Learning the structure of Bayesian networks is a fundamental yet computationally intensive task, especially as the number of variables grows. Traditional algorithms require retraining from scratch when new variables are introduced, making them impractical for dynamic or large-scale applications. In this paper, we propose an extendable structure learning strategy that efficiently incorporates a new variable $Y$ into an existing Bayesian network graph $\mathcal{G}$ over variables $\mathcal{X}$, resulting in an updated P-map graph $\bar{\mathcal{G}}$ on $\bar{\mathcal{X}} = \mathcal{X} \cup \{Y\}$. By leveraging the information encoded in $\mathcal{G}$, our method significantly reduces computational overhead compared to learning $\bar{\mathcal{G}}$ from scratch. Empirical evaluations demonstrate runtime reductions of up to 1300x without compromising accuracy. Building on this approach, we introduce a novel iterative paradigm for structure learning over $\mathcal{X}$. Starting with a small subset $\mathcal{U} \subset \mathcal{X}$, we iteratively add the remaining variables using our extendable algorithms to construct a P-map graph over the full set. This method offers runtime advantages comparable to common algorithms while maintaining similar accuracy. Our contributions provide a scalable solution for Bayesian network structure learning, enabling efficient model updates in real-time and high-dimensional settings.

A new distributed adaptive fault-tolerant tracking controller is proposed for consensus control of high-order nonlinear multi-agent systems in the presence of actuator faults and unknown time-varying delays. The system is subject to uncertain disturbances under directed topology. The proposed adaptive method is based on dynamic surface control, where the nonlinear dynamics are unknown, and are approximated by a radial basis function neural network. This system model is subject to delay, fault, uncertainty, disturbances, and singularity of the input signal. The typical constraints on time-delay terms, addressed in the related literature, are relaxed. This relaxation, however, can lead to singularity in the controller. This drawback is addressed by using a specific differentiable function in the control law. It is shown that the generated closed-loop signals are uniformly semi-globally bounded, and the tracking errors converge to a neighborhood of the origin. The effectiveness of the proposed controller is verified by simulations. Note to Practitioners—This work focuses on the distributed control of nonlinear multi-agent time-delay systems in the presence of actuator faults. The proposed control design has potential applications in various real-world network systems such as smart grids, transportation networks, autonomous vehicles, etc. Note that many practical multi-agent systems are subject to time delay and parameter uncertainties, and ignoring these in the control design can negatively impact the system performance and even destabilize it. We demonstrate that the resulting closed-loop system is uniformly semi-globally stable and that the tracking error converges to a neighborhood of the origin. The boundedness assumption on time delay terms, often considered in the multi-agent system literature, is relaxed here. Consequently, the obtained results closely align with practical situations. In addition, the explosion of complexity problem has been avoided by using the dynamic surface control method. The dynamic system represents an unknown-structure dynamic model. This type of dynamic model is applicable to various practical engineering systems. Particularly, robotic systems, chemical reactor systems and spring-mass-damper systems can be described by a model similar to this dynamic system.