Theoretical study of interval-valued fuzzy cognitive maps: inference mechanisms, convergence, and centrality measures

Fuzzy cognitive maps (FCMs) have garnered significant attention for modeling and analyzing complex systems, owing to their ability to effectively handle uncertainty and nonlinear relationships. Interval-valued fuzzy cognitive maps (IVFCMs), as an extension of FCMs, further enhance this capability by incorporating interval-valued fuzzy numbers to better represent system uncertainty and complexity. Despite their potential, most existing research on IVFCMs has focused on applications, with limited advancement in their theoretical foundations. This study established a comprehensive theoretical framework for IVFCMs, addressing critical gaps in their mathematical basis and dynamic behavior. The key contributions are as follows: (1) Formalization of the inference mechanism, including a rigorous definition of state updates using interval-valued fuzzy operations, and a proof that IVFCMs reduce to conventional FCMs as interval widths approach zero; (2) establishment of convergence guarantees by deriving sufficient stability conditions through spectral radius analysis and the Banach fixed-point theorem for sigmoid activation functions; (3) the proposal of novel centrality measures, introducing interval-valued degree and closeness centrality, supported by an optimized Dijkstra algorithm for efficient identification of key nodes; and (4) empirical validation through a time series prediction case study, demonstrating the model's superior performance in managing uncertainty and noise across different datasets. Overall, this work addressed fundamental theoretical challenges related to inference, convergence, and structural analysis in IVFCMs, while also demonstrating their practical utility in complex system modeling.