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Spatial structure of urban settlement systems: stability versus changeability

Structure spatiale des réseaux urbains
V.A. Shuper et A.L. Valesyan


Postulates of W. Christallers central-place theory are discussed with an introduction of a new sixth postulats. Under consideration are specific cases of settlement systems which can not be described by the apparatus of the classical central-place theory. It is shown that besides external causes responsible for the deformation of Christaller grid (anisotropy of the territory, etc.) there are some inner, immanent causes of it which have spontaneous nature. The paper formulates basics of the relativist version of the central-place theory, introduces notions of "the degree of stability of urban settlement systems" and “isostatic equilibrium" (as an attractor). It presents the results of analysis of the degree of stability of settlement systems of Estonia, Armenia, Georgia, and Central Russia. It is substantiated that it is possible to create a theory of the Great Unification by applying the synthetic approach to both the relativist central-place theory and phasic conceptions of spatial development.

théorie des lieux centraux,

theory of central places

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1In the basic work of W. Christaller (Christaller, 1966) the central-place theory did not achieve complete logical harmony. Christaller laid down the foundations of the central-place theory for he formulated the most essentiel of its basic principles and stimulated so far further research. The theory became axiomatic thus allowing to get non-trivial results using deduction in 1960s only with primary respect to the works of M. Dacey and his school (Dacey and al., 1974). Among the other researchers' works those of J. Parr (Parr, 1969 and 1978) have to be mentioned.

2Numerous scholars' efforts have succeeded in formulating the basic postulates of this theory. Its first postulate is the one of the isotropy of space. The space in the central-place theory is equally permeable in every direction via the transport, homogenous with respect to the rural population density, natural environment, and the distribution of all possible resources ; in other words it is absolutely homogenous except the distribution of urban population which is to be described by the theory itself.

3The second postulate of the theory is the one of the infinity of space. No fragments can be isolated from the Christaller grid. If we denied the infinite spacial transmission of the grid we would inevitably perceive the edge effects the theory can neither take into account no describe. The very principles of the theory's construction would be violated along the edges of the central-place system thus the forth postulats to be discussed below would not be implemented under any circumstances. The impossibilite of an Ilisolated state" results from the second postulats of the centralplace theory.

4The third postulate is the one of maximum compactness of the zones. According to this postulate the reason central-place systems make hexagonal grid is that the maximum compact geometrical figure is a circle, and a regular hexagon is the f igure the most close to the circle geometrically which allows solid packing in the two-dimensional space. The question of validating the space structure of central-place systems was discussed in details by A. Losch (Losch, 1959) and, later on, in general form by B. Rodoman (Rodoman, 1970).

5The forth postulate - the optimization principle - stipulates the polymorphism of the central-place systems. The statement that these systems can exist in three versions i.e. with a K value of 3, K value of 4, and 7 is considered by both scientific and educational accounts of the theory. Under condition that K is equal 3 the optimum configuration of market areas is achieved (central places are located in the nodes of the hexagonal grid), and the number of central places serving the region is minimal. K = 4 provides for minimal distances between central places located in the middle points of hexagonal grid's ribs. K = 7 provides for 2 the best administrative division as each central place of the lower hierarchical tier is subordinated to only one central place of the next higher tier comparing with two for K = 4 or three for K = 3.

6The very parameter K has been def ined by many scholars and J. Parr in particular as the number of central places of the next lower hierarchical tier subordinated to one central place of a given tier, plus one. one can describe K parameter in a different way - as a number of zones of the next lower hierarchical tier subordinated to one central place of a given tier.

7Under a strict approach the central places system with K = 7 should be considered a certain misunderstanding (Shuper, 1990). According to the forth postulate no central-place systems with other Ks could exist either for no basic principle has been found for them to be implemented in their spatial structure.

8The fifth postulate of the theory is the assomption of the "rational" behavior of consumers. It suggests that all goods and services are being acquired at the nearest of all central places where they are available. one should not think though that the phenomenon of multi-purposes trips is essentiel for the systems with the optimum configuration of market areas only (K = 3). It is assumed that in the central places system with K = 4 all trips are strictly hierarchical which does not occur in reality. The divergence between real trips and those strictly hierarchical can be perceived as another expression of the phenomenon of multipurposes trips.

9Finally, the sixth postulate of the central-place theory formulated by V. A. Shuper suggests the constance of the share (k) of the population of a given central place in the total population of the area being served by the place, for all hierarchical tiers.

10The necessity to introduce this postulate is dictated by the paradox of k parameter (Shuper, 1984).

11It is widely known that in reality Christaller grids are substantially deformed. Anisotropy of the territory provoked by both natural and anthropogenic factors cannot leave undestroyed the ideal pattern of hexagons. It is far less known about the inner characteristics of any settlement system as well as the grid itself responsable for the deformation of the latter. What are these characteristics ?

12If a central place at a tier m + 1 is located close enough to the central place of a higher tier m, or even m - 1, then the former does not have to possess all set of fonctions appropriate for that level : some goods and services can be acquired at the nearest central place of higher hierarchical tier. Additional profit including informational one which is being received by central places of lower hierarchical tiers due to their closeness to larger centers is unquestionable. Such a concentration is beneficial for a larger center as well for it can use resources of places of lower tiers for its own needs. One can argue that the use of potentiel of numerous satellites-cities is an important source of development of a large city for it does not require the population growth of the latter, and the whole system can "save" on this.

13If some fonctions of places of lower tiers have been transferred to the principal center the same should happen to their populations as well in order to avoid additional costs of transportation. Under such conditions the value of K can not be kept constant even within the same hierarchical tier.

14Now let us examine the very characteristics of the grid which are responsable for its deformation. According to the forth postulate of the theory each central place with its service zone is being divided equally between central places of higher tiers as the former is located at the boundary between their zones. Each central place of higher hierarchical tier controls so far zones of all places of lower tiers. That postulates that any central place can be located at an equal distance from central places of higher tiers with those places belonging to different levels and thus having different size. In real world it could have corresponded to the situation when a small or average-sized city which is located at the same distance from two larger cities would have been equally attracted to both of them even if one of them is 4 times bigger than the former, and the other is 16 or 64 times bigger. Can we imagine that in order to get goods and services available both in Moscow and Tver' residents of Klin which is situated half-way between these cities would be equally likely to go to Moscow and to Tver' ?

15Let us consider a Christaller grid with K = 4 and three tiers of hierarchy. It is obvious that the service zone of the center of the first and highest tier of hierarchy will include the whole hexagone At the second hierarchical tier the service zone of the principal center being now one of central places of the second tier will include not halves but whole zones of the inner circle of third-tier places, thus making the principal centerls zone significantly larger. Now let us consider including a forth tier of hierarchy. Those forth-tier places being located at the borders of the shaded area will gravitate not to the principal center to get services of the second level but to the second-tier central places. They will "take" their service zones with them thus changing the boundaries of the principal center's zone of second-tier service. Further changes will occur if more tiers of hierarchy are to be included. This violates the main principle underlying a system of central places which follows from the third postulate of the theory - the principle of zones' congruence. If not only a zone's size depends upon the number of hierarchical tiers but the very same tier of hierarchy includes zones of different sizes, then the coefficient K becomes meaningless.

16It has been shown so far that some immanent causes exist which contribute to the deformation of a Christaller grid. These deformations are not a result of some external factors like anisotrope of the territory but occur spontaneously according to internal regularities.

17One of the results of the above-described process of deformation of a Christaller grid is a growth of an urban agglomeration surrounding a national or regional capital. Agglomerations so far are the most pronounced examples of concentration of urban settlement systems (except f or the cases when spatial unevenness of settlements' distribution is due to unevenness of distribution of natural resources). Classical central-place theory being spatially expressed by the ideal Christaller grid does not allow for any concentration of population at all for it can result in irregularities of the grid itself. It will also result in the violation of the Beckmann-Parr's equation which was deduced from the grid's characteristics (Parr, 1969).

18Thus a very important fact which nevertheless - to the best knowledge of the authors - has not been reflected by the literature on the central-place theory points out that formation of large urban agglomerations violates not only theoretically predicted proportions of distances between cities but also relative sizes of central places of different tiers of hierarchy.

19As it was mentioned above, one of the postulates of the theory is that of constance of K (with K being the share of a central place in the population of its service zone) for all tiers of hierarchy. Under condition of sharp deformations of a Christaller lattice the value of K can not remain constant not only for different hierarchical tiers but for same tierls central places as well. Consequently one can not apply the Beckmann-Parr equation which describes relationship between sizes of central places of neighboring tiers. Thus unevenness of spatial distribution of an urban settlement system resulting in the formation of large agglomerations represents a different side of the reality which in turn requires substantiel changes to be included into the classical central-places theory to have it all explained.

20There is at least one more phenomenon which can not be explained by the classical central-place theory : the absence of precisely those places in some urban settlement systems which should have occupied second highest tier of hierarchy. Sometimes, but not always such an effect is accompanied by a heavy condensation of central places' web around the principal center.

21It has been shown so far that population distributions between tiers of the Christaller hierarchy when different from those predicted by the Beckmann-Parr equation simply can not be considered by the classical theory. As a result the latter is unable to take into account interrelations between spatial organization of a central-place system and a distribution of population among different tiers of the same system. As opposed to the classical central-place theory the relativistic central place theory developed by V.A. Shuper is aimed at the very interrelation between spatial organization of a central places' system and population distribution among its hierarchical tiers.

22One of the central concepts of the relativistic central-place theory is that of an isostatic equilibrium. The latter is precisely what allows to establish a functional dependency between spatial organization of urban settlements and distribution of population among different hierarchical tiers. Tiers of centralplace hierarchy are being classified into heavy and light ones according to the fact if they have population respectively larger or smaller than it was predicted (by the classical theory ?). A question can arise, that is why there should be any discrepancies at all between observed and predicted populations of a hierarchical tier ? The answer was given above - there are both internal characteristics and external factors allowing for the deformation of an ideal grid.

Let us try to formulate the concept of the isostatic equilibrium. After drawing straight lines from the center of an ideal Christaller grid through all other central places up to the edges of the hexagon (or the principal center's zone) , expressing all distances in shares of the distance between the principal center and the zonels edges, and then calculating average distances between the center and other places for each hierarchical tier the following results have been produced : second tier of hierarchy gets 1.0 on average, third 0.625, forth - 0.672, and fifth - 0.666 (for the system with K = 4). is more than 1., should have the following form : = / , where is a real or empirical population count in all central places of a given hierarchical tier, is a predicted population count of the tier. pulation concentration is not significant populations of different tiers can be predicted using the Beckmann-Parr equation to redistribute total urban population according to predicted shares of it for the parameters K k r the settlement system under consideration. Completely different approach has to be applied to determine values of ut are equal K K K ven under substantiel deformations of the ideal Christaller grid. Using the relativistic method to determine but first become equal. Population of the principal center is getting onethird bigger than population of any other levels when K K is equal 3.  :/ = m - 1

23Under the condition that isostatic effects are being completely compensated for this parameter has a value of m - 1, where m - 1 is a number of hierarchical tiers in a central places 1 system minus the first tier represented by only one central place. A compliance with the parameter of isostatic equilibrium is the very characteristic of stabilite of an urban settlement system.

24The notion about urban settlement systems as gravitating in their development toward a certain stable state with the latter defined as an isostatic equilibrium has also a certain philosophical meaning. It allows to think about such a state as some kind of an attractor. This introduction of the notion of an attractor is not a mere tribute to a fashionable parlance : if we can assume that a process is determined not by initial conditions but by a f inal state then we suggest both the existence of the equifinality and a possibilite of using different routes between the initial and final conditions , more precisely - between stages of the process which were defined as such conditions.

25The idea of equifinality of the development of largest cities has been known in the science since mid-60s (Haggett, 1983). It states that gigantic cities show much more of a resemblance with each other than do those small cities they have grown up from. Some difficult philosophical questions arise here : it can be supposed that moving to similar conditions require similarite of initial conditions and f actors affecting it, or it can be assumed that it is determined by a certain "construction" such as a potentiel form which is to be revealed in the process of a system's development.

26Such a teleology can be quite well explained scientif ically and logically. For example, K. Popper has shown that Darwin's natural selection must have a future expediency as motive forces of evolution (Popper, 1982). In this case the expediency will reveal itself in the future only but it needs certain spendings today.

27The very same approach is being applied in this research for there is yet no clear understanding of mechanisms determining development of settlement systems. Our knowledge allows to analyze past development of settlement systems but prohibits from formulating any analytical laws allowing to predict future states of these systems. That is why the notion of isostatic equilibrium as an attractor for the development of urban settlement systems possesses a few clear advantages. It happens because the most stable state and directions for the process are given with the latter always aiming to reach the former even if the system has diverged from the most stable state under the influence of external factors.

28An important question has to be raised on the correlation between the classical and relativistic central-place theories. Let us consider this correlation from the position of I.V. Kuznetzov who formulated a partial principle of correspondance as it follows :

29"A new theory's mathematical apparatus containing a certain characteristic parameter which values are different in the old and new areas of phenomena passes to the old theory's mathematical apparatus with a proper value of the characteristic parameter" ("Principe...", 1979).

30It seems obvious that such a characteristic parameter connecting the classical and relativistic theories is represented by the Beckmann-Parr equation. The very parameter K becoming meaningless under strong relativistic effects transforme the apparatus of the classical theory into the relativistic one.

31Another important factor determining the correspondance between the classical and relativistic theories is presented by the existence of a certain transitional zone where relativistic effects take place. Relativistic effects gradually penetrate into the classical theory : there are two poles. Firstly, the ideal Christaller grid described by the classical theory and the central-place system with no second hierarchical tier. The latter does not fit to the classical theory at all and can only be described by the relativistic theory. There is a wide "area" between those poles which includes the vast majority of urban settlement systems existing now or existed before.

32It is obvious that a close-to-ideal Christaller grid can exist in a very rare situation. This is true even for the central places' system of South Germany. A corresponding figure in the classical book by W. Christaller (Christaller, 1966) unequivocally shows that only the existence of a preconceived idea let its author to see on the map something looking like a regular hexagonal grid.

33Finally, the very state of isostatic equilibrium expressed by the equation (1) represents a very important parameter which ensures the correspondance between the classical and relativistic theories. If isostatic equilibrium is complets then the equation (1) is true for both an urban system completely corresponding to the ideal Christaller grid (if one can imagine that such a system exists) and one with high concentration of cities near the principal center and no second hierarchical tier.

34Let us now discuss stability of spatial structures of real settlement systems. It has to be emphasized that hierarchical structuralization of urban settlement systens does not happen at once. A hypothesis was earlier suggested by V.A. Shuper that urban settlement systems first had been formed as a whole while their compliance with predictions of the Zipf's rule getting improved, and then a distinct hierarchical structure was formed thus improving their agreement with predictions of the central-place theory and worsening their compliance with ones of the Zipf's rule.

35As if substances pass with time from amorphous to crystalline conditions, settlement systems also pass from a quasi-amorphus state described by the Zipf's rule to a quasi-crystalline state characterized by predictions of the central-place theory.

36This hypothesis has been successfully proved by such diverse settlement systems as those of Estonia, Armenia, Georgia and Central Russia (computing programs were created by N. Aznauryan).

37Table 1 presents values of a parameter which was calculated using the method by Yu.V. Medvedkov (Medvedkov, 1964) and is equal to the average of squared deviations from the natural logarithm function approximating the population distribution of first 16 largest cities for corresponding regions. As it can be seen from the Table 1 all four settlement systems began improving their agreement with the Zipf's model during some initial period and then changed this trend to the opposite one. Increasing values of the parameter can be seen as an indicator of growing integrity of each system.

38It seems to be quite logical that in Central Russia transition from a quasi-amorphus state into a quasi-crystalline one happened as early as in the middle of nineteenth century. The following dynamics of the parameter show a constant growth. By the time when Estonian Republic proclaimed itself independent and left Russia Estonian settlement system had been already quite mature. Not surprisingly the latter had developed as a kind of integrity earlier than Armenia and Georgia did it. Being more compact than its northern neighbor Armenia left Georgia behind for the latter was witnessing clear centrifugai tendencies. The latest decline of the parameter in Estonia might be understood as a legitimate fluctuation provoked by the process of re-establishing its independence.

39Let us nowanalyze dynamics of the parameter which characterizes the degree of correspondance of a system with the state of isostatic equilibrium (Table 2).

40In the case of Estonia the parameter under consideration got quite close to its predicted value of 3 (for 4 tiers of hierarchy) by the year 1959, and in 1965 it became equal with the ideal. The subsequent behavior of the parameter can be interpreted as follows : under the forces of inertia of its own movement the system achieves the value of 3.18, and after that it starts moving back. The hypothesis on oscillatory type of dynamics of the parameter has been already suggested (Valesyan, 1991). To obtain the data supporting such a hypothesis we have to analyze many examples of real settlement systems.

41A new splash occurring in Estonian settlement system after 1989 might seem quite strange. But we should not forget about stormy political collisions survived by Estonia during the period of painful re-establishing of its national independence for they for sure influence the development of its urban settlement system.


Central Russia



































































































Table 1. Changes in the degree of agreement of some settlement systems with the rule of "rank-size".







2.01 (4)


3.11 (3)

1.76 (4)



2.58 (3)







1.73 (4)



2.40 (3)




2.41 (4)







1.61 (3)

1.92 (4)



2.84 (4)

1.55 (3)

2.79 (4)


3.07 (4)

3.00 (4)

2.33 (4)

2.95 (4)


3.32 (4)

3.09 (4)

2.68 (4)

3.05 (4)


3.63 (4)

3.14 (4)

2.99 (4)

2.82 (4)



3.18 (4)






3.22 (4)



3.55 (4)

3.13 (4)

3.14 (4)

2.90 (4)


3.57 (4)

3.18 (4)






3.37 (4)


Table 2. Changes in the degree of spatial stabilite of some urban settlement systems.

Numbers in parentheses represent the number of hierarchical tiers.

42As if on purpose Estonia has been created by nature to help verify different approaches in the field of central-place theory. Settlement systems which lack the same degree of maturity and integrity fit this goal to a much lesser extent. Nevertheless an attempt to apply either classical or relativistic central-place theories to them can be quite interesting.

43The only one of three Baltic republics Estonia is an ideal region for the development of the central-place theory. Latvia's capital, Riga, is too big for such a country for the city became a major transportational center ; Lithuania historically possesses two capitals. Both Ukraine and Byelorussia as if have been built of two distinctive pieces : their settlement systems are far from integrity, and this affects proportions of these systems. There are two capitals in Ukraine as well ; in Byelorussia the urban settlement system is not only divided into eastern and western parts but also has disproportional hierarchical tiers : it lacks, in particular, a sufficient development of a network of small cities which would have contributed to the forth hierarchical tier.

44In this connection republics of Caucasia are of a great interest, and especially those of Armenia and Georgia as the territory of Azerbaijan has been torn apart. Let us consider the data on Armenia. As it can be seen f rom the Table 2, changes in the degree of stability of spatial structure of the urban settlement system in Armenia have been following the trend similar to one of Estonia : at the beginning there was a poor agreement with predictions of the theory, but since approximately mid-sixties when a forth hierarchical tier appeared in the system the latter began to get closer to the isostatic equilibrium and even passed it "farther". Unlike Estonia with its more mature settlement system Armenia has not seen any returning movement of the "pendulum" yet, and the 1989 value reflects the impact of the major earthquake on its settlement system. It can be suggested that in the near future such a oscillatory trend will show up in Armenia as well.

45A somewhat different trajectory describes the changes in the degree of spatial stability of the settlement system of Georgia. Having reached the isostatic equilibrium by the end of 1960s the parameter of stability went to the opposite direction after that. For now it is hard to say if the settlement system of Georgia follows its own way or it will repeat that of Estonia in a few years with the parameter of stability decreasing after having reached an ideal agreement with the theory. If the second option comes true its later realization can be explained by the lower level of integrity of the settlement system of Georgia : centrifugal tendencies could and still can be seen in it quite clearly.

46Though all three settlement systems under consideration can be described by the classical central-place theory it is not the case for the territory of Central Russia. The settlement system of Central Russia lacks those cities which had to have formed the second tier of hierarchy, that it why in order to evaluate the degree of agreement of the system with the isostatic equilibrium we use the apparatus of the relativistic central-place theory.

nstant growth of the parameter went on until the beginning of 1980s. If we disregard a small perturbation in recent years the parameterls further behavior can be described as the "pendulum" returning back since late 1980s. (Let us recollect that according to the theory the value of the parameter has to be equal to the number of hierarchical tiers minus one. The settlement system of Central Russia had five hierarchical tiers with a missing second tier over the period under t/

47Let us shift our attention from real values of the parameter of stabilite of settlement systems to its methodological importance. At a first approximation the latter can be used as a good indicator of changes in stages of spatial evolution of settlement. A hypothesis on the oscillatory nature of changes in urban settlement systems' stabilite was suggested earlier (Valesyan, 1991). The time points where the direction of oscillatory movements changes may turn out to be the very critical points where the system moves from one state with, for example, predominantly centripetal tendencies, to the other state with predominantly centrifugai ones. Monitoring trajectories of changes in stabilite of settlement systems with different level of maturity together with studying spatial evolution of urbanisation in those systems can lead to quite fruitful results.

48We have analyzed some possibilities of synthesis of those conceptions which deal with settlement systems only. The search for these possibilities seems to be quite logical. It might be a lot less trivial of an idea to suggest the existence of a certain synchronisa in the changes of stages which describe heterogeneous spatial phenomena. An inner connection can exist between, say, an appearance of a new topological level in the transportation network, development of a new hierarchical tier in the corresponding urban settlement system, a change in directions of spatial components of urbanisation (from centripetal to centrifugal and vise versa) and an emergence of a next wave of concentration or deconcentration in the regional development. If enough of serious evidence could be found to support this statement it would be possible to discuss possibilities to create a theory which would allow to embrace all existing conceptions of phasic spatial development. By analogy with physics this future theory can be named with some sense of humor a theory of the Great Unification. The idea to create such a theory capable of unifying research paradigms which are methodologically related but still significantly separated from each other will allow to reach a better understanding of the drama of spatial connections of different phenomena and to reveal their concordance.

49At a first approximation we can intuitively suggest that the invariant which characterizes the degree of stabilite of spatial structure of urban settlement systems can become such a pivot to "thread" the above-named conceptions. This means that determining the trajectory of changes in stabilite of a system a researcher can "catch" different stages of spatial self-development. It seems to have a deep meaning for the urban settlement system of a given region reflects in one way or another a whole palette of the spatial self-development "from geology (location of largest cities at morpho-structural nodes) to ideology (cities as focuses of political and cultural life)" (Guberman, 1987).

50Suggesting the above-named invariant as one of foundation-stones of the future theory of the Great Unification is a logical step from the methodological point of view for the growth of the theoretical knowledge in any areas of the natural science starts from the search for specific invariants, or parameters which are stable with regard to certain transformations (Ovchinnikov & Shuper, 1987).

51It might seem at the first sight that the suggested idea on the possibilite to create the theory of the Great Unification is somewhat improbable or even fantastic. But it is known that the verisimilitude is a false criterion of the correctness of a theory.

52Strict mathematical statements existing in the theory will allow to empirically verify its conclusions in the future. As it was noted by K. Popper, "we try to find similar features in objects and interpret them according to laws invented by ourselves. Not waiting for all premises to be at our disposai we immediately formulate conclusions. They can be rejected later on if the observation will prove their error" (Popper, 1959).

53As a conclusion let us express the hope that the theory of the Great Unification will not lack attention of geographers. For if a conception, as it was righteously noticed by W. Heisenberg, "is so good that it allows to combine a lot of most different phenomena which look "the same" or closely related in a certain aspect then this conception will be accepted for the very unifying power of its own" (Heisenberg, 1991, p.77).


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V.A. Shuper et A.L. Valesyan, « Spatial structure of urban settlement systems: stability versus changeability  », Cybergeo: European Journal of Geography [En ligne], Systèmes, Modélisation, Géostatistiques, document 88, mis en ligne le 26 mars 1999, consulté le 27 février 2024. URL : ; DOI :

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V.A. Shuper

Institut de Géographie, Académie des Sciences de Russie, Moscou, Russian Federation

A.L. Valesyan

Institut de Géographie Académie des Sciences de Russie, Moscou, Russian Federation

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