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associated with this relationship is an indicator of the degree of hierarchization of
the settlement system. Many researchers have focused on the universal character of
this relationship and there are numerous empirical works showing its adequacy
[BAT 01, MOR 93, PUM 06, GUE 95, PEA 80, SMI 05, SCH 09] for example).
Others have sought instead to characterize and interpret the deviations from this
regularity [DRE 04, SAN 12]:
- a primary distribution (concave curve, Figure 3.8(b)) describes a situation
where the second largest city is smaller than what would be expected from the law;
- a convex distribution (Figure 3.8(a)) characterizes, on the contrary, a system of
cities where the cities following the largest city are of a larger size than expected.
The first case corresponds to systems where a few forces are involved and where
the largest city polarizes the system. The second describes a system with
little integration. The adequacy to Zipf's law is however interpreted as a situation
where the forces are numerous and are acting in many varied directions, reflecting a
system whose entities are well-integrated in the system.
Figure 3.8.
Theoretical shape of the curve crossing cities' rank in the abscissa,
and the number of inhabitants in ordinate (source [GRI 11])
The shape of the rank-size distribution can therefore be used to characterize and
interpret the change of the hierarchical organization of a settlement system between
different dates. Figure 3.9 thus illustrates the persistence of a regular distribution for
the case of South-African cities with more than 5,000 inhabitants during the 20th
Century [GIR 09]. On the contrary, Griffin is interested in the oscillation over time
between convex and concave distributions of the urban centers of the Tiwanaku
valley between 1500 BC and AD 1000 [GRI 11]. The concavity period corresponds
to the emergence of an urban system well integrated around a main pole, whereas
the return to a convex distribution marks the decline of Tiwinaku that was reflected
by a dispersion of the population.
