Many complex systems are hierarchical; societies, economies, ecosystems, infrastructures, languages, they all develop hierarchies of their elements that emerge from networked interactions within the system and with the outside. These hierarchies evolve according to system-dependent mechanisms of interaction, such as selection in evolutionary biology, or rules of performance in human sports, and reflect the relevance of elements in performing a function in the system. However, it is still unclear whether the temporal evolution of hierarchies solely depends on the driving forces and characteristics of each system, or if there are generic features of hierarchy stability that allow us to model and predict patterns of hierarchical behavior across systems. We explore this question by analyzing $30+$ datasets of social, nature, economic, infrastructure, and sports systems of diverse observed size $N_0$ ($10^2$--$10^5$) and time scale of dynamics (days to centuries) and find that, despite their various origins, the elements in these systems show remarkably similar stability depending on their position in the hierarchy.
By classifying systems from closed to increasingly open, we capture their hierarchy evolution in a simple diffusive model without system-specific mechanisms of interaction. We find closed systems to be symmetric since highly/lowly ranked elements are stable in time, while in open systems the least relevant elements show the most fluctuations in rank. We also use the model to estimate the unknown size N of each system and a microscopic time scale of hierarchy evolution. This allows us to make predictions on unobserved data, such as the likelihood of an unknown element (R > N0) climbing high in the hierarchy (surprise), or the time scale over which an element can maintain its relevance in the system (success). Our results may be crucial in further understanding why hierarchies evolve similarly in seemingly unrelated areas, and give clues on how to promote stability in the complex socio-technical systems of our day.
