  Virtually anything can be and is ranked; people and animals, universities and
countries, words and genes. Rankings reduce the components of highly complex
systems into ordered lists, aiming to capture the fitness or ability of each
element to perform relevant functions, and are being used from socioeconomic
policy to knowledge extraction. A century of research has found regularities in
ranking lists across nature and society when data is aggregated over time. Far
less is known, however, about ranking dynamics, when the elements change their
rank in time. To bridge this gap, here we explore the dynamics of 30 ranking
lists in natural, social, economic, and infrastructural systems, comprising
millions of elements, whose temporal scales span from minutes to centuries. We
find that the flux governing the arrival of new elements into a ranking list
reveals systems with identifiable patterns of stability: in high-flux systems
only the top of the list is stable, while in low-flux systems the top and
bottom are equally stable. We show that two basic mechanisms - displacement and
replacement of elements - are sufficient to understand and quantify ranking
dynamics. The model uncovers two regimes in the dynamics of ranking lists: a
fast regime dominated by long-range rank changes, and a slow regime driven by
diffusion. Our results indicate that the balance between robustness and
adaptability characterizing the dynamics of complex systems might be governed
by random processes irrespective of the details of each system.
