The universal scaling relationship between an attribute and the size of a system is widespread in nature and society and is known as allometric growth. Previous studies have explained that the allometric growth exponent of single-source systems is uniquely determined by the dimension. However, the phenomenon that the exponent shows diversity in some systems, such as rivers, freight transportation and gasoline stations, lacks a reasonable explanation. In this paper, we hold the view that allometric growth may originate from efficient delivery from sources to transfer sites in a system and propose a multisource transportation network model that can explain diversified allometric growth exponents. We apply this model to some multisource systems, and the results show that our model successfully reproduces the diversity of the allometric growth exponent.

Heavy trucks which undertake the majority of freight volume play an important role in urban freight systems. By analyzing heavy truck trip data, we find a superlinear scaling relationship for heavy truck trips and a sublinear scaling relationship for heavy truck numbers relative to urban population size. Although these allometric scaling relationships that widely appear in nature and social systems have been explained by many models, a simple model that can cover a wide range of scaling exponents in these systems is still lacking. Here, we develop a partially mixing city operation model by quantifying the mixability of the urban population to explain why the superlinear and sublinear scaling exponents are in the range of 1 and . This simple model not only helps us understand the mechanism of allometric scaling of urban freight systems, but also provides a new framework for other superlinear and sublinear scaling relationships in cities.