The Distance-Decay Function of Geographical Gravity Model: Power Law or Exponential Law?

TL;DR

This paper investigates whether the geographical gravity model's distance-decay function is best represented by a power law or exponential law, resolving theoretical dilemmas through mathematical and empirical analysis, and concludes the power law is preferable.

Contribution

The paper demonstrates that the power law decay in the gravity model is fundamentally linked to fractal dimensions and derives it from entropy principles, resolving the dimension dilemma.

Findings

Power law decay aligns with fractal dimension concepts.

Empirical data from Chinese cities supports power law decay.

Exponential decay contradicts the principle of spatial interaction.

Abstract

The distance-decay function of the geographical gravity model is originally an inverse power law, which suggests a scaling process in spatial interaction. However, the distance exponent of the model cannot be explained with the ideas from Euclidean geometry. This results in what is called dimension dilemma. In particular, the gravity model based on power law could not be derived from general principles by traditional ways. Consequently, a negative exponential function substituted for the inverse power function to serve for a distance-decay function for the gravity model. However, the exponential-based gravity model goes against the first law of geography. This paper is devoted to solve these kinds of problems by mathematical reasoning and empirical analysis. First, it can be proved that the distance exponent of the gravity model is essentially a fractal dimension. Thus the dimensional dilemma of the power-based gravity model can be resolved using the concepts from fractal geometry. Second, the exponential function indicates locality and localization, which violates the basic principle of spatial interaction. The power function implies action at a distance, which is the necessary condition of geographical gravitation. Third, the gravity model based on power law decay can be derived from the entropy- maximizing principle by introducing a proper postulate. The observational data of China's cities and regions are employed to verify the theoretical inferences, and the results support power-law distance decay. A conclusion can be reached that the preferred form of geographical gravity model is its original form, which is based on an inverse power law rather than a negative exponential law.