From urban segregation to spatial structure detection

Abstract

We develop a "multifocal" approach to reveal spatial dissimilarities in cities, from the most local scale to the metropolitan one. Think for instance of a statistical variable that may be measured at different scales, eg ethnic group proportions, social housing rate, income distribution, or public transportation network density. Then, to any point in the city there corresponds a sequence of values for the variable, as one zooms out around the starting point, all the way up to the whole city -as if with a varifocal camera lens. The sequences thus produced encode in a precise manner spatial dissimilarities: how much they differ from perfectly random sequences is indeed a signature of the underlying spatial structure. We introduce here a mathematical framework that allows to analyze this signature and we provide a number of illustrative examples.

Segregation through the multiscalar lens

“…Establishing the theoretical properties of this null model is not straightforward, as the trajectories obtained when moving from distributions to KL divergence are not readily modeled. In a simpler framework (23), when working with single-group proportions, one may approximate the corresponding trajectories by generalized Brownian bridges (28,29) and transform the problem into a first-passage one for Brownian motion (30). The general version presented here does not seem amenable to the same techniques and poses an interesting mathematical challenge for future research.…”

“…This is a generalization of a method introduced in ref. 23: Given a starting point and an aggregation procedure (e.g., a nearestneighbor rule), one may sequentially aggregate all individuals in the city. The aggregated sequence then encodes the order in which someone, starting from that particular point, would encounter the city's inhabitants.…”

“…However, single-number indices as well as singlescale indices cannot capture the complexity of spatial patterns that arise in actual cities. Only recently, this has led to the exploration of multiscalar approaches (14)(15)(16)(17)(18)(19)(20)(21)(22)(23), including multidimensional, matrix indices (24). We present in this paper a framework that allows one to carry out fully multiscalar analyses of spatial dissimilarities and to define multiscalar null models.…”

Quantifying ethnic segregation in cities through random walks

“…Spatial heterogeneity is a characteristic aspect of a variety of complex systems, from urban areas to ecosystems [1,2], and the presence of non-trivial spatial patterns in the organisation of such systems has a substantial impact on their functioning and dynamics [3]. This is the main reason why the quantitative characterisation of complex spatial patterns has received much attention in different fields, from urban studies to biology, from geography to economics, from transportation to engineering [4][5][6][7].…”

Multiscale Segregation: Multilevel Modeling of Dissimilarity—Challenging the Stylized Fact That Segregation Is Greater the Finer the Spatial Scale

“…Recent advances in computing power and the development of geographical information systems associated with improved data sources -some of them geocoded -have allowed segregation's intensity, and its spatial organisation, to be explored at a range of spatial scales (Wong et al, 2018). Some of the spatial structures deployed are created by census and other agencies; others use bespoke formats created within software systems (see, for example, Lee et al 2008;Reardon et al 2008Reardon et al , 2009Östh et al 2015;Clark et al 2016;Randon-Furling et al, 2018). Several of those authors have noted that identification of varying levels of segregation at different scales invites investigations of the multi-scalar location-decision processes in operation, leading to possible evaluations of the relative intensity of, say, macro-, meso-and micro-levels of segregation.…”