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where P(i, t) is the residential population in the municipality i, x
i
is the distance
between the municipality i and the center of the Valence and t is the observation
date (the distance is measured by the Euclidean distance and is therefore supposed
constant in time). The two parameters “a” and “b” represent, respectively, the
density in the center and the dissuasive effect of the distance. They evolve over time,
and the b(t) parameter of the model can be used as an indicator of the shape of the
densification process around the center whereas the sequence of the values of a(t)
over the period is associated to the growth of population in the center. The
combination of the two illustrates the different components and timescale of
the “spreading” process associated: first it is possible to observe a strong growth of
the urban population at the center of the region and in its surroundings. Between
1850 and 1950 the growth is very low, and after 1950 growth it picks up due to rural
exodus: the center becomes denser whereas surrounding places emptied
(Figure 3.7)
Figure 3.7.
Spatial model and follow-up over time of a structure (according to [MAT 00])
3.2.1.2.
The rank-size relationship as an indicator of the hierarchical
organization of a system of cities
The rank-size
6
distribution reflects regularity in the relationship between the rank
of a city and the number of inhabitants who live there. The slope of the line
6 The rank-size distribution expresses the regularity of the relation between the cities' sizes
and their rank in the urban hierarchy (one refers to Zipf's law (1949) [ZIP 49] when the slope
of the line is equal to 1, and to rank-size rule otherwise). The empirical applications concern
both contemporary and historical periods and more distant periods. In archaeology, this
framework is often used to characterize settlement systems whose settlements are not cities,
but much smaller agglomerations.
