Multi-Agent Trustworthy Consensus under Random Dynamic Attacks

TL;DR

It is proved that the consensus process converges almost surely despite the existence of malicious agents and the deviation from the nominal consensus value ideally occurring when there are no malicious agents in the system is characterized.

Abstract

In this work, we study the consensus problem in which legitimate agents send their values over an undirected communication network in the presence of an unknown subset of malicious or faulty agents. In contrast to former works, we generalize and characterize the properties of consensus dynamics with dependent sequences of malicious transmissions with dynamic (time-varying) rates, based on not necessarily independent trust observations. We consider a detection algorithm utilizing stochastic trust observations available to legitimate agents. Under these conditions, legitimate agents almost surely classify their neighbors and form their trusted neighborhoods correctly with decaying misclassification probabilities. We further prove that the consensus process converges almost surely despite the existence of malicious agents. For a given value of failure probability, we characterize the deviation from the nominal consensus value ideally occurring when there are no malicious agents in the system. We also examine the convergence rate of the process in finite time. Numerical simulations show the convergence among agents and indicate the deviation under different attack scenarios.

Figures (3)

• Figure 1: Trusted neighborhood learning algorithm for a legitimate agent $i\in\mathcal{L}$ is illustrated. Legitimate neighbors are shown in blue and malicious neighbors in red. Aggregate trust values are placed on a number line with larger values to the right. The green bracketed region represents the trusted region $\xi_t$ from Algorithm \ref{['alg_trust_neig']}. (a) The accumulated trust value $\beta_{ij}(t)$ falls at least $\xi_t$ to the left of $\beta_{in}(t)$, implying that its gap to $\beta_{i\bar{j}}(t)$ is also at least $\xi_t$. (b) Since $\beta_{im}(t)$ lies within $\xi_t$ distance of $\beta_{i\bar{j}}(t)$, other aggregate trust values can either lay on its left or remain within $\xi_t$ distance on its right.
• Figure 2: Average misclassification errors over and attack rates over $100$ runs. (Left) Average misclassification rates of legitimate neighbors (Middle) misclassification rates of malicious neighbors (Right) Average probability of attacks $\frac{1}{|{\mathcal{M}}|}\mathbb{P}(f_m(t)=1)$ over time.
• Figure 3: Trustworthy consensus over $100$ runs. (Left) Maximum distance to average of agents' values $\max_{i \in {\mathcal{L}}} |x_i(t)-\frac{1}{|{\mathcal{L}}|}\sum_{l \in {\mathcal{L}}} x_l (t)|$ (Right) Maximum deviation from the nominal consensus value. $\max_{i \in {\mathcal{L}}} |x_i(t)-\mathbf{1} \bar{v} x_{\mathcal{L}} (0)|$, where $\bar{v}$ is the left eigenvector of $\widebar{W}_{\mathcal{L}}$ corresponding to eigenvalue $1$.

Theorems & Definitions (42)

• Lemma 1: Hoeffding's Lemma (concentration-ineq2013, Lemma 2.2, pg. 27
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Definition 1: $T_f$