Contribution to regional division of Slovakia based on the application of the Reilly's model 237 Hungarian Geographical Bulletin 61 (3) (2012) 237–255. Contribution to regional division of Slovakia based on the application of the Reilly’s model Marián HALÁS and Pavel KLAPKA1 Abstract The objective of the present article is an application of three versions of the Reilly’s model in a regional division of Slovakia at two levels: from the point of view of potential natural gravity towards regional centres, and from the aspect of an alternative proposal of ad- ministrative division of Slovakia at the regional level. In this contribution the geometric version of the Reilly’s model serves only as a complementary tool for an assessment of a regional organisation of Slovakia. A crucial part lies in the application of the topographic version; the oscillatory version possesses a correcting and refi ning role. Functional urban regions have been used as basic spatial zones in our analysis. The geometric version of the model is used for a preliminary assessment of possible infl uences of centres. Let alone several exceptions caused by physical geographical conditions (e.g. an overrated sphere of infl uence of Žilina) this version provides a relatively realistic image of the Slovak regional system. Keywords: regional division, potential spatial interactions, Reilly’s model, Slovakia Introduction Geographical space is not a homogeneous entity and individual geographi- cal components are not distributed uniformly in the space, i.e. they are rep- resented by diff erent intensities in distinct regions. In most cases there is a natural tendency to balance these diff erences. As an example from physical geography the occurrence of horizontal fl ows of various types can be put forward. For instance in case of diff erent values of atmospheric pressure the “polarity” is balanced by the fl ow of air masses. Such fl ows posses a character of vector or gradient. An analogical situation can be witnessed also in case of human geo- graphical partial components. Horizontal fl ows (of people, products, informa- 1 Palacký University, Faculty of Science, Department of Geography, 17. listopadu 12. 779 00 Olomouc, Czech Republic. E-mails: marian.halas@upol.cz, pavel.klapka@upol.cz 238 tion etc.) occur in the social and economic environment as well. Geography mostly refers to them as the spatial interactions. They represent an aggregation of individual mobility and contacts. As for the fl ows of people they are condi- tioned by the activities of individuals. The spatial behaviour of individuals is aff ected by their needs and eff orts to optimise spatial movements (or spatial location) in order to gain economic and social benefi ts. Socio-economic spatial interactions substantially aff ect the geographical organisation of the society and express the interdependence among the sections of geographical space (regions) of diff erent hierarchical levels. The primary information on spatial mobility of the population and spatial interactions is based on the migration data, particularly on the labour and school commuting. The labour commuting, which is the basic platform for regionalisation tasks, was recorded in the former Czechoslovakia for the fi rst time in the 1961 census. Since then the data on main commuting fl ows have been available only in ten years intervals. Other data on the spatial in- teractions (for instance on passenger traffi c volumes, att endance of shopping malls etc.) are very complicated to acquire and in some cases they are subject to business secret. If such insuffi ciencies cannot be supplied by questionnaire surveys (these would be very demanding and practically not applicable on the whole state territory), geographers frequently resort to the use of potential spatial interactions when handling the regionalisation tasks. These approaches rely on the assumed non-homogeneity of geographical space and have been in- spired by the laws of physics, for instance by the Newton law of universal gravitation. The partial objective of the article is to apply basic geometric and topo- graphic versions of the Reilly´s model on the territory of Slovakia. Spheres of infl uences of individual sett lements reached by the geometric version will serve as a base for the selection of the centres and subsequent application of the topographic version of the Reilly´s model with a modifi ed parameter. The primary objective of the article is, then, a delineation of regions in Slovakia aimed at putt ing forward an alternative proposal of an administrative division of Slovakia at the level of regions (i.e. NUTS 3), and their grouping into the NUTS 2 regions. This analysis will employ the potential spatial interactions. The proposal will att empt to take into account the rule of spatial justice and the size of the NUTS 2 regions recommended by the EU. The delineation of the NUTS 3 regions in Slovakia is not correct and is subject to persistent harsh criticisms from the independent scientifi c public (for instance deliberate distribution of the Hungarian minority between the Bratislava, Trnava and Nitra regions, inadequate demarcation of the Spiš his- torical territory etc.). Our alternative proposal has made use of the Reilly´s model since the higher levels of the administrative system (NUTS 2 and NUTS 239 3 regions) cannot be delineated according to real commuting or migration fl ows (for instance the gravity zone of Bratislava reaches far eastern Slovakia). This claim is supported by the fact that such a task has not been undertaken by the Slovak geography so far. One of a few options of such a construction based on the quantitative data processing is presented in the article. Theoretical background and methods Geography (or spatial science) saw the fi rst application of simple models in- spired by the Newton law of gravitation in the 19th century (Ravenstein, E.G. 1885). The issue of spatial interaction modelling had further developed during the inter-war period when William J. Reilly (1929, 1931) defi ned the so called law of retail gravitation, which was based on the real interactions observed in Texas during the second half of the 1920s. However, the theoretical explana- tion of socio-economic spatial interactions based on the gravitation concept occurred aft er the Second World War, when Stewart, J.Q. (1948) formulated his concept of social physics and used the term demographic force as an analogy to gravitation force used in the natural sciences. Stewart´s work refl ected the conclusions reached by Zipf, G.K. (1947) regarding the principle of minimum eff ort which is very important for the spatial interaction modelling since it is closely related to a “resistance” exerted by the geographical space towards the spatial interactions. The law of retail gravitation has been further modifi ed on the basis of the above mentioned works by Converse, P.D. (1949) and Huff, D.L. (1964). Converse mathematically expressed the breaking point between the spheres of infl uences of neighbouring centres, whereas Huff was the fi rst to express the theoretical probability of choice of shopping centre by customers (not ap- plied on the concrete territory). Derivation of the interaction models has not been limited to a mere analogy to the Newtonian physics. Wilson´s deriva- tion is based on the entropy maximising method inspired by the second law of thermodynamics (Wilson, A.G. 1970) and it shows that the Reilly´s model and equation identifying the breaking point is only a special case of the so called unconstrained interaction model. Originally the Reilly´s model was constructed as a tool identifying the retail att raction and was based on purely formal relations. It was applied mainly to determine the tendencies of the population to travel to selected cen- tres in order to reach diff erent types of services and to identify the borders of infl uence between centres within simple graphical schemes of the sett lement system (e.g. Fotheringham, A.S. and O´Kelly, M.E. 1989). A diff erent type of task was and mainly currently is a delineation of tributary areas of shopping centres, i.e. points carrying the masses are not conceived as sett lements (e.g. 240 Lee, M.L. and Pace, R.K. 2005; Baray, J. and Cliquet, G. 2007). A note should be made here that not all mentioned works are products of geography but of spatial economy as well. Lee and Pace (2005) deal with a spatial distribution of retail sales between the shopping centres in relation to their mutual location in Houston, Baray and Cliquet (2007) besides the application of the gravity models discuss the possibility of mathematical morphological analysis for delineation of shopping centres’ tributary areas. In Slovakia the Reilly´s model was used by Očovský, Š. (1973) to deline- ate the tributary areas of shopping centres in Slovakia, in the Czech literature the topic was discussed by e.g. Maryáš, J. (1983), Hlavička, V. (1992) or Řehák, S. (2004). Although originally the Reilly´s model was intended to identify the retail att raction, currently it can also be used – despite some objections made by Berry, B.J.L. (1967), regarding however rather its original use – for the assessment of geographical organisation of a territory, suitability of its administrative division, of its historical or future development, and for gen- eral regionalisation tasks (Hubáčková, V. and Krejčí, T. 2007; Halás, M. and Klapka, P. 2010; Klapka, P. and Niedźwiedźová, K. 2010). The law of retail gravitation formulated by Reilly, W.J. (1929) states that the portion of realised shopping visits in two competing centres (sett le- ments) depends on the size of these centres (increasing size of a centre brings increasing portion of visits in it) and on the distance between these centres (increasing distance from a centre brings decreasing portion of visits in it). Mathematical expression of this relation is: , where BA, BB are number of visits in centres A, B from the place (sett lement) examined; MA, MB are masses (weights) of the centres A, B given by their population; dA, dB are distances from centres A, B. Parameter N was set by Reilly to 1 and parameter n to 2, which is of course the full analogy to the gravitation law. Border of the infl uence spheres of two competing centres A, B is made by a set of points possessing the equal number of visits to the centres, i.e. BA/BB = 1. If we keep N = 1 we get by adjustments the equation valid in case that MA ≥ MB where dAB is a distance of centres A, B; dB is a dis- tance of smaller of the two centres from the infl uence breaking point of these centres along their shortest link. In practice it means that the border between 241 infl uences of two centres is a set of points whose distance from the centre A is a k-multiple of the distance from the centre B. In case that MA ≠ MB we get a circle as a dividing set of points. Its construction is described in detail by Řehák, S., Halás, M. and Klapka, P. (2009). Weights of centres MA, MB can be defi ned in various ways according to the character of phenomena to be modelled. In original works the weights were given either by the populations of centres or by the fi nancial expres- sion of the retail sales. In case of a delineation of the economic infl uence the econometric indices can be used, such as the size of shopping area, number of occupied job positions or the number of entrepreneurial subjects (used e.g. by Hubáčková and Krejčí, 2007). The results are basically analogous to those reached by the use of the population as the weight of a centre (see conclusions of several reports cited by Maryáš, 1983). Löffler, G. (1998) in his complex outline states that generally the pop- ulation, number of fi rms in the service sector, number of job opportunities and the so called “commuting balance” can be used as weights. However, it is the population that is the simplest and most universal factor expressing the weight of a centre. It is most suitable for general tasks of approximation of complex human geographical regionalisation of a territory and for proposals of administrative division of a territory as well (other indices can be distorted by a functional specialisation of some centres). In the present applications a potential (theoretical) att raction to a centre at a general level will be applied, therefore the most complex and universal factor, which is the population of the centres as of January 1, 2009, will be used. According to the way of ar- eal delineation three basic versions of the Reilly´s model have been defi ned: geometric, topographic and oscillatory (Řehák, S., Halás, M. and Klapka, P. 2009), each of them having its reason in a particular orientation and phase of the research. The simplest geometric version of the Reilly´s model works in isomor- phic space, i.e. with distances in air kilometres, without the communication network being taken into account. The border of the infl uence spheres of two centres is always a circle, in the case of centres of equal weight this border is straight line. The advantage of the geometric version is seen particularly in cases of preliminary assessment of possible infl uences of centres when sur- veying larger territories, with a well designed communication network and without distinct natural barriers. This version can also be very well used for the identifi cation of the infl uence spheres crossing the national border and in historically conceived tasks for generalised retrospective analyses of the sett lement system. The topographic version of the model does not work with an isotropic plain but with some more or less tangible geographical characteristics of a territory, for instance with communication network, which also refl ects the 242 physical geographical conditions of a surveyed territory to a certain extent. This version already employs the spatial zones (e.g. municipalities) and with distances separating the centres of these spatial zones along the elements of the communication network, such as roads or railways. The border between the infl uence spheres is confi ned to the borders of the spatial zones, when each zone can be unequivocally assigned. The topographic version can be used for classic regionalisation tasks and for testing the suitability of the spatial divi- sion of a territory. The oscillatory version of the Reilly´s model is not inherently aimed at the regionalisation but at the identifi cation of transitory belts. The construction of these areas has its reason particularly when employing the spatial zones as in the preceding version. It identifi es those zones that are located near the border of the infl uence spheres of the centres. Delineation of this transitory belt can be achieved by sett ing the span of the belt, for instance in a form (0.9 · k; 1/0.9 · k ). The oscillatory version of the model can be applied at the beginning of detailed study of att raction, but also at the fi nal stages of research as a correction of the resulting regionalisation. A special att ention should be paid to the n parameter choice in the basic equation of the Reilly´s model. Maryáš (1983) reminds a long time discussion about the values of this parameter (e.g. also Schwartz, G. 1962) and claims that for centres of lower orders the needed tributary area had been reached by application of the distance parameter with the value at least three. The construction of the model shows that it can be calibrated by the n parameter estimation, while the classic version of the Reilly´s model had worked with the value n = 2. Reilly had chosen two having said that the mode of his sample had belonged to the interval from 1,51 to 2,50. If we strive to work exactly and take into account the att raction/gravity as the analogy to the laws of physics we have to use the value two. However, the parameter choice has to refl ect the character of a phenomenon to be approximated or the nature of the application task (if we focus on the practical use). The selection of centres is an important part of the Reilly´s model application. It can follow several criteria. The simplest is selection according to the size criterion where the population number of the centre or potential population of its infl uence sphere can be taken into account. The selection of the centres can use existing versions of regionalization based on the real interactions when it is possible (at a given hierarchical level) to adopt these centres. In this article the geometric version of the model will serve only as a supplementary tool of the assessment of the regional division of the Slovak territory. Observed infl uence spheres of individual centres will be taken into account when selecting later the centres for individual variants of the possible territorial divisions. The application of the topographic version will bear the 243 major signifi cance (see below), the oscillatory version will serve a practical purpose as a correcting and refi ning tool for the results of the topographic version. Since the Reilly´s model will be used for regionalisation with several diff erent objectives, details of the method will be provided in the ensuing text in the places where it is applied. Existing regionalisations of Slovakia, basic spatial units The literature presents several regionalisations of Slovakia. One of the most important of them is the delineation of the so called daily urban systems, generally known as the functional urban regions (FUR further on). Bezák, A. (2000) has delineated two versions of functional urban regions (FUR 91-A and FUR 91-B) on the basis of 1991 labour commuting data having employed a relatively sophisticated method. The second system (marked as FUR 91-A), except for the inner coherence and outer separation, has met the criterion of minimal population size of the region (35,000 inhabitants) as well. In this article (precisely when applying the topographic version of the Reilly´s model) Bezák´s FURs 91-B have been used as basic spatial units. They had been delineated only on the basis of a scientifi c data processing without inclusion of a minimum size of a region, and at the same time considered to present the primary regionalisation of the Slovak territory convenient for the analyses of the spatial data or for further geographical processing. Slovak dis- tricts cannot be used as basic regions, since their delineation was purposefully and strongly political and the criterion of spatial justice was not obeyed. In the north of Slovakia the status of a district capital was assigned even to the small centres (such as Bytča, Kysucké Nové Mesto, Turčanské Teplice) and these districts do not match their counterparts in the southern Slovakia in terms of their size (for instance Nové Zámky, Levice, Rimavská Sobota, Trebišov). In geographical scientifi c literature no regionalisations of Slovakia have been found at a hierarchical level higher than the system of FUR, that would aim directly at the proposals of the territorial administrative division. It can be considered as a relatively serious defi ciency, since particularly this level (NUTS 3) acts as regional self-governments that are delineated in a very poor and biased way in Slovakia (their delineation in 1995–1996 was strongly infl uenced by the political intentions of the government coalition of the time). Hierarchically higher regional system delineates NUTS 2 regions, i.e. basic ter- ritorial units of the EU countries serving (among others) for the distribution of main fi nancial fl ows into the regional policy of the EU. We register only several scientifi c proposals of the division to the regions, i.e. NUTS 3 level (e.g. Bačík, V. and Sloboda, D. 2005; Sloboda, D. 2006), which are all already bett er than current territorial administrative division, but are not supported 244 by the exact quantitative processing of the data on real or potential population fl ows. Moreover, these proposals employ the districts delineated in 1996 as basic spatial units, which had not been constructed very transparently either (see the preceding paragraph). Application of the geometric version of the Reilly´s model on the Slovak territory The geometric version of the model serves in the article only illustrative purposes and presents a preliminary view of the possible application of the Reilly´s model. As the parameter we have set the most frequently used value n = 2 and the selection of centres followed a simple population criterion. The original intention for the centre selection has been the level of 25,000 inhabit- ants, which has been later lowered to 24,000. The reason for this has been an inclusion of Rimavská Sobota (its population was 24,446 as of January 1, 2009), which could be in further applications, regarding its location in less exposed areas in terms of the population, seen as a potential regional capital. In this chapter we turn our att ention to a brief commentary of the geometric version results of the model (Figure 1) in relation to major geomor- phologic barriers or to the comparison of the results of the topographic version of the Reilly´s model. Despite the fact that the geometric version of the Reilly´s model cannot be completely ideally applied on the dissected territory in terms of landforms, it is generally able to approximate the sett lement patt ern and regional struc- ture of Slovakia. Even when using the direct geometric distance it is possible Fig. 1. Potential infl uence of centres of sett lement system in Slovakia , 2009 245 to identify the main direction of the so-called central Slovak communica- tion barrier (more in Lukniš, M. 1985) mainly along the borders of infl uence spheres of Banská Bystrica or Prievidza (in the north-eastern part), Martin, Ružomberok and Liptovský Mikuláš. In comparison to the real situation the Prievidza infl uence sphere is actually reaching southwards (the infl uence of Žiar nad Hronom and Žarnovica, both having less than 24,000 inhabitants, has not been taken into account). The same situation holds true for the Poprad and Spišská Nová Ves infl uence spheres, when the infl uences of Rožňava and Revúca have not been considered. The Bratislava infl uence sphere is relatively compact according to the geometric version, forming circa 50 km wide crescent reaching approximately 80 km to the north and 80 km to the southeast. On the contrary the Košice infl uence sphere forms several lobes or projections towards Rožňava, Stará Lubovňa, Medzilaborce, and Trebišov (all towns below 24,000 inhabitants). The spheres of infl uence of Bratislava and Košice have the so called exclaves, i.e. they do not obey the contiguity principle. The infl uence of Bratislava, inter- sected by the sphere of infl uence of Komárno, reaches the region of Štúrovo; the infl uence of Košice, intersected by the spheres of infl uence of Humenné and Michalovce, reaches the north-easternmost part of the country. The geometric version overrates the infl uence sphere of Žilina and pushes it signifi cantly to the north-east, where it comprises a large part of the Orava region. This territory would actually belong to Martin or Ružomberok (unless the infl uence of Dolný Kubín is taken into account), particularly as a result of very poor transport connection of the Kysuce region (and thus Žilina) to the Orava region. In other instances the geometric version of the Reilly´s model relatively aptly approximates the basic features of the Slovak regional system (e.g. central Považie region etc.) and in general off ers a relatively relevant information. Application of the topographic version of the Reilly´s model on the Slovak territory: potential natural interactions Att empting to approximate the potential natural interactions by the topo- graphic version of the Reilly´s model, it is necessary to use the variant directly following the Newton´s law of gravitation, i.e. the variant with the value of the parameter n = 2. This fact has been tested on the territory of the Czech Republic (Halás, M. and Klapka, P. 2010), when the areas of the municipalities with extended authority (further on referred to as MEA areas) have been used as basic spatial units. The comparison of potential natural interactions reached by the topographic version of the Reilly´s model (using the parameter n = 2) to the real commuting interregional fl ows (i.e regionalisation by Hampl, M. 2004) 246 has provided a high degree of correlation. The majority of the MEA areas has been a part of the same mezzo-region; 94.8% of the population belonged to the same mezzo-region according to both methods. Moreover, out of remain- ing 5.2% of the population that has been assigned to diff erent mezzo-regions, almost one half (exactly 2.5% of the population) is concerned with the trade- off s between the Hradec Králové and Pardubice mezzo-regions. It is caused by the advantageous location of the city of Pardubice on the major railway line, while this study takes into account the road distances only. As it has been already mentioned, the basic spatial units (zones) that have been tested in terms of their affi nity towards centres are FURs 91-B, when the road distances among their centres have entered the model as a dis- tance variable between the zones. The model thus has used the road distances between our centres and FUR 91-B centres. The distances have been set by the Škoda Auto route planner (www.skoda-auto.com/cz) and they have had a character of the fastest connection in terms of the time needed. Each FUR 91-B has been assigned in the way described in the method above to the so-called mezzo-region (approximately the level of the NUTS 3 region). During the fi rst phase 8 regional capitals have been defi ned as the mezzo-regional centres (Figure 2), later four others have been supplemented (Figure 3). The main results are generally comparable to the geometric version of the model. The dominance of larger centres documents a regular co-existence of Bratislava with such centres as Trenčín, Trnava and Nitra in the west of Slovakia and similar situation occurs in the case of Košice and its co-existence with other centres in the east. Bratislava forms the infl uence sphere reaching along the bor- ders with the Czech Republic, Austria and Hungary (from the town of Myjava to the town of Šahy). This sphere is signifi cantly conditioned by the eccentric position of Bratislava within the state territory of Slovakia. The infl uence sphere of Košice comprises almost the whole eastern Slovakia with the exception of small tributary area of Prešov that is made up by fi ve FURs only. The patt ern in the central Slovakia (cities of Žilina, Banská Bystrica and partly also Trenčín and Nitra) approximates the regional administrative division quite well, since this territory lacks a signifi cantly dominant centre (Figure 2). The cities of Trenčín and particularly of Trnava form reduced infl u- ence spheres in comparison to other regional capitals. Therefore other centres capable of forming the infl uence spheres comparable to those of Trnava and Trenčín have entered the second phase of the topographic version of the Reilly´s model (aimed at potential natural interactions). The cities of Martin, Poprad and Michalovce have not been doubted as regional centres in this respect. In the south of Slovakia a decision has had to be made whether insert the city of Lučenec or Rimavská Sobota in the set of the regional centres, since their spheres of infl uence have considerably overlapped. The fi nal decision has favoured Lučenec, since it has been able to att ract more population in this version. 247 Fig. 3. Regional division of Slovakia based on potential natural interactions (12 centres) Fig. 2. Regional division of Slovakia based on potential natural interactions (8 centres) When compared to the preceding situation the regional patt ern in the west of Slovakia has not been adjusted and the central Slovak communication barrier has remained in its extent and position as well. The larger part of the infl uence sphere of Žilina has been att racted by the city of Martin (includ- ing the whole Orava region and the west of the Liptov region) and the FUR Liptovský Mikuláš has joined the regional centre of Poprad. The south of the infl uence sphere of Banská Bystrica has been taken over by the city of Lučenec, which has also att racted the FUR of Rimavská Sobota originally being a part 248 of the Košice region. In the eastern Slovakia the sphere of infl uence of Prešov has retained its extent. The part of the Spiš region, originally belonging to the regional centre of Košice, has been transferred to the regional centre of Poprad. Finally the Zemplín region has joined the sphere of infl uence of Michalovce. Should the contiguity constraint of the delineated region be obeyed, the results in the north east of the state territory had to be corrected (the FURs Medzilaborce and Snina). These two exclaves are primarily att racted to the in- fl uence sphere of Košice according to the potential natural interactions (this fact is witnessed in the geometric version of the model as well – fi g. 1). Their fi nal regional affi liation has been determined by their secondary affi nity, then. The population characteristics of the delineated regions are presented in Table 1. Application of the topographic version of the Reilly´s model on the Slovak territory: proposals of territorial division The att empt to propose the alternatives to the territorial administrative divi- sion (further on only territorial) of Slovakia by the application of the topo- graphic version of the Reilly´s model has employed a higher value of the parameter n. The application of the higher parameter is necessary because when constructing a possible territorial administrative division, the rule of spatial justice has to be obeyed. It ensures that the extreme most locations of municipalities in individual regions have the comparable distance from their regional capitals. The value of the parameter has been set on the basis of extensive statistical testing. Table 1. Population characteristics of regions of Slovakia (potential natural interactions) Regional capital Population in 1,000 Share of capital in population, %Total Capital Hinterland Bratislava Nitra Košice Žilina Poprad Banská Bystrica Prešov Trenčín Martin Michalovce Trnava Lučenec Total 1,026.4 738.5 667.9 528.7 408.6 361.8 344.2 333.3 298.0 256.3 241.7 208.2 5,413.5 428.8 84.1 233.7 85.3 54.6 80.1 91.3 56.8 58.4 39.5 67.7 27.5 1,307.9 597.6 654.4 434.2 443.3 354.0 281.7 253.0 276.5 239.6 216.7 173.9 180.6 4,105.6 41.8 11.4 35.0 16.1 13.4 22.1 26.5 17.0 19.6 15.4 28.0 13.2 24.2 249 In the preceding study on the Czech Republic (Halás, M. and Klapka, P. 2010), the F-test (with the level of signifi cance 0,05 used for the comparison of the current territorial division and regionalisation according to the topographic version of the Reilly´s model and increasing the values of n gradually by 0,1) has produced the optimal value of the parameter n for the purpose of territo- rial administrative division (n = 5). Statistical sets in both compared divisions have been acquired as the maximum distances of the MEA area capital from the selected regional centres. The level of signifi cance 0.05 has been reached also at the parameter with the value 5.0 (test criterion F has counted for 3.04 and for the last time it has been lower than the critical value of F-distribution for m x m degrees of freedom F = 3.28; the number of regional centres has been m + 1. While F-test demands for a comparison with other (preferably existing) patt ern, its application on the Slovak territory would not be correct. The current territorial administrative division of Slovakia, including the selection of eight regional capitals, can be considered as insuffi cient from the scientifi c point of view and thus cannot be used for statistical testing. Therefore, the value n = 5, statistically tested in the Czech Republic, has been applied in case of Slovakia as well. This approach appears to provide an optimal compromise between the elimination of increased infl uence of the largest centres (Bratislava and Košice) and the partial fulfi lment of the principles of spatial justice. The argument for the use of the same parameter lies in the fact that both countries (Czech Republic and Slovakia) have a similar disproportion in hierarchy of their largest centres in terms of their order (seen for instance also in the Zipf curve etc.). These centres serve as potential capitals of NUTS 2 and NUTS 3 regions, then. The selection of regional capitals (centres) is an important question. During the fi rst phase the current regional capital, except for Trnava and Trenčín, have been employed. The two exceptions mentioned are not capable, due to their relative location to Žilina and particularly Bratislava, of forming the infl uence spheres comparable to the remaining six regional capitals. This variant has also substituted the city of Zvolen for Banská Bystrica. The reason for this step has been twofold: 1) Zvolen is capable of generation of larger tributary area, in comparison to the city of Banská Bystrica it att racts the FUR Rimavská Sobota in addition; 2) the variant with six resulting regions asks that Zvolen/Banská Bystrica should comprise the FURs in the Jihoslovenská kotlina basin and the city of Zvolen possesses a more advantageous central location within the resulting region, i.e. the sum of distances (simple and weighted by the population) from all centres of FURs to Zvolen is lower than to Banská Bystrica and thus the city of Zvolen appears to be a bett er option for the regional capital in terms of the principle of spatial effi ciency. The result (Figure 4) can be considered as optimal: six selected centres (regional capitals) manage to form comparable natural tributary areas and generally this variant of the possible territorial administrative divisions seems 250 Fig. 4. Proposal of possible administrative division of Slovakia (regional level, 6 centres) to be balanced. The population characteristics of the delineated regions are presented in Table 2. Even though the territorial division of Slovakia with six regions can be seen as appropriate, it unfortunately does not match the EU criteria of the recommended population size of the NUTS 3 regions. It should vary accord- ing to the rules between 150,000–800,000 inhabitants. The optimal number of regions in Slovakia following the EU normatives is somewhere between 12 and 16. Therefore, the Reilly´s model for such a patt ern of regional division has been applied once more. The possibility of merging the NUTS 3 regions into the NUTS 2 regions has been taken into account in the examination as well. The NUTS 2 regions should have the population between 800,000 and 3,000,000 according to the EU recommendations. Table 2. Population characteristics of proposed regions of Slovakia (potential modifi ed interactions, 6 centres) Regional capital Population in 1,000 Share of capital in population, %Total Capital Hinterland Žilina Nitra Bratislava Košice Prešov Zvolen Total 1,129.0 1,118.9 994.1 824.8 763.4 583.3 5,413.5 85.3 84.1 428.8 233.7 91.3 42.5 965.7 1,043.7 1,034.8 565.3 591.1 672.1 540.8 4,447.8 7.6 7.5 43.1 28.3 12.0 7.3 17.8 251 The fi rst important step is a selection of centres, again. Eight regional capitals entered the model, the city of Banská Bystrica has not been substituted for by the city of Zvolen this time, since neither of the two arguments used in the variant with six regions has remained valid. Without any doubt the cities of Martin, Poprad, Michalovce and one of the pair Lučenec and Rimavská Sobota has had to be included in the set of the regional capitals. Using n = 5 the model has produced such a regional patt ern that has had to be refi ned in several cases. First, the tributary area of Nitra has appeared to be too large and it exceeds together with the city of Nitra itself the level of 800,000 inhabitants. It seems to be necessary to form a region with a centre in Komárno or Nové Zámky, then. This adjustment makes the region of Nitra to fulfi l the demanded size level and proves that the cities of Komárno/Nové Zámky are capable of forming the area comparable to other potential regions (i.e. exceeding the level of 250,000 inhabitants) as the only other centres in Slovakia. Finally the city of Nové Zámky seems to act as a more advantageous centre of the region due to its central location in the formed region and due to the fact that it meets bett er the demands of the spatial effi ciency (the sum of distances weighted by the population from all municipalities is lower in case of Nové Zámky than in case of Komárno). No other city or town is able to form a tributary area comparable to the above-mentioned examples, therefore 13 regional capitals appear to be an optimal number taking into account the recommended size criteria. All other transfers of the FURs from one region into another have been based on the application of the oscillatory version of the Reilly´s model only, which means in cases when the tested FUR has been located at the boundary of the infl uence spheres of two regional capitals within the span (0.9 · k; 1/0.9 · k) (see methodology). Thus the region of Trnava has been joined by the FUR Skalica (otherwise the region of Trnava would have been too small and the region of Bratislava too large) and by the FUR Myjava. The FUR Liptovský Mikuláš has been transferred from the region of Poprad to the region of Martin, while preserving the historical border between the Liptov and Spiš regions, and, moreover, this transfer has enabled a simpler division to the hierarchical higher NUTS 2 regions. The choice of the regional centre between the cities of Lučenec and Rimavská Sobota has remained question that has not been easy to answer. The city of Lučenec manages to form a litt le larger area though (the diff erence is in the FUR Veľký Krtíš - the variant of the centre in Lučenec, and in the FUR Revúca – the variant of the centre in Rimavská Sobota), but the FUR Veľký Krtíš oscillates between Banská Bystrica and Rimavská Sobota, and the FUR Revúca has a poor accessibility to the proposed regional centre in Lučenec and never can be assigned to its region. There is just the only way to solve these problems satisfactory. If the city of Rimavská Sobota is made a regional capital than both questioned FURs Rimavská Sobota and Veľký Krtíš remain in the region. Other variants only raise other and more even problematic questions to cope with. 252 The resulting alternative proposal of the territorial administrative divi- sion with 13 regions is presented in Figure 5. It also shows an alternative divi- sion to the NUTS 2 regions: the current names West, Central and East can be preserved. All three proposed regions manage to follow the demanded popula- tion span, individual treatment of the city Bratislava is considered as irrelevant. Bratislava is not a metropolis with a million of inhabitants and in this phase it is not necessary to approach to the division of the NUTS 2 regions in an expedient manner in order to maximise the fi nancial infl ow from the EU funds. The popu- lation characteristics of the delineated regions are presented in the Table 3. Tab. 3. Population characteristics of proposed regions Slovakia (potential modifi ed interactions, 13 centres) Regional capital Population in 1,000 Share of capital in population, %Total Capital Hinterland Bratislava Nitra Martin Michalovce Košice Žilina Trnava Banská Bystrica Prešov Trenčín Poprad Nové Zámky Rimavská Sobota Total 735.6 630.8 515.3 436.5 430.0 386.8 372.6 361.8 344.2 333.3 333.2 267.5 265.8 5,413.5 428.8 84.1 58.4 39.5 233.7 85.3 67.7 80.1 91.3 56.8 54.6 40.5 24.2 1,340.5 306.8 546.8 456.8 397.0 196.3 301.5 304.9 281.7 253.0 276.5 278.6 227.0 241.5 4,073.0 58.3 13.3 11.3 9.1 54.3 22.1 18.2 22.1 26.5 17.0 16.4 15.1 9.1 24.8 Fig. 5. Proposal of possible administrative division of Slovakia (regional level, 13 centres) 253 Conclusion From the standpoint of natural conditions and prerequisites and from the standpoint of socio-economic regional disparities Slovakia is a considerably di- versifi ed country, therefore its regionalisation at any hierarchical level is a very demanding challenge. The results reached can be infl uenced to a considerable extent by the choice of regionalisation criteria, therefore these tasks cannot be taken as completed and could become a subject of further discussions. The regionalisation employing the potential spatial interactions and the comparison of the results to the current territorial division of Slovakia (or to its proposals) has shown according to our opinion the viability of the Reilly´s model and its versions also in the solution of current regionalisation tasks. The results can be used also for the assessment of the characteristics of the sett lement system in Slovakia and regional infl uence of its centres. The application of the Reilly´s model can vary according to the nature of the solved task. The resulting regionalisations have confi rmed several well-known facts. The specifi c position of the cities of Bratislava and Košice in the regional and sett lement systems of Slovakia remains unchallenged. The capital city of Bratislava, despite its eccentric location, can be seen as a natural centre of the country, while the city of Košice is able to supply some functions of the country´s capital for the territory of the eastern Slovakia. Lukniš´ macro-re- gionalisation into two central and two corridor regions has been confi rmed as well, while the central Slovak communication barrier can be identifi ed in all results of the regionalisation tasks. Other centres, excluding Bratislava and Košice, are classifi ed at the minimum as one or two hierarchical levels lower. None of them signifi cantly dominates since their development is limited either by natural barriers or the location within the vicinity of two largest centres. The absence of one dominat- ing centre and signifi cant geomorphologic barriers causes that the regionalisa- tion of the central Slovakia is less problematic. Therefore the smallest number of the diff erences between individual proposals of this article and between current and past territorial divisions is witnessed in this territory. The resulting regionalisations are signifi cantly dependent on the se- lection of centres, while this issue is considered as a key question in the re- gionalisation tasks. The variant choices of centres can considerably aff ect the potential territorial divisions (e.g. Poprad, Michalovce etc.). In this place 12–13 years old discussion of the potential “Komárno region” can be mentioned when the main argument raised has been homogeneity instead of nodality. The alternative proposal of 13 regions put forward in this article comprises this independent region in the south of Slovakia (with the centre in the city of Nové Zámky). However, a note should be made here that the method presented in 254 this article is a mere theoretical expression of spatial interactions and is not able to reveal a possible infl uence of national diff erentiation of the territory on the existing nodal relations. The results that have been reached in the article are in our opinion interesting in any case. They confi rm the insuffi cient and problematic territo- rial delineation of current regional self-governments including the selection of their centres. A comparison of the results reached by the use of potential spatial interactions with the regionalisations constructed on the basis of real commuting or migration relations would be a very interesting research task for the future, then. Acknowledgement: This study was supported by the Grant Agency of AS CR under the contract KJB 300860901 (Quantitative methods and synthesizing graphic methods in approximation, projection and modelling of geographical phenomena) and the contract IAA 301670901 (Spatio-temporal organization of daily urban systems: analysis and assessment of selected regions). REFERENCES Bačík, V. and Sloboda, D. 2005. Župný variant 2005. Návrh na zmenu územného členenia SR. Bratislava, Konzervatívny inštitút M. R. Štefánika. Baray, J. and Cliquet, G. 2007. Delineating store trade areas through morphological analysis. European Journal of Operational Research 182. 886–898. Berry, B.J.L. 1967. Geography of market centres and retail distribution. 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The hypothesis of the `minimum equation' as a unifying social principle: with att empted synthesis. American Sociological Review 12. (6): 627–650. 256 Hungary in Maps Edited by Károly Kocsis and Ferenc Schweitzer Geographical Research Institute Hungarian Academy of Sciences Budapest, 2009. 212 p. ‘Hungary in Maps' is the latest volume in a series of atlases published by the Geographical Research Institute of the Hungarian Academy of Sciences. A unique publication, it combines the best features of the books and atlases that have been published in Hungary during the last decades. This work provides a clear, masterly and comprehensive overview of present-day Hungary by a distinguished team of contributors, presenting the results of research in the fi elds of geography, demography, economics, history, geophysics, geology, hydrology, meteorology, pedology and other earth sciences. The 172 lavish, full-colour maps and diagrams, along with 52 tables are complemented by clear, authoritative explanatory notes, revealing a fresh per- spective on the anatomy of modern day Hungary. Although the emphasis is largely placed on contemporary Hungary, important sections are devoted to the historical development of the natural and human environment as well. In its concentration and focus, this atlas was intended to act as Hungary's 'busi- ness card', as the country's résumé, to serve as an information resource for the sophisticated general reader and to inform the international scientifi c community about the foremost chal- lenges facing Hungary today, both in a European context and on a global scale. Examples of such intriguing topics are: stability and change in the ethnic and state territory, natural hazards, earthquakes, urgent fl ood control and water management tasks, land degradation, the state of nature conservation, international environmental confl icts, the general popula- tion decline, ageing, the increase in unemployment, the Roma population at home and the situation of Hungarian minorities abroad, new trends in urban development, controversial economic and social consequences as a result of the transition to a market economy, pri- vatisation, the massive infl ux of foreign direct invest- ment, perspectives on the exploitation of mineral re- sources, problems in the energy supply and electricity generation, increasing spatial concentration focused on Budapest in the fi eld of services (e.g. in banking, retail, transport and telecommunications networks), and fi nally the shaping of an internationally competi- tive tourism industry, thus making Hungary more att ractive to visit. This project serves as a preliminary study for the new, 3rd edition of the National Atlas of Hungary, that is to be co-ordinated by the Geographical Research Institute of the Hungarian Academy of Sciences. ------------------------------------------- Price: EUR 20.00 Order: Geographical Institute RCAES HAS Library H-1112 Budapest, Budaörsi út 45. 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