Fig. 2: Model of rank dynamics reproduces features of real-world ranking lists.

a Model of rank dynamics in a system of N elements and ranking list size N₀. At time t, a random element is moved to a random rank with probability τ. A random element is also replaced by a new element with probability ν. b Probability P_x,t that element in rank r = R/N moves to x = X/N after time t (uppercase/lowercase symbols are integer/normalized ranks). Elements not replaced diffuse around x = r (with probability D_x,t) or perform Lévy walks⁵⁵ (with probability L_t). Eq. (2) recovers simulation results, shown here for τ = 0.1, ν = 0.2, N = 100, and N₀ = 80 at times t = 1, 5 (left/right plots), averaged over 10⁵ realizations. c Time series of rank flux F_t over observation period T for data (lines), and mean flux F from the fitted model (dashes) (all datasets in SI Fig. S3; for fitting see SI Section S5). d Probability P_x,t for t = 1 and varying r (left) and rank change C (right), shown for selected datasets (lines) and fitted model (dashes; τ and ν in the plot) (empirical P_x,t is passed through a Savitzky–Golay smoothing filter; see SI Figs. S6–S9 and SI Table S2). As systems become more open, we lose symmetry in the rank dependence of both C and the height of the diffusion peaks of P_x,t (signaled by curved arrows). Data and model have similar qualitative behavior in all rank measures (for a systematic comparison see SI Fig. S19).