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).
Highlights
There exist countless sets of data detailing magnitudes or sizes of a vast variety of measurable properties from many different fields: astrophysical, geophysical, ecological, biological, technological, financial, urban, social, etc
We described different classes of size-rank distributions N(k) that originate each from a different parent distribution P(N) for data samples of the size random variable N
Exponential, and Gaussian parent distributions to obtain analytic expressions for N(k)
Summary
There exist countless sets of data detailing magnitudes or sizes of a vast variety of measurable properties from many different fields: astrophysical, geophysical, ecological, biological, technological, financial, urban, social, etc. We find that these maps have as a common feature being close to or at tangency with the identity line and their trajectories have zero or small Lyapunov exponent as in the transition in or out of chaos via the tangent bifurcation [14]. Dynamical analogues of rank distributions with a central sector parallel and close to tangency These circumstances translate, respectively, into different expressions for the rank distribution, first the mentioned exponential, power law, logarithmic, or inverse error function forms.
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