Denise Pumain on “Scaling laws and urban systems”
August 6 and 8, 2003
Notes written by Margherita Russo, revised by Denise Pumain
Why urban systems are interesting for scaling laws?
Four reasons:
1. cities have scaling properties (from historical, geographical, sociological and economic
perspectives)
2. multilevel perspective: structuration of space and societies give rise to meso level
(intermediate from micro to macro)
3. several social sciences can contribute to modelling and it is necessary an integration of
knowledge form various fields
4. policy implications.
Are we free to choose how to organize our space; i.e. Is it possible to intervene in the dynamics of
the system?
Now we consider what the urban ontology is. First of all: what is a “city”? A city is usually
conceived as a political entity, a municipality. As a matter of fact, cities emerge.
Slide 1
Slide 1 shows satellite images of built up areas in Europe, just surfaces
Slide 2
Slide 2 shows Berlin in several years (1875, 1920, 1945). At this smallest scale there is as well a
fragmentation of built up area: can we deduce that there are self similarities in the urban system?
MR: are there natural barriers justifying that fragmentation?
DP: not always
A most practical definition of city recommended by UN: a continuous built area and an urban
agglomeration regroups the administrative units which are contiguous; (in the US this is defined as
“urbanised areas”, while “metropolitan areas” are those urban areas which have 50.000 jobs in the
area and 15% of labour force commuting within the area).
An “urban entity” should be defined as evolutionary entity in time and space.
Slide 3
Slide 3 shows population of urban agglomerations in Europe in 1990 (differences between Europe,
Japan and Us would be interesting to discuss)
It has been noted that that this map should be overlapped with environmental data
and historical changes
DP reminds that the European Environmental Agency has set a similar map showing
environmental pressure
Slide 4
Slide 5
Slide 6
Slide 4 (in green) represents the variation of density in European cities larger than 10.000
inhabitants and with a distance of less than 25 km. At a glance we note that Spain and France have
very low and not regularly distributed density; high density are observed in a median zone from
central England to northern Italy; whereas Eastern Europe is characterized by regular pattern of
settlements. Moreover, this map gives an idea of the role of historical administration. Consider the
main difference between France and England: France had a top down creation of the nation
(everything was decided in Paris) while in England the developing of industrial activities gave rise
to larger cites (like Birmingham). Pay attention: the ruler is a critical aspect to be considered in
interpreting the map. In fact, if we change the ruler from a distance of 25 to 50 km (Slide 5) or to
greater size of the population (cites larger than 100.000 inhabitants) (Slide 6) the same pattern can
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be observed, even though in none of the representations Madrid appears (it has no cities around).
This is a proof of the consistency of spatial and time organization of cities.
Slide 7
Slide 7 represents a map of Europe based on a multivariate typology (it has been produced by
Denise for the European Commission): London is similar to Milan and Paris looks like Madrid.
From this map, it emerges a visual idea what a “region” might be. The map considers all settlements
including rural areas. But administrative units are of a different size in the different countries. We
have 80.000 municipalities in Europe but half of them are in France.
Notice that data are referred to higher level of aggregation. In general it is not
possible to have data across Europe at lower level, in US there are disaggregated
data.
Slide 8
Slide 8 presents “A system of cities” with regard to three dimensions: organization levels (micro
level with urban actors, just individuals, meso level with town and cities, macro level with the
system of cities); processes (aggregation and migration linking the micro to the meso level, and
attraction and selection from the meso to the micro level; competition and growth from the meso to
the macro and network integration from the macro to the meso level) and state variables in each
level of organization (family and professional cycles at the micro level; size, function and
morphology at the meso level; hierarchy, specialization and spatial network at the macro level).
Urban actors are individuals.
Competition between cities refers to the process of having access to resources (of
different origin)
DAL: Competition is channelled by hierarchy and specialization
SvdL: competition is also related to innovation.
In commenting this representation, Denise introduces the definition of “function” proposed by W.
Christaller in the 1930’s, as set of specializations (Christaller did not include industrial functions).
Economic structure is made up by the functions of the cities and it is quite stable (40-50 years, but
also for 200 years).
What does it mean hierarchy in this context?
Until the 15th Century hierarchy was linked to political status: the status of citizens was different
according to the type of city they lived in (in Italy there were: città, terre murate, villaggi, …). The
political power of each level in the hierarchy was not necessarily related to the size. Marco Polo
observed that in China there where differences of the same nature: differences among persons
according the status of the place where they “belonged”.
A different description was first provided by Botero who wrote a book on the “La ragione di stato”
and, in 1588, a little book on “Delle cause de la grandezza et magnificenza delle città”. According
to Botero, cities may be ranked according to the number of inhabitants and their possessions
(“possenza loro” which means both power and properties/assets/richness).
There is a paper by Denise Pumain on Botero’s contribution in www.cybergo.presse.fr
DAL: point A - the notion of subordination, inclusion and relations should be set
clearly with respect to the notion of hierarchy. Take the example of Rome and
Bologna in the papal reign: Bologna was economically most important while Rome
was the capital of the state
SvdL: relational hierarchies matter and not just one hierarchy
DP: you can have overlapping networks; in general there are limitations constraining
what a city can do
MR: which kind of limits? Natural resources, natural barriers
DP: yes, and also possibilities to have access to innovation
GW: hierarchy is linked to ranking, but is crucial to define which variables you are
considering
DAL: is it dominance a matter of size?
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DP: no, there are other criteria defining dominance
DAL: point B – the notion of different hierarchies is related to a dynamics, which
one?
MR: what is the meaning of networks in this context?
DP: different types of networks are considered. Since the 1950’s interdependencies
among cities was related to functions
RS: is “hierarchy” in urban hierarchy an ordering notion?
DP: two questions arise. First: is urban hierarchy the result of an optimization
process (but we should better define which are the constraints), or the result of a
purely random growth process? Second: does it represent an equilibrium or the
outcome of an evolutionary process?
Zipf law
Slide 9
Slide 9 shows the relation between population and ranking of cities in France, India, Italy and US
presenting a as a Zipf law. As a matter of fact, since the 1913 Auerbach noted that there was a
constant product between size and rank of cities. Similar observations are in Lotka and Gibrat.
Lotka reacted to Zipf presumption of having discovered a new law. But the success of the Zipf law
was mainly due to the fact that Lotka (and Auerbach) referred to statistical distribution, while Zipf
examined possible explanations of the observed deviations to the model: the success of this
formulation of the law was mainly due to a more detailed discussion of the specific characteristics
of the cities.
The shape of the distribution stays stable over time
DAL: from the slide 9 it would seem that the slope changes in different periods
DP: Fletcher found that the distribution of settlements is similar in archaeological
data
You can use this result of the Zipf law to make approximately reasonable prediction on number and
size of cities in a given country.
Urban hierarchies
Inequality in city size: the value of the slope α in the Zipf relationship is ranging from 0.7 to 1.3.
There is no difference between developing and developed countries, but difference is found
between old and new countries.
Slide 10
Slide 10 shows US with the largest cities, while Europe has the greatest number of cities. Notice
that the average spacing among cities in Europe is 13 km while in the US is 50 km.
RD: there is an explanation why the slope is around 1? the value of the coefficient
has to do with resource constraint?
SS: there are data on settlements distribution of non human?
DAL: the literature he knows relates to primates and regards the maximum size of
population, not the distribution of settlements
A set of indexes is introduced in the discussion
Primacy index: P1 P2 , where P1 is the population of the first (i.e. largest) city: lower in the
developed countries than in developing countries
Macrocephaly index: Max Pi Pi 1 , with 1  i  8 (even in this case: lower in the developed
countries than in developing countries)
Slide 11
Slide 11 plots log Pm against log Pu , where Pm is the population of major cities and Pu is the total
population of urban areas (which is about 80% of total population). It shows the relationship
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between the share of population in major areas Pm Pu  and the size of the urban population of the
country Pu  . The coefficient is less than 1 means that the larger the country, the lesser is the
population in metropolitan areas.
Slide 12
Longitudinal approach: Pm (t )  fPu (t ) . In Gibrat’s model the growth of metropolitan areas would
be the same as the growth of the total urban population, whereas this is not the case for instance for
London, Paris or largest Indian cities.
GW: what is the meaning of Pm ?
DP: Pm might be P1  P2  P3  ...
SS: is Zipf law consistent with the distribution of Pm Pu  ?
GW: actually this result is a direct consequence of Zipf’s distribution
Other ways of looking at urban areas: the contribution of W. Christaller
Slide 13
Slide 13 presents Christaller’s representation of hexagon of hierarchical ranking based on functions
performed by the cities in an homogeneous space (ranks from M to A, K, B and G). More than 50%
of income derives from services to the resident population. In general among services there are also
functions that are not related to a specific area (e.g. financial services). So, even if it is incomplete,
Christaller’s theory still has some relevance for interpreting urban hierarchy.
Notice that services dedicated to local population are very difficult to measure.
DAL: it must be noted that the bigger the more reflective ??? is the city
RS: there is an empirical ground for the hexagon representation of hierarchical
functions?
DP: yes, Christaller was representing a stylized case in Germany. Jean Reynauld
(1841) developed a full theory of organization of space including the hexagon [but
R. didn’t draw them] which was not so well known as the one proposed by
Christaller who had the insight of drawing the symbolic representation of a hexagon.
From Christaller to spatial simulation
Slide 14
Slide 15
Slide 16
Slide 17
Slide 18
Slide 19
Slide 20
Slide 21
Slide 14-21 present simulations from SIMPOP model, a multiagent system adopting an abstract
definition of function and introducing a notion of innovation as new function. There is a lattice
structure in the model.
SS: questions on central place theory
DP: central place theory is a static theory
Christaller developed three models with regard to different functions: market, transportation,
administration
BA: comments on Zipf law and Christaller model
MR: is network defined with regard to functions?
DP: in the first SIMPOP model networks are only roughly defined. Another version
including a more explicit representation of networks is developed in the ISCOM
project.
Slide 22
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Other formulations
Auerbach (1913)
Zipf (1949)
Gibrat (1931) considered the size of firms and cities as having a log normal distribution which is
generated by a growth process. If they grow peoportionally to their size, size is randomly
distributed
PIZZA Break
Growth rates dP Pdt  normally are independent of size
Contribution of Madden (1956) on US cities.
Slide 23
Gibrat’s law hold in general, but there may be correlations between growth rates and size or
between successive growth rates: see data for France on log of size and growth rate. Result:
increase in size inequality.
Slide 24
Slide 25
Contribution by Bairoch: maps of Europe in 1500, 1600, 1700, 1750, 1850.
Pumain and her colleagues elaborated a transition matrix for cities by size class of inhabitants for
the period 1800-1990, in Europe: in that period cities became larger and only in few cases became
smaller.
SS: can you apply this model to log normal distribution of one city across time?
DAL: using the model for predicting distribution in time
BA: when we consider the location of cities we observe that in the past raw materials
had a central role for the economy and hence for the location of economic activity.
Now the knowledge based economy might be based everywhere
DP: but Silicon valley is not everywhere, it is close to universities
GG: in location of cities we should consider that historical accidents are important,
moreover pre historical boundaries might be relevant, e.g. rail road towns had an
economic push
DP: this issue might be discussed later
Evolution of concentration
Slide 26
Slide 26presents data on Zipf law (value of the slope) from 1600 to 1980 for Europe
Slide 27
Slide 27 shows maps of Europe 1500-1850 normalized according to the size of cities
Traveling time
Let us consider the evolution of travelling time in the period 1650-1995.
Slide 28
Data in slide 28 refers to the travelling time between Paris and Marseille from 1650 to 1990. We
know that travelling time affect the interaction space and we observe that in France that faster rail
or road transport technologies provoked a collapse of intermediate towns.
Slide 29
Slide 29 shows the shrinking of topological space in Europe as the outcome of the faster traveling
time in the period 1650-1990.
At the local scale also, changes in the interaction space affects city size. The surfaces occupied by
agglomerations are related to the extension of possibilities of commuting within the area.
Slide 30
In slide 30 data on Valence with normalized space for the commuting time are presented.
Here you can have a new definition of city: it is something growing with respect to surroundings.
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DAL: is the traffic decreasing the travelling time
DP: data show that traffic has not decreased travelling speed, which is around
16 km h in central part of cities (data should be specified in detail)
BA: rent is increasing in centre cities, so the metric should be based not so much on
the time to reach B from A, but to the cost of living in a place
DP: accessibility to activities is a key variable
Expansion is not completely independent from scaling law, if you include daily urban system
[defined with regard to the share of population commuting in the area] you get better scaling law.
So far we have considered “normal” processes. Let us consider now some details on changes in
specialization with regard to relative size.
Slide 31
Slide 31 shows the relative size of Paris and other different towns (Toulouse, Orleans, Lyon, Rouen,
Saint Etienne, Amiens) across time: notice that the towns around Paris simply declined in size, and
increases and decrease had specific historical accidents.
Urban potential
Slide 32
Slide 32 expands the work by De Vries (1984) who tested Braudel’s hypothesis of a shift from
Mediterranean Sea to North. De Vries used the notion of urban potential as the integration of
population over distances of surrounding towns and cities.
Innovation
Slide 33
Slide 33 presents data on multinational subsidiaries in European cities the period 1990-1996 and
shows a process of hierarchical diffusion
An evolution model of scaling law in urban system [complete with the notes for the 8 August seminar]
Slide 34
 Selection of relevant variables:
 urban size
 mobility rate
 activity space
 interaction space
 number, size and spacing of towns and cities
 Articulation of the variables in a model generating two levels in urban system depending on
two critical time scale of the interaction space:
 intra urban space
 inter urban space
note that a fractal dimension in filling the space should be considered
Questions to solve [complete with the notes for the 8 August seminar]
Slide 35
 social interpretation of urban scaling
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 comparing urban systems with Geoff West’s criteria
Geoff West’s criteria
1) Hierarchical fractal like branching network
2)
Space filling
3)
Invariant terminal units
4)
Energy dissipating in minimized
cites
Ok, but there are two levels
House/flats, household, firms or
individual towns
?
Participants (listed by name initials)
BA
Brian Arthur
DAL David Avra Lane
DH
Dirk Helbing
DP
Denise Pumain
GG
George Gumerman
GW Geoff West
IP
Irene Poli
MR Margherita Russo
RD
Ralph Dum
RS
Roberto Serra
SvdL Sander van der Leewe
SS
Spyros Skouras
References
(will be provided by Denise Pumain)
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