Demonstration of duality of fractal gravity models by scaling symmetry

Abstract

A pair of fractal gravity models can be derived from spatial interaction models based on the entropy maximizing principle and the allometric scaling law. The models can be expressed in dual form in mathematics and are important for analyzing and predicting spatial flows in networks of cities. However, the dual relationship of urban gravity parameters has remained an empirical relationship for a long time and lacks theoretical proof. This paper is devoted to proving the duality of fractal gravity models by means of ideas from scaling invariance. The results show that a fractal gravity model can be derived from its dual form. The observational data of interurban spatial flows of the Beijing-Tianjin-Hebei region of China are employed to make a case study, lending further support for the theoretical derivation. It appears that the duality of gravity models rests with the scaling symmetry of fractal structure. This work may be helpful for understanding the theoretical essence and application direction of spatial interaction modeling.