Self-Similar Processes Follow a Power Law in Discrete Logarithmic Space

Cities, wealth, and earthquakes follow continuous power-law probability distributions such as the Pareto distribution, which are canonically associated with scale-free behavior and self-similarity. However, many self-similar processes manifest as discrete steps that do not produce a continuous scale-free distribution. We construct a discrete power-law distribution that arises naturally from a simple model of hierarchical self-similar processes such as turbulence and vasculature, and we derive the maximum-likelihood estimate (MLE) for its exponent. Our distribution is self-similar, in contrast to previously studied discrete power laws such as the Zipf distribution. We show that the widely used MLE derived from the Pareto distribution leads to inaccurate estimates in systems that lack continuous scale invariance such as branching networks and data subject to logarithmic binning. We apply our MLE to data from bronchial tubes, blood vessels, and earthquakes to produce new estimates of scaling exponents and resolve contradictions among previous studies.