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

Abstract

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 selfsimilar 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.

Persistence analysis of velocity and temperature fluctuations in convective surface layer turbulence

“…A well-suited strategy to evaluate the PDFs of such variables is to take the logarithmic transformation and then binning the transformed variables in the logarithmic space. [50][51][52][53][54][55][56][57] More details about the effect of binning strategy on the persistence PDFs is provided in Appendix A.…”

“…The normalized variable (t p u)/z has a large range, given the minimum of t p is restricted by the sampling interval of 0.05 s (20-Hz sampling frequency) and the maximum of t p could go as large as in the order of 10 2 s (see figure 1). A well-suited strategy to evaluate the PDFs of such variables is to take the logarithmic transformation and then binning the transformed variables in the logarithmic space (Christensen and Moloney, 2005;Newman, 2005;Pueyo, 2006;Sims et al, 2007;Benhamou, 2007;White et al, 2008;Dorval, 2011;Newberry and Savage, 2019). More details about the effect of binning strategy on the persistence PDFs is provided in the Appendix A.…”

Estimation of heavy tails in optical non-linear processes

“…with 1{•} denoting the indicator function. In other words, one has to check what proportion of the observations exceed the limit l; l = x (k) yields the first definition, equation (3). If the sample is powerlaw distributed (at least for the largest observations) the ETF should be linear on a log-log plot (again, for the largest observations), with the slope equal to −α (see figure 3).…”

“…Heavy-tailed distributions, and power-law (or Pareto) distributions in particular have been reported from a very broad range of areas, including earthquake intensities [1][2][3], avalanche sizes [4], solar flares [5], degree distributions of various social and biological networks [6][7][8], incomes [9,10], insurance claims [11,12], number of citations of scientific publications [13][14][15], and many more. For financial institutions, the importance of heavy-tailed behavior comes from the fact that a simple Gaussian model severely underestimates the risks associated with different products or investment strategies, which in turn results in considerable losses.…”

Polygon-Based Hierarchical Planar Networks Based on Generalized Apollonian Construction

“…Importantly, that degree distribution is not a true power law but rather a log-periodic distribution consisting of a sequence of atoms at points 3 × 2 n and a power-law envelope. This means that the network is scale-invariant only with respect to certain discrete renormalizations and thus do not have the full set of properties of a true power law distribution; see [30] for a recent discussion. One natural generalization is a random Apollonian network [31][32][33], which is constructed, instead of a regular generation-by-generation process, by sequential partitioning of arbitrarily chosen triangles.…”