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.

Figure 1

(a) Tail distributions of power laws [cf. Eq. (1)]. Power laws and self-similarity appear on log-log plots as regular slopes and repeating patterns. The self-similar discrete distribution shows stair steps (allowed values) at regular intervals in logarithmic space, in contrast to Zipf’s law, the classical discrete power law. (b) Our distribution draws node values $x>x_m$ uniformly from a self-similar branching network wherein each node with value $x$ has one parent with value $\lambda x$ and $n$ children of value $x/\lambda$.

Figure 2

Comparison of estimated $\widehat{\alpha }$ and the true value ($\alpha =2$) versus assumed $x_m$ in synthetic data. The data are (a) 10 000 samples from our distribution (discrete) with $x_m=1$, $n=2$, $\lambda =\sqrt{2}$, and (b) 10 000 samples with replacement from a simulated imperfect branching network (continuous) where each child node introduces a random 2.5% proportional error, $\lambda \sim \sqrt{2}\text{Lognormal}[0,\log (1+\epsilon )],$ independent, $\epsilon =0.025$, starting from node value 100. The Pareto MLE ${\widehat{\alpha }}_c$ [Eq. (5), dots and lines] over or under estimates $\alpha$ for many assumed values of $x_m$. Our MLE ${\widehat{\alpha }}_d$ [Eq. (6), diamonds] CIs include the true $\alpha$ for all feasible assumed $x_m$. The CIs of the Pareto ${\widehat{\alpha }}_c$ often exclude the true value, suggesting more data would not improve the estimate. These parameter choices represent the Da Vinci model for tree limbs.

Figure 3

Tail distributions of (a) earthquake magnitudes and (b) bronchial diameters with slopes derived from different estimates of $\alpha$ (Table 1). On earthquake magnitudes (a) the Pareto MLE slope $-{\widehat{\alpha }}_c$ (gray dashed line) is visibly incorrect due to failure to account for discrete data. Bronchial measurements (b) are continuous but deviate from continuous scale invariance. Thus, discretizing by powers of 2 (dotted green line) and using the discrete MLE ${\widehat{\alpha }}_d$ produces a slope (green line) that is visually consistent with the data and consistent with theory (black bar, Table 1) in contrast to the continuous ${\widehat{\alpha }}_c$ (dashed gray line).