  Previous works have shown the universality of allometric scalings under
density and total value at city level, but our understanding about the size
effects of regions on them is still poor. Here, we revisit the scaling
relations between gross domestic production (GDP) and population (POP) under
total and density value. We first reveal that the superlinear scaling is a
general feature under density value crossing different regions. The scaling
exponent $\beta$ under density value falls into the range $(1.0, 2.0]$, which
unexpectedly goes beyond the range observed by Pan et al. (Nat. Commun. vol. 4,
p. 1961 (2013)). To deal with the wider range, we propose a network model based
on 2D lattice space with the spatial correlation factor $\alpha$ as parameter.
Numerical experiments prove that the generated scaling exponent $\beta$ in our
model is fully tunable by the spatial correlation factor $\alpha$. We
conjecture that our model provides a general platform for extensive urban and
regional studies.
