A Scaling Approach to Evaluating the Distance Exponent of Urban Gravity Model

TL;DR

This paper introduces a fractal-based scaling approach to evaluate the distance exponent in the urban gravity model, linking it to fractal dimensions and Zipf's law, and applies it to Chinese cities.

Contribution

It proposes a novel fractal geometry method to interpret and estimate the distance exponent in the gravity model, addressing theoretical and methodological issues.

Findings

Distance exponent equals the average fractal dimension of city sizes.

Scaling exponent can be derived from Zipf's law and spatial fractal dimensions.

Empirical results from Chinese cities support the fractal model interpretation.

Abstract

The gravity model is one of important models of social physics and human geography, but several basic theoretical and methodological problems remain to be solved. In particular, it is hard to explain and evaluate the distance exponent using the ideas from Euclidean geometry. This paper is devoted to exploring the distance-decay parameter of the urban gravity model. Based on the concepts from fractal geometry, several fractal parameter relations can be derived from scaling laws of self-similar hierarchies of cities. Results show that the distance exponent is just a scaling exponent, which equals the average fractal dimension of the size measurements of the cities within a geographical region. The scaling exponent can be evaluated with the product of Zipf's exponent of size distributions and the fractal dimension of spatial distributions of geographical elements such as cities and towns. The new equations are applied to China's cities, and the empirical results accord with the theoretical expectations. The findings lend further support to the suggestion that the geographical gravity model is a fractal model, and its distance exponent is associated with a fractal dimension and Zipf's exponent. This work will help geographers understand the gravity model using fractal theory and estimate the distance exponent using fractal modeling.