Intuitionistic fuzzy sets extend traditional fuzzy sets by incorporating degrees of membership, non-membership, and indeterminacy, making them particularly useful in contexts where uncertainty and hesitancy are prevalent. Rough soft sets combine rough sets' approximation capabilities with soft sets' flexible, parameterized approach to managing uncertainty. This study introduces Intuitionistic Fuzzy Rough Soft (IFRS) sets, integrating these advantages to create a robust framework for handling uncertainty, vagueness, and ambiguity in complex decision-making environments. The paper meticulously defines operations, operators, and measures between IFRS sets, establishing their characteristic properties through rigorous mathematical demonstrations. An innovative algorithm is proposed to address multi-criteria decision-making problems within this framework. The algorithm's effectiveness is thoroughly evaluated through comparisons with state-of-the-art algorithms using reputable datasets in medical consultation, agricultural land evaluation, educational support, and sensitivity analysis experiments. The results demonstrate the proposed algorithm's superior performance and robustness in complex decision-making scenarios, highlighting its potential as a valuable practical tool.

For many years, topological data analysis (TDA) and deep learning (DL) have been considered separate data analysis and representation learning approaches, which have nothing in common. The root cause of this challenge comes from the difficulties in building, extracting, and integrating TDA constructs, such as barcodes or persistent diagrams, within deep neural network architectures. Therefore, the powers of these two approaches are still on their islands and have not yet combined to form more powerful tools for dealing with multiple complex data analysis tasks. Fortunately, we have witnessed several remarkable attempts to integrate DL-based architectures with topological learning paradigms in recent years. These topology-driven DL techniques have notably improved data-driven analysis and mining problems, especially within graph datasets. Recently, graph neural networks (GNNs) have emerged as a popular deep neural architecture, demonstrating significant performance in various graph-based analysis and learning problems. Explicitly, within the manifold paradigm, the graph is naturally considered as a topological object (e.g., the topological properties of the given graph can be represented by the edge weights). Therefore, integrating TDA and GNN is considered an excellent combination. Many well-known studies have recently presented the effectiveness of TDA-assisted GNN-based architectures in dealing with complex graph-based data representation analysis and learning problems. Motivated by the successes of recent research, we present systematic literature about this nascent and promising research direction in this article, which includes general taxonomy, preliminaries, and recently proposed state-of-the-art topology-driven GNN models and perspectives.

Topological data analysis is a new theoretical trend using topological techniques to mine data. This approach helps determine topological data structures. It focuses on investigating the global shape of data rather than on local information of high-dimensional data. The Mapper algorithm is considered as a sound representative approach in this area. It is used to cluster and identify concise and meaningful global topological data structures that are out of reach for many other clustering methods. In this article, we propose a new method called the Shape Fuzzy C-Means (SFCM) algorithm, which is constructed based on the Fuzzy C-Means algorithm with particular features of the Mapper algorithm. The SFCM algorithm can not only exhibit the same clustering ability as the Fuzzy C-Means but also reveal some relationships through visualizing the global shape of data supplied by the Mapper. We present a formal proof and include experiments to confirm our claims. The performance of the enhanced algorithm is demonstrated through a comparative analysis involving the original algorithm, Mapper, and the other fuzzy set based improved algorithm, F-Mapper, for synthetic and real-world data. The comparison is conducted with respect to output visualization in the topological sense and clustering stability.