Dynamical analogues of rank distributions

Abstract

We present an equivalence between stochastic and deterministic variable approaches to represent ranked data and find the expressions obtained to be suggestive of statistical-mechanical meanings. We first reproduce size-rank distributions N(k) from real data sets by straightforward considerations based on the assumed knowledge of the background probability distribution P(N) that generates samples of random variable values similar to real data. The choice of different functional expressions for P(N): power law, exponential, Gaussian, etc., leads to different classes of distributions N(k) for which we find examples in nature. Then we show that all of these types of functions can be alternatively obtained from deterministic dynamical systems. These correspond to one-dimensional nonlinear iterated maps near a tangent bifurcation whose trajectories are proved to be precise analogues of the N(k). We provide explicit expressions for the maps and their trajectories and find they operate under conditions of vanishing or small Lyapunov exponent, therefore at or near a transition to or out of chaos. We give explicit examples ranging from exponential to logarithmic behavior, including Zipf’s law. Adoption of the nonlinear map as the formalism central character is a useful viewpoint, as variation of its few parameters, that modify its tangency property, translate into the different classes for N(k).