Relationship between Valeriepieris Ring Parameters and Urban Scaling Exponents

Determine the theoretical relationship between the ring-specific power-law exponent a_j in the Valeriepieris concentric ring model—where the radial density within ring j scales as r^{a_j−2}—and the scaling exponents used in urban scaling laws, as well as parameters in other mathematical models of cities, so that these quantities can be interpreted and connected across the Valeriepieris ring framework and established urban theory.

Background

The paper introduces a method that transforms two-dimensional spatial data into a one-dimensional profile via Valeriepieris circles, then models the data as a set of concentric rings whose radial densities follow a power-law form. The slopes of piecewise-linear segments in the VP profile determine ring-specific parameters a_j, which describe how density decays with radius within each ring.

In the Conclusion, the authors discuss the broader theoretical implications of this ring model for the science of cities, including its potential connections to urban scaling laws and other mathematical models of cities. They explicitly note that the precise relationship between the ring model parameters a_j and standard exponents or parameters in urban theory remains unknown, and they highlight understanding this connection as future work.