  Urban inequality is a major challenge for cities in the 21st century. This
inequality is reflected in the spatial income structure of cities which evolves
in time through various processes. Gentrification is a well-known illustration
of these dynamics in which the population of a low income area changes as
wealthier residents arrive and old-settled residents are expelled. Less
understood but very important is the reverse process of gentrification through
which areas of cities get impoverished. Gentrification has been widely studied
among social sciences, especially in case studies, but there have been fewer
quantitative analyses of this phenomenon, and more generally about the spatial
dynamics of income in cities. Here, we first propose a quantitative analysis of
these income dynamics in cities based on household incomes in 45 American and 9
French Functional Urban Areas (FUA). We found that an important ingredient that
determines the evolution of the income level of an area is the income level of
its immediate neighboring areas. This empirical finding leads to the idea that
these dynamics can be modeled by the voter model of statistical physics. We
show that such a model constitutes an interesting tool for both describing and
predicting evolution scenarios of urban areas with a very limited number of
parameters (two for the US and one for France). We illustrate our results by
computing the probability that areas will change their income status in the
case of Boston and Paris at the horizon of 2030.
