An exponent tunable network model for reproducing density driven superlinear relation

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.