  The process of urbanization is one of the most important phenomenon of our
societies and it is only recently that the availability of massive amounts of
geolocalized historical data allows us to address quantitatively some of its
features. Here, we discuss how the number of buildings evolves with population
and we show on different datasets (Chicago, $1930-2010$; London, $1900-2015$;
New York City, $1790-2013$; Paris, $1861-2011$) that this `fundamental diagram'
evolves in a possibly universal way with three distinct phases. After an
initial pre-urbanization phase, the first phase is a rapid growth of the number
of buildings versus population. In a second regime, where residences are
converted into another use (such as offices or stores for example), the
population decreases while the number of buildings stays approximatively
constant. In another subsequent phase, the number of buildings and the
population grow again and correspond to a re-densification of cities. We
propose a stochastic model based on these simple mechanisms to reproduce the
first two regimes and show that it is in excellent agreement with empirical
observations. These results bring evidences for the possibility of constructing
a minimal model that could serve as a tool for understanding quantitatively
urbanization and the future evolution of cities.
