  A pair of fractal gravity models can be derived from the spatial interaction
models based on entropy maximizing principle and allometric scaling law. The
models can be expressed as dual form in mathematics and are important for
analyzing and predicting spatial flows in network of cities. However, the dual
relationship of urban gravity parameters has been 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
Beijing-Tianjin-Hebei region of China are employed to make a case study,
lending a further support to the theoretical derivation. A conclusion can be
reached that the duality of gravity models rests with scaling symmetry of
fractal structure. This work may be helpful for understanding the theoretical
essence and application direction of spatial interaction modeling.
