Rank clocks.

Many objects and events, such as cities, firms and internet hubs, scale with size in the upper tails of their distributions. Despite intense interest in using power laws to characterize such distributions, most analyses have been concerned with observations at a single instant of time, with little analysis of objects or events that change in size through time (notwithstanding some significant exceptions). It is now clear that the evident macro-stability in such distributions at different times can mask a volatile and often turbulent micro-dynamics, in which objects can change their position or rank-order rapidly while their aggregate distribution appears quite stable. Here I introduce a graphical representation termed the 'rank clock' to examine such dynamics for three distributions: the size of cities in the US from ad 1790, the UK from ad 1901 and the world from 430 bc. Our results destroy any notion that rank-size scaling is universal: at the micro-level, these clocks show cities and civilizations rising and falling in size at many times and on many scales. The conventional model explaining such scaling on the basis of growth by proportionate effect cannot replicate these micro-dynamics, suggesting that such models and explanations are considerably less general than has hitherto been assumed.