Modelling the interdependence of spatial scales in urban systems

The multitude of interwoven spatial scales and their relevance for urban systems has been of interest to the complexity science of cities since its conception. Today, we are well aware that urban environments are being simultaneously shaped and organised through actions at all levels. However, the fundamental question of how to reveal and quantify the interdependence of processes in between various spatial and temporal scales is less often addressed. Deepening our theoretical understanding of the multiscale spatiotemporal complexity of urban systems demands a transdisciplinary framework and the deployment of novel and advanced mathematical models. This article performs a multiscale analysis of urban structures using a large dataset of rent price values in the Ruhr area, Germany. We argue that, due to their many interacting degrees of freedom, urban systems exhibit similar features as other strongly correlated systems, for example, turbulent flows, notably the occurrence of extreme small-scale fluctuations. This analogy between urban and turbulent systems, which we support by empirical evidence, allows for the modelling of spatial structures on the basis of concepts and methods from turbulence theory. We demonstrate how by identifying the main turbulence-borrowed characteristics of an arbitrary two-dimensional urban field, it can be fully reproduced with a small number of prescribed points. Our findings have theoretical implications in the way we quantify and analyse scales in urban systems, model small-scale urban structures as well as potential policy relevance on understanding the evolution and spatial organisation of cities.