The quest for a theory of cities that could offer a quantitative and systematic approach to managing cities represents a top priority. If such a theory is feasible, then its formulation must be in a mathematical way.

As a contribution to organizing the mathematical ideas that deal with such a systematic way of understanding urban phenomena, we review the main theoretical models present in the literature that aim at explaining the origin and emergence of urban scaling.

We intend to present the models, identify similarities and connections between them, and find situations in which different models lead to the same output. In addition, we report situations where some ideas initially introduced in a particular model can also be introduced in another one, generating more diversification and increasing the scope of the original works.

The models treated in this paper explain urban scaling from different premises, i.e. from gravity ideas, densification and cites’ geometry to a hierarchical organization and social network properties.

We also investigate scenarios in which these different fundamental ideas could be interpreted as similar — where the similarity is likely but not obvious. Furthermore, concerning the gravity model, we propose a general framework that includes all analyzed models as particular cases. We conclude the paper by discussing perspectives of this field and how future research designs and schools of thought can build on the ideas treated here.

F.L. Ribeiro, D. Rybski, Mathematical models to explain the origin of urban scaling laws, Physics Reports 1012 (2023) 1-39.