Modeling forbidden ordinal transition patterns in multi-span ordinal transition networks: From probabilistic analysis to system determinism

Ordinal pattern analysis has been widely applied in detecting determinism in complex systems due to its sensitivity to nonlinear structures. However, most existing studies focus on the absence of individual patterns, capturing only local deterministic features, while the deeper dynamical information embedded in transitions between patterns remains underexplored. To address this gap, we propose a determinism analysis framework grounded in multi-span ordinal transition networks (MOTN). By defining multi-span forbidden ordinal transition patterns (MFOTP) and their corresponding probabilities, the framework aims to capture hidden deterministic structures across multiple temporal scales. Furthermore, the evolution of MFOTP probabilities with respect to span reveals scale-induced deterministic behaviors that are largely independent of parameter choices. To characterize this phenomenon, a stretched exponential model is introduced, from which two key parameters-the decay rate and the stretching exponent-are extracted to describe the system's cross-scale variation and uncover latent determinism embedded in its multi-scale evolution. Experiments conducted on seven chaotic and seven stochastic systems demonstrate the method's stable performance in distinguishing deterministic from stochastic dynamics across diverse scenarios, including under noise contamination and limited data lengths. Applications to real-world datasets-such as heart rate variability and ship-radiated noise-further validate the method's generalization ability and robustness. This study highlights the importance of incorporating span information to enhance the identification of latent deterministic structures, offering a novel ordinal-based tool for multi-scale dynamic modeling in complex systems.