Foundations of Neutrosophic MR-Metric Spaces with Applications to Homotopy, Fixed Points, and Complex Networks

Abstract

This paper introduces and explores the concept of Neutrosophic MR-Metric Spaces (NMR-MS), which generalize classical metric spaces by incorporating neutrosophic logic to handle uncertainty, indeterminacy, and truth-membership in complex systems. We define the structure of NMR-MS, including MR-metrics, neutrosophic functions, and associated algebraic operations. Key contributions include the introduction of the Neutrosophic Graph MR-Metric Space, where graph-theoretic concepts such as shortest paths and betweenness centrality are integrated with neutrosophic degrees. We establish several fundamental results, including contraction lemmas, robustness under targeted attacks, and the existence of fixed points for neutrosophic centrality mappings. Furthermore, we define Neutrosophic MR-Homotopy and the associated Neutrosophic MR-Fundamental Groupoid, proving that it forms an equivalence relation and possesses a well-defined algebraic structure. The theoretical framework is applied to diverse real-world networks, including social, neural, and internet topologies, demonstrating its utility in network analysis, robustness testing, and path planning. Computational algorithms and performance metrics are provided, showcasing the practical applicability of the proposed framework.