The growth of cities has traditionally been studied from a population perspective, while urban sprawl - its spatial growth - has often been approached qualitatively. However, characterizing and modeling this spatial expansion is crucial, particularly given its parallels with surface growth extensively studied in physics. Despite these similarities, approaches to urban sprawl modeling are fragmented and scattered across various disciplines and contexts. In this review, we provide a comprehensive overview of the mathematical modeling of this complex phenomenon. We discuss the key challenges hindering progress and examine models inspired by statistical physics, economics and geography, and theoretical ecology. Finally, we highlight critical directions for future research in this interdisciplinary field.

In the quest for more environmentally sustainable urban areas, the concept of the 15-minute city has been proposed to encourage active mobility, primarily through walking and cycling. An urban area is considered a ``15-minute city" if every resident can access essential services within a 15-minute walk or bike ride from their home. However, there is an ongoing debate about the effectiveness of this model in reducing car usage and carbon emissions. In this study, we conduct a large-scale data-driven analysis to evaluate the impact of service proximity to homes on CO$_2$ emissions. By examining nearly 400 cities worldwide, we discover that, within the same city, areas with services located closer to residents produce less CO$_2$ emissions per capita from transportation. We establish a clear relationship between the proximity of services and CO$_2$ emissions for each city. Additionally, we quantify the potential reduction in emissions for 30 cities if they optimise the location of their services. This optimisation maintains each city's total number of services while redistributing them to ensure equal accessibility throughout the entire urban area. Our findings indicate that improving the proximity of services can significantly reduce expected urban emissions related to transportation.

Community of researchers & practitioners in human-centric data science, broadly defined, like network science, computational social science, geospatial data science.

Analyzing 9,000 urban areas' street networks, we identify properties, including extreme betweenness centrality heterogeneity, that typical spatial network models fail to explain. Accordingly we propose a universal, parsimonious, generative model based on a two-step mechanism that begins with a spanning tree as a backbone then iteratively adds edges to match empirical degree distributions. Controlled by a single parameter representing lattice-equivalent node density, it accurately reproduces key universal properties to bridge the gap between empirical observations and generative models.

Accessibility is essential for designing inclusive urban systems. However, the attempt to capture the complexity of accessibility in a single universal metric has often limited its effective use in design, measurement, and governance across various fields. Building on the work of Levinson and Wu, we emphasise that accessibility consists of several key dimensions. Specifically, we introduce a conceptual framework that defines accessibility through three main dimensions: Proximity (which pertains to active, short-range accessibility to local services and amenities), Opportunity (which refers to quick access to relevant non-local resources, such as jobs or major cultural venues), and Value (which encompasses the overall quality and personal significance assigned to specific points of interest). While it is generally beneficial to improve accessibility, different users and contexts present unique trade-offs that make a one-size-fits-all solution neither practical nor desirable. Our framework establishes a foundation for a quantitative and integrative approach to modelling accessibility. It considers the complex interactions among its various dimensions and facilitates more systematic analysis, comparison, and decision-making across diverse contexts.