The Hegselmann-Krause (HK) model was first introduced in the field of opinion dynamics to describe the opinion evolution of individuals who interact with others and whose opinions are influenced by those of the people around them [1]. In the HK model, the individuals update their opinions over time by taking the average of the opinions of all their neighbors whose opinions are close enough to their own. This closeness is determined by a bounded confidence threshold, such that agents influence each other's opinion only if their opinions lay within the confidence threshold. Though initially proposed in the context of opinion dynamics, the HK model captures a fundamental self-organizing mechanism in complex systems. Beyond its original application, it has also been adopted as a basic game learning algorithm [2, 3] and has found widespread use in diverse fields, including demand response programs in smart grids [4] and hybrid energy storage management [5]. Among the many properties, one of the interesting features of the model is that it can be synchronized by random noise. This phenomenon, also known as "noise-induced order" in self-organizing systems, was first found in some simulation studies [6-8]. Then the analysis of the phenomenon was considered based on some noisy HK-type models.

Memory effects play a crucial role in social interactions and decision-making processes. This paper proposes a novel fractional-order bounded confidence opinion dynamics model to characterize the memory effects in system states. Building upon the Hegselmann-Krause framework and fractional-order difference, a comprehensive model is established that captures the persistent influence of historical information. Through rigorous theoretical analysis, the fundamental properties including convergence and consensus is investigated. The results demonstrate that the proposed model not only maintains favorable convergence and consensus characteristics compared to classical opinion dynamics, but also addresses limitations such as the monotonicity of bounded opinions. This enables a more realistic representation of opinion evolution in real-world scenarios. The findings of this study provide new insights and methodological approaches for understanding opinion formation and evolution, offering both theoretical significance and practical applications.

The Hegselmann-Krause (HK) model of opinion dynamics describes how opinions held by individuals in a community change over time in response to the opinions of others and their access to the true value, T, to which these opinions relate. Here, I extend the simple HK model to incorporate an Artificially Intelligent (AI) Oracle that averages the opinions of members of the community. Agent-based simulations show that (1) if individuals only have access to the Oracle (and not T), and incorporate the Oracle's opinion as they update their opinions, then all opinions will converge on a common value; (2) in contrast, if all individuals also have access to T, then all opinions will ultimately converge to T, but the presence of an Oracle may delay the time to convergence; (3) if only some individuals have access to T, opinions may not converge to T, but under certain conditions, universal access to the Oracle will guarantee convergence to T; and (4) whether or not the Oracle only accesses the opinions of individuals who have access to T, or whether it accesses the opinions of everyone in the community, makes no marked difference to the extent to which the average opinion differs from T.

Linguistic fuzzy information evolution is crucial in understanding information exchange among agents. However, different agent weights may lead to different convergence results in the classic DeGroot model. Similarly, in the Hegselmann-Krause bounded confidence model (HK model), changing the confidence threshold values of agents can lead to differences in the final results. To address these limitations, this paper proposes three new models of linguistic fuzzy information dynamics: the per-round random leader election mechanism-based DeGroot model (PRRLEM-DeGroot), the PRRLEM-based homogeneous HK model (PRRLEM-HOHK), and the PRRLEM-based heterogeneous HK model (PRRLEM-HEHK). In these models, after each round of fuzzy information updates, an agent is randomly selected to act as a temporary leader with more significant influence, with the leadership structure being reset after each update. This strategy increases the information sharing and enhances decision-making by integrating multiple agents' evaluation information, which is also in line with real life (\emph{Leader is not unchanged}). The Monte Carlo method is then employed to simulate the behavior of complex systems through repeated random tests, obtaining confidence intervals for different fuzzy information. Subsequently, an improved golden rule representative value (GRRV) in fuzzy theory is proposed to rank these confidence intervals. Simulation examples and a real-world scenario about space situational awareness validate the effectiveness of the proposed models. Comparative analysis with the other models demonstrate our ability to address the echo chamber and improve the robustness.

This paper introduces a new multidimensional extension of the Hegselmann-Krause (HK) opinion dynamics model, where opinion proximity is not determined by a norm or metric. Instead, each agent trusts opinions within the Minkowski sum $\xi+\mathcal{O}$, where $\xi$ is the agent's current opinion and $\mathcal{O}$ is the confidence set defining acceptable deviations. During each iteration, agents update their opinions by simultaneously averaging the trusted opinions. Unlike traditional HK systems, where $\mathcal{O}$ is a ball in some norm, our model allows the confidence set to be non-convex and even unbounded. We demonstrate that the new model, referred to as SCOD (Set-based Confidence Opinion Dynamics), can exhibit properties absent in the conventional HK model. Some solutions may converge to non-equilibrium points in the state space, while others oscillate periodically. These ``pathologies'' disappear if the set $\mathcal{O}$ is symmetric and contains zero in its interior: similar to the usual HK model, SCOD then converges in a finite number of iterations to one of the equilibrium points. The latter property is also preserved if one agent is "stubborn" and resists changing their opinion, yet still influences the others; however, two stubborn agents can lead to oscillations.

However, their computational demands often become a significant barrier as the number of agents and complexity of the simulation increase. Traditional ABM platforms often struggle to fully exploit modern computing resources, hindering the development of large-scale simulations. This paper presents Vahana.jl, a high performance computing open source framework that aims to address these limitations. Building on the formalism of synchronous graph dynamical systems, Vahana.jl is especially well suited for models with a focus on (social) networks. The framework seamlessly supports distribution across multiple compute nodes, enabling simulations that would otherwise be beyond the capabilities of a single machine.