Is this scaling nonlinear?

TL;DR

This paper introduces a probabilistic framework to rigorously test and quantify nonlinear scaling relationships between city size and various indexes, accounting for data fluctuations and heavy-tailed distributions.

Contribution

It develops a comprehensive probabilistic approach to estimate, compare, and test nonlinear scaling models in complex systems, addressing previous limitations.

Findings

Nonlinear scaling evidence depends on data fluctuations and model assumptions.

The framework allows estimation of confidence intervals for scaling exponents.

Heavy-tailed city size distributions influence scaling analysis.

Abstract

One of the most celebrated findings in complex systems in the last decade is that different indexes y (e.g., patents) scale nonlinearly with the population~x of the cities in which they appear, i.e., $y\sim x^\beta, \beta \neq 1$. More recently, the generality of this finding has been questioned in studies using new databases and different definitions of city boundaries. In this paper we investigate the existence of nonlinear scaling using a probabilistic framework in which fluctuations are accounted explicitly. In particular, we show that this allows not only to (a) estimate $\beta$ and confidence intervals, but also to (b) quantify the evidence in favor of $\beta \neq 1$ and (c) test the hypothesis that the observations are compatible with the nonlinear scaling. We employ this framework to compare $5$ different models to $15$ different datasets and we find that the answers to points (a)-(c) crucially depend on the fluctuations contained in the data, on how they are modeled, and on the fact that the city sizes are heavy-tailed distributed.