the presence of faulty agents, while keeping the non-faulty agents' initial states private. For privacy, agents strategically add noise to obscure their original state, while later withdrawing a function of it to ensure accuracy. Besides, and unlikely many consensus methods, our approach does not require each agent to compute the left-eigenvector of the dynamics matrix associated with the eigenvalue one. Moreover, the proposed framework has a polynomial time complexity relative to the number of agents and the maximum quantity of faulty agents. Finally, we illustrate our method with examples covering diverse faulty agents scenarios.

Average consensus with resilience and privacy guarantees without losing accuracy

TL;DR

The paper tackles private and resilient average consensus in discrete-time networks with up to faulty agents, aiming to preserve initial-state privacy while achieving exact average consensus among non-faulty nodes. It introduces a noise-scheduling privacy mechanism integrated with an augmented-state design and a block-structured weight design to maintain accurate averaging across subnetworks, without requiring left-eigenvector computations. The main contributions include a modular framework (weight design, finite-window noise injection, and state-exclusion-based selection) with formal guarantees of soundness and privacy under resilience, plus a polynomial-time complexity bound in and , validated by two illustrative examples. This work has practical implications for secure and robust distributed coordination in networks where faults and eavesdropping risk privacy leakage, offering a scalable alternative to differential privacy while preserving accuracy.

Abstract

This paper addresses the challenge of achieving private and resilient average consensus among a group of discrete-time networked agents without compromising accuracy. State-of-the-art solutions to attain privacy and resilient consensus entail an explicit trade-off between the two with an implicit compromise on accuracy. In contrast, in the present work, we propose a methodology that avoids trade-offs between privacy, resilience, and accuracy. We design a methodology that, under certain conditions, enables non-faulty agents, i.e., agents complying with the established protocol, to reach average consensus in the presence of faulty agents, while keeping the non-faulty agents' initial states private. For privacy, agents strategically add noise to obscure their original state, while later withdrawing a function of it to ensure accuracy. Besides, and unlikely many consensus methods, our approach does not require each agent to compute the left-eigenvector of the dynamics matrix associated with the eigenvalue one. Moreover, the proposed framework has a polynomial time complexity relative to the number of agents and the maximum quantity of faulty agents. Finally, we illustrate our method with examples covering diverse faulty agents scenarios.

Paper Structure

This paper contains 13 sections, 6 theorems, 30 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Average consensus with resilience and privacy guarantees without losing accuracy
  4. Design to achieve average consensus
  5. Design to achieve privacy
  6. State augmentation and initialization
  7. Initial finite window of time noise injection
  8. Scalar state selection
  9. Main algorithm
  10. Illustrative Examples
  11. Example I - a faulty agent in a 5 agents network with non-constant state (ensuring privacy)
  12. Example II - 3 faulty agents in a 10 agents network sharing asymptotically distinct values (ensuring privacy)
  13. Conclusions

Key Result

Lemma 1

Consider Algorithm alg:main, the vector state of agent $v$, $\tilde{x}_v$ and let $\mathcal{S}\in\mathcal{P}([n], 0, f)$. If $|\mathcal{S}|<|\mathcal{F}|$ then where $\beta = \sum_{u\in\mathcal{F}}\alpha_u x_u^{(0)}$, $\alpha_u\geq 0$ and $\sum_{u\in\mathcal{F}}\alpha_u = 1$. $\diamond$

Figures (4)

  • Figure 1: Networks of agents utilized in Examples I--II.
  • Figure 2: Agents consensus evolution of Example I, with network of agents $\mathcal{G}_1$, and $\mathcal{F}=\{1\}$.
  • Figure 3: Subnetworks of $\mathcal{G}_1$ excluding one agent (indicated in each subfigure).
  • Figure 4: Agents consensus evolution of Example VI, with network of agents $\mathcal{G}_2$, and $\mathcal{F}=\{1,2,3\}$.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Theorem 1: Soundness
  • Remark 4
  • Theorem 2: Privacy without resilience
  • Corollary 1: Privacy with resilience
  • Remark 5
  • ...and 2 more