Superlinear and sublinear urban scaling in geographical network model of the city

Using a geographical scale-free network to describe relations between people in a city, we explain both superlinear and sublinear allometric scaling of urban indicators that quantify activities or performances of the city. The urban indicator $Y(N)$ of a city with the population size $N$ is analytically calculated by summing up all individual activities produced by person-to-person relationships. Our results show that the urban indicator scales superlinearly with the population, namely, $Y(N)\propto N^β$ with $β>1$ if $Y(N)$ represents a creative productivity and the indicator scales sublinearly ($β<1$) if $Y(N)$ is related to the degree of infrastructure development. These coincide with allometric scaling observed in real-world urban indicators. We also show how the scaling exponent $β$ depends on the strength of the geographical constraint in the network formation.

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