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10ème Colloque Européen de Géographie Théorique et Quantitative, Rostock, Allemagne, 6-11 septembre 1997

Space-time contraction and the dynamics of urban systems

La contraction espace-temps et les dynamiques des systèmes urbains
Anne Bretagnolle, Denise Pumain et Céline Rozenblat


An attempt is made to identify the dynamics of urban systems during the historical process of their evolution. An illustration is made with the case of European cities between 1200 and 1990, using harmonised historical data bases. First off, simple maps show a general expansion in cities number and size of cities over time, reflecting the ability of the system to continuously adapt its structure over time. A second trend is an increased hierarchisation in city size, which may be related to the continuous improvement in the speed and capacity of transportation means. Because of this space-time contraction, large cities short-circuit small towns, and inequalities in city size are widening. A third trend leading to a spatial reorganisation, emerges from maps of the urban population potentials. Among all possible mathematical equations and parameters values which could describe the intensity and the spatial range of interactions between cities, we choose those which give rather stable results and which are compatible with the analysis of urban historians, such as De Vries, Braudel, Bairoch, Hohenberg and Lees, for the centuries preceding the industrial revolution, by using evaluations of distances in terms of time and cost. Urban systems dynamics is better approached with reference to this relative space than on usual topographic maps.

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The concept of a European urban system

1The concept of system of cities has been introduced a long time ago, to acknowledge both the hierarchical and spatial structure as well as the multiple interdependencies between cities belonging to the same country. Well before Berry (1964) suggested his famous image of " cities as systems within systems of cities ", J. Reynaud (1841, quoted by Robic, 1982) had emphasised regularities in the spatial and functional organisation of " le système général des villes ".

2Urban systems have been mostly described and modelled by geographers after their hierarchical and spatial organisation. Rank-size distribution and central place theory have been universal references for those descriptions, which were mainly developed during the fifties and the sixties. More recently, following the observations made by A. Pred (1977) about the complexity of linkages and interactions within city systems, researches have tried to analyse and formalise the dynamic processes which may simulate the emergence of those structural features. Analogies have been taken in biological processes (Dendrinos and Mullaly, 1985), in physical processes (Sanders, 1992), or in complex automata (Bura et al., 1996). Another research trend is to integrate the emergence of urban systems in economic theory, but then the concept of an urban system seems still more embedded in structural and static connotations than dynamic ones. For instance, the most recent publications by regional economists attempting to formalize the emergence and structuration of urban systems postulates that the structure of urban systems first satisfy an equilibrium, then that this equilibrium is changing over time (Fujita et al., 1994).

3We develop here a complementary and different approach. We suggest that the structure of urban systems is not the result of an equilibrium which would be produced by antagonistic forces or market mechanisms or any optimisation constraint. The spatial and functional structure of urban systems is the product of a complex evolutionary process, involving mainly a competition between cities. Competition was historically oriented first mainly towards the political control of territories, then towards the control of economic and social networks, through trade and under changing technological conditions. In this process, actors located in settlements are both anticipating for innovation and adapting to it by imitation (Allen, 1997). The emergence of innovation and adaptation to innovation are partly random and partly determined by the former structure of the settlement system. This conceptual framework has been developed elsewhere (Pumain, 1997). It is leading to the development of a specific evolutionary theory for urban systems, including historical trends and spatial features into more general formalisms of systems dynamics.

4Several questions remain unsolved when trying to formalise the concept of urban system. It is difficult to decide if cities should be isolated, or considered as part of broader settlement systems : if there is a rather clear functional distinction between rural and urban settlements, the transition between the two types is more or less continuous, and on the other hand, if the population dynamics of villages and towns are very different, historically many towns and cities proceed directly from a former rural village. Changes in functions and extended suburbanisation have nowadays made the distinction more difficult. It is therefore probable that a consistent evolutionary theory can be produced for whole settlement systems rather than for urban systems only. Another problem is the spatial delimitation of urban systems. A. Pred suggested a definition such as " a system of cities is defined as a national or regional set of cities which are interdependent in such a way that any significant change in the economic activities, occupational structure, total income, or population of one member city will directly or indirectly bring about some modification in the economic activities, occupational structure, total income or population of one or more other set members " (1977, p.13). A. Pred however admit large variations in the level of interdependence and closure of such systems. Actually, the most difficult question to solve in transferring the concept of a system to a set of cities is that, inside a regional or national system, cities have systematic variations in the range of their interactions, according to their size and activities.

5The question of identification and delimitation of an urban system reaches a maximum level of difficulty when applying the concept to a large area and over long periods of time. When a European urban system did emerge ? Interdependencies between cities at a European scale have intensified only recently and in a strict sense the application of the concept should probably be limited to the last decades ! Even at the scale of national European countries, like France or Britain, historians deny the reality of city networks before the end of XVIIIth century (Lepetit, 1988, Martin, 1984). Nevertheless, even if direct linkages are obviously not strong and frequent enough for identifying city networks, we suggest that indirectly, towns and cities have always been connected and interacting, at least because they were limited and imitated in their development by their neighbours. Therefore, step by step, the influence of one centre on another was propagated through those local interactions. The probability of developing and meeting far distant influence obviously has increased over time and is always higher for the largest cities and some specialised centres.

6We shall then observe on a long period the emergence and structuration of a set of settlements which may be considered nowadays as the European urban system. A practical problem follows : how to characterise the evolution of the structure, how to measure the transformations through long periods of time in a relevant way ? Especially, concentration is delicate to compare in space and time (Bretagnolle, 1996). Such questions raise difficult methodological problems. We suggest here a set of methods for exploring the transformations of urban systems, which are not simple applications of well established statistical procedures. We try to integrate in the statistical investigation some specific results from urban theory as well as knowledge from historians of urbanisation.

A simple cartographic method

7Good historical data bases are now available for studying the long-term development of the European urban system. After the first attempts by Chandler and Fox (1976) and Davis (1969), historians have collected data, cross-checked them, controlled the validity of figures and now provide information which allow a good overview of the evolution of population size for all European agglomerations with 10 000 inhabitants or more. De Vries (1984) established such figures for the period 1500-1800 and Bairoch, Batou and Chèvre (1988) from 850 to 1850. Minor differences remain between those two data bases, which give however similar results regarding large urbanisation trends. F. Moriconi-Ebrard (1994) completed in his base Geopolis the information for the more recent period, 1950-1990. The three data bases rely on the same concept of urban agglomeration for ensuring the validity and comparability of urban population figures in space and time.

8Population figures remain the more readily available information regarding the importance of cities. The meaning of this indicator is not only demographic, since the number and variety of the urban functions and the general level of complexity of urban society and culture are highly correlated with population size. For a long time, the city size has been a good indicator of urban sustainability, e.g. the capacity of an urban place to ensure a living to a large population on a permanent basis. Cities of the ancient world were growing mainly through inmigration from the countryside. They were competing for capturing resources, as territories first through wars, then markets through trade and industry. They competed for acquiring central positions in their surrounding region first, then in more distant networks. Their attraction on rural populations, their demographic growth and their size eventually reflected the ability of the urban actors located in a city to build durable profits from their activities, to innovate and to imitate profitable innovations.

9Demographic expansion over time is a first characteristic of the evolution of urban systems. It is of course linked with increase in productivity and innovation. This trend affects all parts of urban systems. This may be illustrated by a series of maps of Europe where each city having 10 000 or more between year 1000 and 1990 is represented by a dot whose surface is proportional to city size, with the same legend for all maps (figure 1). The result of mapping city sizes in absolute numbers is mainly a progressive filling of the map, from quasi white to almost black, due to both an increase in size of each city and the emergence of new cities completing the network. This is a new demonstration of the process of population growth in urban systems over long periods of time, which is now rather well known. Most empirical results, as those brought by Robson (1973), Pumain, (1982), Guérin-Pace (1993), are comforting the main hypothesis first made by Gibrat (1931) : on the whole, in a closed and relatively homogeneous set of settlements, urban growth is proportional to city size, urban growth rates are randomly distributed regarding city size. Cities have constantly the same probability for growing as other cities and over a long period their growth rates are on average the same, with a range of variation. New towns also appear in the system, mostly in the lowest size class, at a rate which may vary over time (de Vries, 1984) but their contribution to the total increase of urban population remains in a range not exceeding usually 20%. Interpretation of this quasi stochastic process of growth regarding city size and spatial distribution may be linked with the high connectivity of urban systems, which facilitates the diffusion of innovations. The apparent stochasticity of the process actually refers to the adaptive ability of the urban system, which maintains through its connectivity its structural properties over time.

Figure 1 : evolution of the size of European cities from 1000 to 1990 – absolute figures

10The general expansion of the urban system does not allow to visualize the possible differential growth of some cities, which could lead to a spatial restructuring of the system. We suggest a simple cartographic method for a better representation of those changes in the spatial structure. The mapping method on figure 2 uses different legends for each date: the same circle size corresponds to a population size which vary from one map to the other. This appears as a different population figure in the legend. For each city, the size of the circle represents the size which would have been reached if cities of that size had increased according to the mean urban growth rate during the period. Each map then represents directly the relative size of the cities, at a given date, irrespective of the general expansion of the urban system. If all cities were growing at the same rate, the maps should be the same whatever the date.

Figure 2 : evolution of the size of European cities from 1000 to 1990 – Relative figures 

11On the contrary, the maps of figure 2 are different and demonstrate two processes. The first is the appearance of new cities as new elements in the system: urban systems are open systems. The second is the increase of the inequalities in city sizes. This trend is perceptible as soon as between 1600 and 1700 and become especially important after the first industrial revolution. Another way of describing and measuring it is to notice the sharp increase in the variance of city sizes (the variance of their logarithms is more significant, as the shape of those distributions is generally of the lognormal type) or in the slope of rank-size distribution. This hierarchisation process, when measured, appears to be much more intense than it would be expected from a pure stochastic process of growth following all Gibrat’s hypothesis. One has to explain this increase in the hierarchisation of the system, which has also been observed in other parts of the world during the urban transition. Obviously it could be related mainly to the transformations which have happened since the revolution in industry and transportation means in the XIXth century.

Effect of space-time contraction at the system level

12Due to major innovations in communication technologies, the social meaning of physical space has changed dramatically in the last centuries. From 1500 to 1990, the mean ordinary travel time for crossing Europe from Ireland to Southern Italy has been reduced from two months to one day! If the same topographical backcloth is used to figure the spatial configuration of the urban system, it is actually a false representation, as far as existing interactions in a system of cities are concerned. To give an idea of the evolution of the possible interaction field of cities, one should draw maps with a constant scale according to mean ordinary travel time. Figure 3 gives an example of the tremendous global change in relative interurban distances.

Figure 3 : Inter-urban circulation space in Europe 

13The process of space-time contraction (Janelle, 1969, Juillard, 1972) has been durable and universal, even if this evolution is not linear. For a very long period of time, transports remain very slow. Relying on walking, horses, or wind force on the sea, the average travelling speed did not exceed 5 km/h. The main discontinuity in the evolution has coincided with mechanic energy replacing human and animal energy, in the second half of XIXth century. A little before, on the eve of XIXth century, a few innovations had brought about a gradual shrinkage of space: better roads and coaches had fasten the travels over land whereas various technical improvements also had speeden-up journeys over sea. In France, the average speed of coaches was brought from 5 km/h in 1800 to 10 km/h in 1850. But it is the mecanisation which started an exponential increase in transportation speed: averaging at each date the mean speed of the most rapid transportation means according to the frequency of their use, it can be estimated at 20 km/h in 1850 (coach and train), 80 km/h in 1950 (car and train), 250 km/h in 1990 (rapid train and aircraft).

14The adaptation of the urban system to this major change in the technological context is remarkable. A few rather large new cities have emerged in the industrial areas on coal and iron mining basins. But elsewhere the spatial configuration of the urban system has not been significantly altered by the revolution in transportation technology. The main observation when comparing the maps on figure 2 for 1700 and 1850 or 1990 is the general change in the relative sizes of cities, which are much more contrasted at the end of the period than they used to be before. There is a slight trend towards this major transition as soon as 1800. The emergence of a few large industrial cities in north-west Europe appears as a minor change, compared to this tremendous transformation in the internal structure of the system.

15A simple but powerful explanation of the reinforcement of inequalities in city size has then to be searched in the technological progress, especially communication technology. Many empirical studies have demonstrated that facilities of circulation can hardly be thought of as determining urban growth : there is not a linear but on the contrary circular causality between gain in accessibility and urban development. Nevertheless, the speed-up of communications has had strong and systematic effects on the hierarchical organisation of the urban systems.

Space-time contraction and local urban dynamics

16Map comparison does not allow to analyse in detail the possible local changes in the spatial pattern of cities. Another method is necessary for a more precise description of what happened at a regional scale or even at the scale of each city. The method most frequently used for assessing the relative situation of cities in an urban system is to compute and map their population potential. As it considers both the population of a city and its distances to other cities according to their size, the potential may illustrate through its variations in time how local positions are changing respective to the hierarchical and spatial structure of the urban system. We suggest a few improvements to this method by including in the computation measurements of the changes in the time-space of the urban system and in the accessibility of individual cities.

17The population potential may be conceived as a tool for describing the relative situation of a city in an urban system. De Vries (1984) wanted to demonstrate on a quantitative basis Braudel’s hypothesis of a reversal of the location of the main centre from Mediterranean sea to North sea between 1500 and 1800. He was unable to prove it when using as empirical evidence the gross population size figures to measure the relative importance of cities in the world economy of the time. However, the hypothesis was supported when a combination of size and of the relative geographical situation of cities was evaluated after population potential computations. Two mathematical functions are used classically by geographers for computing populations potentials: power and exponential functions. A multiplicity of choices among various forms of functions and parameters values are possible and may produce very different results (Fotheringham, O’Kelly, 1989). Very little theoretical justifications can be provided for determining the choice of the best model. In order to be able to establish comparisons with the former work of historians, like de Vries (1984), we have chosen the power function. Our main hypothesis is that the highest population potential values should coincide with the location of what Braudel named a " center of world-economy ".

18The equation of the model is:

19Pi = mi /dii + ∑ mj / kij dij

20where Pi is the population potential of city i, mi its population size, mj population of city j, dij the physical distance between city i and j. The parameters which are used to adapt the function over time are dii, the measure of half the distance of city i to its next neighbour, kijt is a weighting factor of the physical distances according to their social meaning (in terms of time or cost). It can take twenty different values, according to the existing transportation facilities which exist at a given time between cities i and j. Therefore, all parameters vary according to time.

21We did not however retain the parameter dii chosen by de Vries, a constant value that he had empirically fixed at 20 km, because it did not give for every dates results which were compatible with the observations of other historians. Nevertheless, we have followed the same method than de Vries for weighting the physical distances between cities, using his codification from 1500 to 1800 and prolonging it afterwards. Various transportation infrastructures have been taken into account at each date for computing the weights: from 1200 to 1800 we have given different codes to cities located along navigable rivers, canals or coastlines, for 1850 we have encoded the cities having an access to navigable canals, and railways; for 1990 high speed trains and airports have been considered.

22The introduction of weighted distances, whose values are determined according to the existing transportation facilities between cities, in the computation of the values of population potential, improves the quality of the estimations (Bretagnolle, 1998). Compared to unweighted measures of potentials, it mainly reinforces the inequalities between cities: for instance, cities which have just become accessible by a canal or by train get a significant increase of their position in the hierarchy of urban potentials. The maps of figure 4 have been drawn after the values of the each city’s weighted potential, represented as a percentage of the maximum value. (They were generalised by isolines, in order to allow the comparison at different dates. In order to improve the legibility of the last map (1990), we rose the size threshold of cities to 50 000 inhabitants instead of 10 000 on the maps of the other periods).

23There are clear but slow modifications of the spatial pattern of the relative importance of cities, in the potential urban interaction field defined by the structure of the urban system (figure 4). The migration of the highest potential values from northern Italy to the North sea around 1600 is clear. Venise and Milan have been progressively replaced by Bruges, Anvers and Amsterdam at the top of the urban potential interaction field, and Londres has become the main centre in 1800, with a stronger advantage on any other cities than it was observed in the former centuries. On the map, the contrast between the potential in London and the rest of Europe illustrates the effects of the concentration of the industrial revolution at this first stage, whereas in 1850 and after we can observe a diffusion of the maximum potential values towards continental Europe. If one excepts this period of very concentrated innovation, the maps illustrate the regional character of the transfer of population potential peaks, which are generally not located in an isolated city but on a subset of well and closely interconnected cities.

Figure 4 : Urban population potentials between 1200 and 1990 (variable distances over time)

24Maps of figure 4 also give an idea of the time scale which is necessary for observing a reorganisation of urban system. It is a very slow process, which is developing rather continuously in space and time.


25The study of the evolution of urban systems over long periods of time is complex for several reasons. First, the available data do not allow to take into consideration the whole settlement systems of the past, but only the largest urban centres for which population is documented. Second, the general expansion of the system, in the number of cities and in their size, hides the structural transformations of the system. Taking into account the relative size of cities, either by computing the share of their population in the total population of the system, or by mapping solutions, put forward the impressive accentuation of city size inequalities, which affects all parts of the system, whatever the region or the hierarchical level under consideration. Historically, the hierarchical structure of urban systems has been transformed according to this trend which enlarges the variety of city sizes and functions in the system.

26Space-time contraction due to the speeding-up of transportation means is considered as an important factor responsible for that reorganisation process. The possible spatial transformations of urban systems have then to be observed, not in a physical space, but in a time (or cost)-space which is relevant for defining the real possibilities of interaction between cities. Following works by historians, we suggested to introduce in measurements of classical urban population potentials one parameter which accounts for the general and local modifications in the accessibility of cities. The maps of urban population potentials which are produced illustrate in a synthetic way the spatial effects of differential growth in a system of cities. The evolution of the spatial configuration of European urban system shows a regular and slow adaptation of its structure to major changes in energy sources and trade roads.

27One should make a remark about the possible generalisation of this very simple method for comparing urban systems, world-wide and for different periods in time. The measurement of urban success after population growth is valid only for populations which have a rather homogeneous level of economic development. A measure after the growth of GNP would give different results, for instance if one tries to assess the relative expansion of urban metropolises nowadays between developed and developing world. Again, the notion of an urban system has to be examined preliminary to any comparative analysis.

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Anne Bretagnolle, Denise Pumain et Céline Rozenblat, « Space-time contraction and the dynamics of urban systems », Cybergeo : European Journal of Geography [En ligne], Dossiers, document 61, mis en ligne le 10 juillet 1998, consulté le 28 septembre 2020. URL : ; DOI :

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Anne Bretagnolle

Equipe P.A.R.I.S., URA1243, CNRS-Université Paris I, 13 rue du Four, 75006 Paris, France

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Denise Pumain

Equipe P.A.R.I.S., URA1243, CNRS-Université Paris I, 13 rue du Four, 75006 Paris, France

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Céline Rozenblat

Université de Montpellier, France

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