This paper analyzes a fuzzy multiplex from a logical perspective in a way that has not been formalized so far. A fuzzy multiplex is a nested structure with inner nodes representing sub-system level agent traits and with outer nodes representing system agents; all while the ensemble is the system under consideration. Moreover, a mathematical framework is necessary to describe that structure which is formulated and then utilized. The system is firstly initialized using fuzzy set theory [2], inspired by Fuzzy Cognitive Maps [1]. Then a criterion that describes the structure is devised to implement a multiplex instead of a map [7] [8], and lastly system optimization is achieved. Furthermore, the theoretical context behind the multiplex is expounded in an attempt to establish a formal way of handling implications within a closed system using human intelligence. The paper is organized in sections following the reasoning process behind this unique idea. 1

Federated learning (FL) is an emerging distributed artificial In the context of Fuzzy Cognitive Maps (FCMs), federated intelligence framework that enables privacy-preserving machine learning (FL) is employed to address several intrinsic challenges learning by synthesizing local models instead of sharing associated with these models. FL offers an effective actual data [1]. The general fundamental process can be outlined approach to managing these challenges, enhancing the performance as follows [2]: the federation process is initiated by a and applicability of FCMs. One necessary issue in the single server or participant who provides an initial model for FCMs is the decentralised nature of data sources and the need individual participants to train using their local data. These to preserve data privacy and security while enabling collaborative participants then share the model's weights or gradients with model development. FCM models frequently rely on the server (or other participants) for aggregation, typically using data distributed across multiple locations or organisations.

Fuzzy General Grey Cognitive Map (FGGCM) and Fuzzy Grey Cognitive Map (FGCM) are extensions of Fuzzy Cognitive Map (FCM) in terms of uncertainty. FGGCM allows for the processing of general grey number with multiple intervals, enabling FCM to better address uncertain situations. Although the convergence of FCM and FGCM has been discussed in many literature, the convergence of FGGCM has not been thoroughly explored. This paper aims to fill this research gap. First, metrics for the general grey number space and its vector space is given and proved using the Minkowski inequality. By utilizing the characteristic that Cauchy sequences are convergent sequences, the completeness of these two space is demonstrated. On this premise, utilizing Banach fixed point theorem and Browder-Gohde-Kirk fixed point theorem, combined with Lagrange's mean value theorem and Cauchy's inequality, deduces the sufficient conditions for FGGCM to converge to a unique fixed point when using tanh and sigmoid functions as activation functions. The sufficient conditions for the kernels and greyness of FGGCM to converge to a unique fixed point are also provided separately. Finally, based on Web Experience and Civil engineering FCM, designed corresponding FGGCM with sigmoid and tanh as activation functions by modifying the weights to general grey numbers. By comparing with the convergence theorems of FCM and FGCM, the effectiveness of the theorems proposed in this paper was verified. It was also demonstrated that the convergence theorems of FCM are special cases of the theorems proposed in this paper. The study for convergence of FGGCM is of great significance for guiding the learning algorithm of FGGCM, which is needed for designing FGGCM with specific fixed points, lays a solid theoretical foundation for the application of FGGCM in fields such as control, prediction, and decision support systems.

While Agent-Based Models can create detailed artificial societies based on individual differences and local context, they can be computationally intensive. Modelers may offset these costs through a parsimonious use of the model, for example by using smaller population sizes (which limits analyses in sub-populations), running fewer what-if scenarios, or accepting more uncertainty by performing fewer simulations. Alternatively, researchers may accelerate simulations via hardware solutions (e.g., GPU parallelism) or approximation approaches that operate a tradeoff between accuracy and compute time. In this paper, we present an approximation that combines agents who `think alike', thus reducing the population size and the compute time. Our innovation relies on representing agent behaviors as networks of rules (Fuzzy Cognitive Maps) and empirically evaluating different measures of distance between these networks. Then, we form groups of think-alike agents via community detection and simplify them to a representative agent. Case studies show that our simplifications remain accuracy.

The interpretability of machine learning models is critical, as users may be reluctant to rely on their inferences. Intuitionistic FCMs (iFCMs) have been proposed as an extension of FCMs offering a natural mechanism to assess the quality of their output through the estimation of hesitancy, a concept resembling to human hesitation in decision making. To address the challenge of interpretable image classification, this paper introduces a novel framework, named Interpretable Intuitionistic FCM (I2FCM) which is domain-independent, simple to implement, and can be applied on Convolutional Neural Network (CNN) models, rendering them interpretable. To the best of our knowledge this is the first time iFCMs are applied for image classification. Further novel contributions include: a feature extraction process focusing on the most informative image regions; a learning algorithm for data-driven determination of the intuitionistic fuzzy interconnections of the iFCM; an inherently interpretable classification approach based on image contents. In the context of image classification, hesitancy is considered as a degree of inconfidence with which an image is categorized to a class. The constructed iFCM model distinguishes the most representative image semantics and analyses them utilizing cause-and-effect relations. The effectiveness of the introduced framework is evaluated on publicly available datasets, and the experimental results confirm that it can provide enhanced classification performance, while providing interpretable inferences.

In the quest for accurate and interpretable AI models, eXplainable AI (XAI) has become crucial. Fuzzy Cognitive Maps (FCMs) stand out as an advanced XAI method because of their ability to synergistically combine and exploit both expert knowledge and data-driven insights, providing transparency and intrinsic interpretability. This letter introduces and investigates the "Total Causal Effect Calculation for FCMs" (TCEC-FCM) algorithm, an innovative approach that, for the first time, enables the efficient calculation of total causal effects among concepts in large-scale FCMs by leveraging binary search and graph traversal techniques, thereby overcoming the challenge of exhaustive causal path exploration that hinder existing methods. We evaluate the proposed method across various synthetic FCMs that demonstrate TCEC-FCM's superior performance over exhaustive methods, marking a significant advancement in causal effect analysis within FCMs, thus broadening their usability for modern complex XAI applications.

Distributed Artificial Intelligence is attracting interest day by day. In this paper, the authors introduce an innovative methodology for distributed learning of Particle Swarm Optimization-based Fuzzy Cognitive Maps in a privacy-preserving way. The authors design a training scheme for collaborative FCM learning that offers data privacy compliant with the current regulation. This method is applied to a cancer detection problem, proving that the performance of the model is improved by the Federated Learning process, and obtaining similar results to the ones that can be found in the literature.

Among various soft computing approaches for time series forecasting, fuzzy cognitive maps (FCMs) have shown remarkable results as a tool to model and analyze the dynamics of complex systems. FCMs have similarities to recurrent neural networks and can be classified as a neuro-fuzzy method. In other words, FCMs are a mixture of fuzzy logic, neural network, and expert system aspects, which act as a powerful tool for simulating and studying the dynamic behavior of complex systems. The most interesting features are knowledge interpretability, dynamic characteristics and learning capability. The goal of this survey paper is mainly to present an overview on the most relevant and recent FCM-based time series forecasting models proposed in the literature.

Increasing complexity comes from some factors including uncertainty, ambiguity, inconsistency, multiple dimensionalities, increasing the number of effective factors and relation between them. Some of these features are common among most real-world problems which are considered complex and dynamic problems. In other words, since the data and relations in real world applications are usually highly complex and inaccurate, modeling real complex systems based on observed data is a challenging task especially for large scale, inaccurate and non stationary datasets. Therefore, to cover and address these difficulties, the existence of a computational system with the capability of extracting knowledge from the complex system with the ability to simulate its behavior is essential. In other words, it is needed to find a robust approach and solution to handle real complex problems in an easy and meaningful way [1]. Hard computing methods depend on quantitative values with expensive solutions and lack of ability to represent the problem in real life due to some uncertainties. In contrast, soft computing approaches act as alternative tools to deal with the reasoning of complex problems [2]. Using soft computing methods such as fuzzy logic, neural network, genetic algorithms or a combination of these allows achieving robustness, tractable and more practical solutions. Generally, two types of methods are used for analyzing and modeling dynamic systems including quantitative and qualitative approaches.

Fuzzy Cognitive Maps (FCMs) have emerged as an interpretable signed weighted digraph method consisting of nodes (concepts) and weights which represent the dependencies among the concepts. Although FCMs have attained considerable achievements in various time series prediction applications, designing an FCM model with time-efficient training method is still an open challenge. Thus, this paper introduces a novel univariate time series forecasting technique, which is composed of a group of randomized high order FCM models labeled R-HFCM. The novelty of the proposed R-HFCM model is relevant to merging the concepts of FCM and Echo State Network (ESN) as an efficient and particular family of Reservoir Computing (RC) models, where the least squares algorithm is applied to train the model. From another perspective, the structure of R-HFCM consists of the input layer, reservoir layer, and output layer in which only the output layer is trainable while the weights of each sub-reservoir components are selected randomly and keep constant during the training process. As case studies, this model considers solar energy forecasting with public data for Brazilian solar stations as well as Malaysia dataset, which includes hourly electric load and temperature data of the power supply company of the city of Johor in Malaysia. The experiment also includes the effect of the map size, activation function, the presence of bias and the size of the reservoir on the accuracy of R-HFCM method. The obtained results confirm the outperformance of the proposed R-HFCM model in comparison to the other methods. This study provides evidence that FCM can be a new way to implement a reservoir of dynamics in time series modelling.