The Role of Confidence for Trust-based Resilient Consensus (Extended Version)

TL;DR

This work addresses resilient consensus in multi-agent networks with malicious agents and uncertain channel-derived trust signals $\alpha_{ij}(t) \in [0,1]$. It introduces a resilience mechanism that blends channel trust with a decaying confidence parameter $\lambda_t = c e^{-\gamma t}$ in a time-varying weighting scheme, avoiding the need for an initial trust window $T_0$ and enabling asymptotic consensus. The authors prove almost-sure convergence under connectivity and trust-recovery assumptions and derive a non-monotonic bound on the steady-state deviation from nominal consensus, highlighting an optimal $\gamma$ that balances learning time against misclassification risk. Numerical simulations on sparse graphs illustrate how the optimal decay rate depends on trust statistics $E_\mathcal{L}$ and $E_\mathcal{M}$, demonstrating practical guidance for tuning $\gamma$ to minimize steady-state error in the presence of adversaries.

Abstract

We consider a multi-agent system where agents aim to achieve a consensus despite interactions with malicious agents that communicate misleading information. Physical channels supporting communication in cyberphysical systems offer attractive opportunities to detect malicious agents, nevertheless, trustworthiness indications coming from the channel are subject to uncertainty and need to be treated with this in mind. We propose a resilient consensus protocol that incorporates trust observations from the channel and weighs them with a parameter that accounts for how confident an agent is regarding its understanding of the legitimacy of other agents in the network, with no need for the initial observation window $T_0$ that has been utilized in previous works. Analytical and numerical results show that (i) our protocol achieves a resilient consensus in the presence of malicious agents and (ii) the steady-state deviation from nominal consensus can be minimized by a suitable choice of the confidence parameter that depends on the statistics of trust observations.

Figures (2)

• Figure 1: Profile of $-\ell$ in \ref{['eq:leg-bound-prob']} as a function of $\gamma$ with $T_\text{f} \in \{2,\dots,10\}$ (the arrow indicates how the curve varies as $T_\text{f}$ grows). Recall that the bound on (probability of) deviation due to legitimate agents is proportional to $-\mathbb{E}\left[\ell\right]$.
• Figure 2: Steady-state deviation from nominal consensus value (right) and contributions due to legitimate agents $\tilde{x}_{\infty}^{i,\mathcal{L}}$ (left) and to malicious agents $\tilde{x}_{\infty}^{i,\mathcal{M}}$ (middle) averaged over 1000 Monte Carlo runs. As predicted by the bound \ref{['eq:leg-bound-prob']}, the deviation term due to misclassification of legitimate agents is minimized by a (small) positive value of $\gamma$ that decreases as the trust scores get more uncertain, while the deviation term due to malicious agents steadily increases as $\gamma$ grows, according to bound \ref{['eq:state-mal-error-bound-prob']}.

Theorems & Definitions (14)

• Definition 1: Trust variable $\alpha_{ij}$
• Lemma 1: Decaying misclassification probability Yemini22tro-resilienceConsensusTrust
• Corollary 1: Yemini22tro-resilienceConsensusTrust
• Lemma 2: Yemini22tro-resilienceConsensusTrust
• Remark 1
• Lemma 3
• proof
• Lemma 4: Yemini22tro-resilienceConsensusTrust
• Lemma 5
• Lemma 6