This Research Bulletin has been published in International Journal of Pattern Recognition and Artificial Intelligence, 21 (3), (2007), 439-462.
Although the lineage of world city studies can be traced back to well before 1980 (e.g. Hall, 1966), it is only in the last two decades that a concerted research effort has emerged. The most seminal contributions to the recent upsurge in world city research are Friedmann and Wolff's (1982) and Friedmann's (1986) identification of 'command centers' that control and articulate the 'new international division of labor' being created by multinational corporations. This reflected the later recognition by Amin and Thrift (1992) of a shift from an international to a global economy, characterized by increasingly integrated global networks of production and services. World cities are then the basing points in these multifarious networks, and their specification therefore relates to the identification of a 'global network of cities' (King, 1990, p. 12, emphasis added).
Apart from the on-going discussions on more formal definitions and conceptualisations (see for instance, Smith, 2003), one the most pressing problems in world city research is the lack of assessments of less connected nodes in this global urban network (Beaverstock et al., 1999; Taylor et al., 2002b). While it is obvious that cities like London and New York are world cities, there has hardly been a consensus as to the status of less significant cities in this context. As a consequence, somewhat vague discourses on patterns of global competence in the outer reaches of the world city network have dominated the debates. Some authors have for instance reified the lower ranks of the hierarchy as 'sub-global cities' (e.g. Daly and Stimson, 1992; O'Connor and Stimson, 1995), while this is particularly problematic given the pervasive nature of globalization. As Knox (1996) has pointed out, these 'medium cities' have just as much need to respond to globalization trends as their larger neighbours. It is evident that every city operates as an integral part of the global system as a producer and marketplace for global goods and services, and as a hub in the flow of people, remittances, finance and ideas. A lower amount of global capital served does not imply a lack of global connections, and its thus indispensable to include a large number of cities in an analysis of the world city network.
Previous assessments of lower-ranked world cities have however been limited to ad hoc taxonomies that all in all remain limited to the higher ranks of the hierarchy (e.g. Friedmann, 1986: Knox, 1995; Beaverstock et al., 1999). Apart from the lack of an undisputed definition of world cities and the associated dearth of data, the main reason for these somewhat eclectic approaches is the fact that classical data analysis techniques cannot assess the sparse information on the less connected nodes in the world city network. Taylor et al. (2002b, p. 2378), for instance, restricted the exploratory analysis of 316 cities to the top 123 cities "because the data matrix becomes increasingly sparse [.] and thus becomes increasingly unsuitable for analysis." In the present paper, it is shown that a fuzzy c-means clustering algorithm copes better with the sparse objects in the data matrix, which enables us to analyse less connected nodes in the world city network, most of which never figured in discussions in this context. The resulting classification provides a comprehensive and detailed taxonomy of the world city network, where both general tendencies and more specific profiles can be explored.
This paper is divided in three sections. First, we examine the theoretical and empirical rationale behind explorations of the world city network. It is demonstrated that assessments of less connected nodes have been largely absent from previous explorations, which is due to the sparseness of information on their world city-ness. The second section discusses the outset of a classical clustering algorithm and its fuzzy counterpart. This discussion of classification algorithms commences with the outset of the more classical crisp algorithm for two reasons. First, applying the crisp version of the algorithm enables us to show why more multivariate techniques fail to grasp the paucity of information on less connected nodes in the world city network, while the outset of the methodology will also serve as a guide to the fuzzy clustering algorithm. The third section, then, ascertains a detailed taxonomy of the world city network. Both general tendencies and more specific outcomes of the fuzzy set analysis will be discussed.
ASSESSMENT OF A NETWORK OF WORLD CITIES
From Central Place System to World City Network
Urban geographers have long sought to unravel and describe the systematic nature of the spatial arrangement of urban centers, as Christaller's central place-theory (1933) and Lösch's (1954) extensions of this central place-theory show. Most of the studies oriented towards the description of the spatial arrangement of such an urban system inherit their physical boundaries from their definition as an integrated economy. Since the beginning of the twentieth century, the world-economy is truly global (Wallerstein, 1983), and hence all cities can be thought of as participating in a single central place system (Chase-Dunn, 1985). However, although the predicaments of these classical central place systems may still do a reasonably good job in describing the urbanization patterns on a regional scale, they are not suited to explain patterns of global urbanization. At the most basic level, there are at least three (heavily intertwined) alterations that should be taken into account with respect to the assessment of a global urban system:
The first and most obvious difference between more classical conceptions of urban systems and urbanization processes in a global economy lies in the fact that the original hierarchy needs to be supplemented by some additional levels, i.e. the world cities this paper is concerned with (Hall, 2001)1. An exact definition of a world city is difficult to pin down, since it is itself subject to a great deal of controversy (Knox, 1995; Hill and Kim, 2000; Smith, 2003). Friedmann (1995) has however pointed out that a formal definition of a 'world city' is not necessary, since it is a paradigm rather than a well-defined theoretical object. As such, a general approach of world cities as the 'command and control centers' and 'basing points' of the world-economy is a good starting point for its operationalization, for this notion captures the importance of assessing interdependencies and flows between world cities.
Second, the recent evolution of technology and economy implies that there are new services that should fit in the model of central place functions described by Christaller (1933). One of the most important features of these new central place functions lies in the steady decline or slowdown in manufacturing, which contrasts with both a sharp rise in computing and telecommunication and a sharp growth in producer services such as accounting, legal, advertising and consulting services. The fundamental reason for this explosion of producer services can be traced back to the increased service intensity in the organization of all industries. Whether in manufacturing or in warehousing, all firms are increasingly outsourcing legal, financial, advertising, consulting and accounting services, which stimulate the growth of a whole gamut of new businesses with their proper location strategies (Sassen, 2000). The combined and mutual reinforcing effects of (i) an explosive growth in computing, telecommunication and knowledge production, and (ii) a sharp rise in producer services, have led to different spatial patterns than those implied in classical conceptions of central place systems. That is, the new global economy is more than merely another layer of economic activity on top of the existing production process: it has induced a profound change in location strategies, for it restructures all economic activities based on goals and values introduced by the aggressive exploitation of new productivity potentials of advanced information technology. Producer service firms do not follow the demand side of the market; they have their own global strategies for selling their products in a world service market (Taylor, 2000). These changing geographies have been explored in more detail by other researchers (e.g. Daniels and Moulaert (1993), and Moulaert and Todtling (1995)), but it was Sassen (1991, 2000) who asserted the explicit link to world city formation. This association between fragmentation and agglomeration is the articulation of the reorganisation of existing processes into new patterns and the repositioning vis-à-vis the new highly productive sectors. Moreover, these evolutions in technology and economy induce a growth dynamic that seems to be somewhat independent of the broader regional economy, a sharp change from the past where a city was deeply articulated with its hinterland. It seems likely that a strengthening of transnational ties between leading financial and business centers is accompanied by a weakening of the ties between each of these cities and its hinterland and national urban system.
Third, apart from hierarchical tendencies at the global level, world cities have the tendency to expose functional specialisations. This is very different from Christallerian patterns, where cities occupying the same rank in the hierarchy by definition exhibit functional equivalence. Camagni (1998) has reminded us that the relations between cities provide more insight when assessments of these relations are based on functional specializations rather than mere hierarchies, while Knox (2002) has argued that different world cities fulfil different functions within the world-system, and will hence differ in both the nature and the degree of their world-city-ness. For instance, in the United States, New York leads in banking, accounting and advertising, while Washington D.C. leads in legal services, R&D and membership organizations (Sassen, 2000). Moreover, Taylor and Hoyler (2000) have shown that cities that are less thoroughly connected in the world city network will assume more regional roles, while the upper rungs of the network will have global specificities that prevent them from being very similar to nearby cities. For instance, British cities appeared very similar in the way they respond to globalization with the exception of London, which had a totally different profile as one of the most connected cities in the world city network.
This brief overview of the most salient features of an emerging network of world cities has a profound impact on its assessment, i.e. empirical explorations of its spatial outline should be focus on the following elements:
Specification and Measurement of the World City Network
Drawing on the guidelines set out in the previous section, the Globalization and World Cities Group and Network (GaWC) has developed a methodology for studying world city network formation by analysing the location strategies of advanced producer firms (Beaverstock et al., 1999; Taylor, 2001; Taylor et al. 2002a). The world city network is hereby conceptualised as an inter-locking network with three levels: a network level (the world city network), a nodal level (world cities), and a sub-nodal level (the firms providing the advanced producer services). It is at the latter level that world city network formation takes place, and GaWC thus follows Sassen (1991, 1995, 2000) in her treatise of world cities as global service centres. Based on this formal specification, Taylor (2001) has outlined the construction of connectivity matrices that systematically measure relations between each pair of world cities through the aggregation of the cross products of all of the firms for any pair of cities. Starting from a service value matrix Vij, which contains 'service values' vij ranging from 0 to 5 for all cities i and firms j, the basic relational element for each pair of cities is rab,j , which can be aggregated to produce an inter-city link rab:
These inter-city links are a good surrogate for measuring the actual flows of information and knowledge between the cities when two assumptions are made. First, offices generate more flows within a firm's network than to other firms in the sector. Although not formally empirically tested, this assumption is plausible, for flows of information and knowledge are indispensable for a seamless service. Second, the larger the office, the more flows will be generated, which will have a multiplicative effect on inter-city relations (Taylor et al., 2001).
The advantage of this precise specification of the world city network is that elementary network analysis van be used. The most basic measure is city's connectivity in relation to all other cities in the matrix. This measure departs from the observation that each city has a link with every other city, and aggregating all the links of a city produces the global network connectivity (GNC) of a city
(a ¹ i) 
The limiting case is a city that shares no firms with any other city, so that all of its elemental links are 0 and it has zero connectivity. These overall GNC measures will be used below to order our results hierarchically, where they will be expressed as proportions of the largest computed connectivity in the data, thus creating a scale from 0 to 1.
This theoretical rationale has been used to analyse the location strategies of a total of 46 advanced producer services across 55 world cities (Beaverstock et al., 1999). A new data collection along these lines covered 100 firms and 316 cities (Taylor et al., 2002a), and was used to investigate patterns in the 123 most connected cities (Taylor et al., 2002b). The latter study was confined to the most connected cities because as the size of the data matrix increases (i.e. inclusion of more cities) it becomes relatively 'sparse' (lots of zero entries) which makes "[the matrix] increasingly unsuitable for analysis" (Taylor et al., 2002b, p. 2378)2.
Although this analysis of 123 cities is a huge step forward compared to earlier ad hoc classifications (Friedmann, 1986; Knox, 1995), it had a number of serious flaws. First, the analysis still leaves several regions relatively unrepresented, e.g. there are only two cities in inter-tropical Africa (Lagos and Nairobi) that qualify in the top 123 connected cities. Second, the analysis was based on the presence scores for firms rather than the inter-city matrices constructed in Taylor (2001). As such, the various groups derived in Taylor et al. (2002b) infer relations rather than using the relational measures deemed crucial in world city network analysis. Third, the analysis was based on a principal component analysis, and this factor analytic approach implied that a significant number of the 123 cities were not allocated to any of the components.
The present paper aims to move beyond this and other analyses by (i) explicitly drawing on the matrices that assert the relations between cities, and by (ii) including many more cities (234) than has been the case thus far. Moreover, by using a cluster analysis, every city will be classified. Combining Taylor's (2001) specification and Taylor et al.'s (2002a) measurement in a single framework, we have computed measures of inter-locking connectivity between 234 cities across the world. This matrix then gives way "to various forms of analysis available to simpler types of network. This means the wide repertoire of network techniques from elementary derivation of indices to scaling, ordinating, factoring, clustering and blocking" (Taylor, 2001, p. 192).
SEARCHING FOR CLUSTERS: CRISP AND FUZZY CLASSIFICATIONS
The main purpose of unsupervised classification (clustering) of a set of objects is to detect subgroups (clusters) based on the (dis)similarity between objects. All members belonging to the same group or cluster have certain properties in common, and the classification has the effect of reducing the dimensionality of a data table by reducing the number of rows. The aim of a cluster analysis is thus to partition a given set of data or objects into clusters, with the following properties (Everitt et al., 2001):
In this classical crisp classification approach, the state of clustering is expressed by an n x C matrix U=(uic), where uic=1 if object i belongs to the cluster c, otherwise uic =0. To ensure that the clusters are disjoint and non-empty, uic must then satisfy the following conditions (Sato et al., 1996):
This classification scheme has certain distinct advantages. For one thing, results are clear-cut, and possible cumbersome interpretations of in-between values are expelled from any analysis since there is no overlap in cluster membership. When applied to the classification of cities, regions or countries based on certain criteria, this implies that the only admissible spatial boundaries are unambiguous ones (MacMillan, 1995; e.g. Arrighi and Drangel, 1986; Van Rossem; 1996; Dezzani, 2001).
Defined more formally, the outset of the clustering problem can be stated as follows (Chi et al., 1996). Let:
be a set of samples to be clustered into C classes. The clustering process can be considered as an iterative optimization procedure. Suppose that the samples have already been partitioned into c classes, be it by random assigning the data points to clusters or through theoretical considerations on potential clusters. The task at hand, then, is to adjust the partition so that the similarity measure, based on a Euclidean distance function, is optimized. The criterion function for this optimisation procedure is equal to:
where vi is the center of the samples in cluster i, and
The classification of the data is thus based upon a measure of dissimilarity between the different data points in the matrix3, where the Euclidean distance is the most simple and common measure of dissimilarity4.
In order to improve the similarity of the samples in each cluster, we can minimize this criterion function so that all samples are more compactly distributed around their cluster centers. Setting the derivative of J(V) with respect to vi to zero, we obtain
Thus, the optimal cluster center of cluster center vi is
where ni is the number of samples in class i and Ci contains all samples in class i.
Starting with the initial clusters and their center positions, the samples can now iteratively be regrouped so that the criterion function J(V) is minimized. Once the samples have been regrouped, the cluster centers need to be recomputed to minimize J(V). This process then continues for the new cluster centers: the samples are regrouped in order to reduce J(V), which yields a new classification with associated cluster centers, and so forth. This iterative process can be repeated until J(V) cannot be further reduced or drops below a pre-defined small number. Obviously, the criterion function is minimized if each sample is associated with its closest cluster center. This means that xk will be reassigned to cluster i so that (xk-vj)² is minimum when j=i. Up to this point, each sample xk appears only once, that is, it is associated with only one cluster center.
Applying this crisp c-means algorithm to our 234 x 234 matrix for C = 2056, the classification result reveals primarily hierarchical patterns (Table 1). By ordering the clusters based on the average GNC of the cluster members, it can be observed that because of the smaller GNC gaps between lower-ranked world cities the latter tend to be classified in larger groups. In fact, while the 49 most connected cities are scattered over 16 clusters, the 185 cities least connected cities in the world city network are assigned to merely four clusters. Apart from the remarkable distinctiveness of US cities (clusters 5, 9, 12, 15, 18) and, albeit to a lesser degree, German cities (cluster 16), the outer reaches of the world city network are thus classified based on their overall connectivity. Taylor et al.'s (2002b) remark that sparse evidence on world city formation prevents a defendable classification thus holds sway, since this classification of lower-ranked world cities does not add information to the most elementary network measure of overall connectivity. Moreover, Table 1 unveils that this crisp classification also brings about problematic results in the upper rungs of the world city network. Since top ranked world cities have very distinctive patterns, they tend to be classified in a large number of very small clusters, which makes them in turn very hard to interpret soundly. The meaning of dyads such as Amsterdam-São Paulo (cluster 6), Zurich-Buenos Aires (cluster 8) and Barcelona-Houston (cluster 14) is for instance hard to infer, while it is unclear to what degree Singapore is different from Hong Kong and Tokyo (clusters 2-3). And finally, in the first section we have stressed that classifications should focus on intertwining hierarchical, regional and functional tendencies, while the crisp separation purported by a classical clustering algorithm can hardly taken these possible hybrid tendencies into account.
A Fuzzy-set Approach Towards Classification
The theory of fuzzy sets was formally introduced by Zadeh (1965), and addressed problems in which the absence of sharply defined criteria is involved. In particular, fuzzy sets aim at mathematically representing the vagueness and lack of preciseness, which are intrinsic in linguistic terms and approximate reasoning. As such, through the use of the fuzzy set theory, ill-defined and imprecise knowledge and concepts can be treated in an exact mathematical way (Tzafestas, 1994). However, this does not imply that fuzziness is mere ambiguity or stems from total or partial ignorance. Rather, fuzzy sets aim to provide the mathematical underpinnings for the specification of the inherent vagueness, and are therefore formally defined as "classes of objects with a continuum of grades of membership" (Zadeh, 1965, p. 338). Fuzzy sets are thus characterized by a membership function which assigns to each object of the set a grade of membership ranging from zero (non-membership of the set) to one (full membership of the set). Apart from the apparent fuzziness in standard linguistic terminology and everyday events, vagueness is also a problem in classification schemes framed upon the unravelling of patterns in large data sets (Bezdek, 1981; Pal and Majumder, 1986; Bezdek and Sankar, 1992; Pal and Mitra, 1999), including large geographical databases7.
One of the major advantages of the use of a fuzzy set-theoretical approach lies in the fact that it is possible to capture various aspects of vagueness in a single framework (Everitt et al., 2001). Fuzzy sets can for instance capture both the vagueness due to the sparsity of data and vagueness due to the lack of theoretically defined pre-existent categories. In general, four of the main useful features of fuzzy set methodologies are (Chi et al., 1996; Höppner et al., 1999):
A fuzzy set approach to clustering thus dispenses with unambiguous mapping of the data to clusters, and computes degrees of membership specifying to what extent objects belong to clusters. This implies that the second part of equation  is replaced by
When this general premise is introduced in the criterion function for the crisp clustering algorithm, equation  can be replaced by its fuzzy notion counterpart (Chi et al., 1996; Höppner et al., 1999; all drawing on the seminal work by Bezdek, 1981), based on the iterative minimization of
for i=1,2,.,C and k=1,2,.,n.
The altered objective function is the sum of the squared Euclidean distances between each input sample and its corresponding cluster center, with the distances weighted by the fuzzy memberships. The algorithm is iterative and makes use of the following equations:
For the calculation of a cluster center, all input samples are considered in accordance with their membership value. For each sample, its membership value in each cluster depends on its distance to the corresponding cluster center. Following Chi et al. (1996), the clustering procedure consists of the following steps:
Note that the crisp c-means clustering algorithm can be considered as a special case of the fuzzy c-means clustering algorithms. If m ik is 1 for only one class and zero for all other classes in equation , then the criterion function J(U,V) used in the fuzzy c-means clustering algorithm is the same as the criterion function J(V) used in the crisp c-means cluster algorithm. This is the so-called extension-principle.
A DETAILED TAXONOMY OF THE WORLD CITY NETWORK
Analysing the 54756 pieces of relational information with the fuzzy c-means algorithm requires a comprehensive look at the content of the clusters, and we therefore summarize our results by designating cities based on their membership degrees in various clusters:
The diverse tendencies in the world city network are summarized in Table 2 and Figure 18. Table 2 asserts the affiliations of each of the 234 cities, while Figure 1 visualizes these results through ordering the clusters both hierarchically and regionally, i.e. based on their average connectivity and their approximate geographical latitude. In order to aid in the initial interpretation of the results, Figure 1 also shows two distinctive cities for each cluster, in addition to a number of hybrid cities.
Cluster 1 unveils London and New York as a wholly distinctive dyad that ascertains the most important link in the world city network in terms of connectivities. Clusters 2 and 4 show important cross-regional links between major European, Pacific Asian and Latin American cities, which replaced the somewhat odd dyads in the crisp classification. These trans-regional links show that European cities are pivotal in linking to other world regions, which contrasts enormously with the relational patterns of US cities. Cluster 3, for instance, includes just the three US cities that rank below New York. This cluster thus only contains US cities, and articulates an idiosyncratic pattern that re-appears for lower-ranked world US cities (clusters 7 and 14). Apart from New York, as half of the NY-LON dyad, and Miami, as an important hybrid city linking to Latin America (clusters 7 and 11), there are no continental US cities that have important relations beyond their own country. We interpret this as relating to the sheer scale of the US economy and its long-developed, massive market in financial and business services that provides less of an incentive for firms 'to go global' to the same degree as global service forms from other world regions. Although it will become clear that most clusters that bring together lower-ranked world cities retain an important regional dimension, this tendency is nowhere as distinct as in the case of US cities.
Lower-ranked European cities, for instance, appear to be thoroughly interlinked. With the exception of a cluster that only contains secondary German cities (cluster 6), lower-ranked European clusters share a lot of hybrid members (clusters 17, 18, 19). Moreover, a number of European cities seem to perform gateway functions to other world regions, e.g. Stockholm as gateway city between Western Europe and Eastern Europe (clusters 8 and 10), and Bratislava and Sophia between western Europe and the Middle East (clusters 8 and 9). Note that in the classical clustering algorithm all cities are lumped together in a single cluster, which does not allow assessing these gateway functions.
Cluster 5 brings together the most important Pacific Asian cities beyond Hong Kong, Tokyo and Singapore. The average connectivity of this cluster unveils that Pacific Asia is a third main 'globalization arena' next to Europe and the US, while the lack of hybrid cities and trans-regional links furthermore shows the systematic interconnectedness between second-ranked Pacific Asian cities. It is clear that cities in southern Asia and the Middle East (clusters 9, 13, and 16) play a far less important role in this transnational urban network.
Beyond the three dominant world-regions, the relational patterning of Latin American cities reveal the ambiguous role it has come to play in the world-economy. There are a limited number of Latin American cities that can be found among the most important world cities (Mexico City, São Paulo, and Buenos Aires), but the relational prowess of these cities contrasts with the marginal position of the other Latin American cities, especially Central American cities, which are found in lower-ranked clusters (clusters 11, 15, and 20). This implies that although these major Latin American cities are found along the likes of Zurich and Brussels, their position is embedded in a totally different regional context. That is, the significant connectivity gap between major and minor Latin American cities is very distinct from Europe, the USA and Pacific Asia, where there is a markedly even distribution of cities across most connectivity levels. Following Friedmann's (1986) initial suggestions on world city categories, this suggests a semi-peripheral pattern as nodes that articulate surplus transfers to the core of the world-economy.
The remarkable regional dimension beyond the highest rungs of the world city network is broken by a cross-regional cluster that covers the old British Commonwealth (cluster 12). This 'cultural' historical throwback arena replicates previous findings with fewer cities (Taylor et al. 2002b), and is in fact the only urban arena that adds a politico-cultural configuration by providing a slightly different historical twist to the story of the otherwise dominating hierarchical and regional tendencies in the world city network. Membership covers Australian, Canadian, New Zealand and South African cities not found in the clusters with the most connected cities, in addition to a number of British cities that link up with lower-ranked European cities.
Cluster 20 brings together the less connected nodes in the world city network. With the exception of Nairobi, all inter-tropical African cities are distinctive members of this cluster. Singular cities include a number of relatively unimportant Asian and Latin American cities, hybrid cities include Baltic cities, which link up with minor European cities (clusters 17 and 19), and a number of South Asian cities that have a hybrid pattern with cluster 16. Nairobi is the only sub-Saharan city that is not a contained in cluster 20. Its affiliation in cluster with South Asian cities may seem quite odd from the perspective of the strong regional tendency of the arenas, but it is the only sub-Saharan African city that moves a marginal connections to the wider world city network. Likewise, Mumbai is the only South Asian city that escapes the otherwise marginal position of this region through its affiliation with a cluster with most important cities in the Middle East a Band II arena, This also indicates that Mumbai rather than New Delhi is the leading Indian city in the world-economy.
Our fuzzy-set approach has allowed unveiling a number of hierarchical and regional tendencies beyond the top-ranked cities, but it also sets aside to 'individual' profiles. Miami is the exemplary case here with its hybrid profile in clusters 7 and 11. The former contains secondary US cities, the latter is the arena of major Latin American cities. Although Miami is not a major world city according to Beaverstock et al. (1999), our analysis is picking up the suggestion that it does perform a very important regional role through its articulation role between the USA and Latin America. As a city it has indeed been designated as unusual before, the 'most foreign city' in the USA (Nijman, 1997), a contingent political (CIA) creation (Grosfoguel, 1995), a sort of 'extra-mural capital' of Latin America (Brown et al. 2002), with totally distinctive connections (Taylor and Walker, 2001), and our analysis lends further empirical support for these theses. Furthermore, it can be read from Table 2 that the odd linkage Houston/Barcelona and Washington's distinctive pattern are now replaced with a pattern which makes intuitively more sense, i.e. both North American cities now linking up with their 'hinterland'.
The present paper has moved beyond previous attempts of world city categorization by addressing the methodological problems related to assessments of less-connected nodes in a global urban network. The end-result is a complementation of previous specifications and measurements of the world city network with a detailed taxonomy of relational profiles of cities across the globe.
The use of a fuzzy c-means algorithm, which computes membership degrees rather than mere membership and allows influencing the results through choosing the exponent weight factor m, provides more flexibility for classifying objects. A comparison between Tables 1 and 2 clearly unveils that the latter is more fitted to serve as a detailed taxonomy of the world city network. In addition to being less sensitive to bias in the dataset, the fuzzy classification reveals more than just hierarchical patterns for less connected cities, while the idiosyncratic patterns in more thoroughly connected nodes are replaced with more knowledgeable patterns (i.e. larger clusters). And finally, the fuzzy clustering result is more useful since the identification of multiple patterns was the specific goal of our analysis.
Controlling the degree of fuzziness through manipulation of m has thus allowed transforming the small clusters/large clusters dichotomy of the crisp results to a distinct clusters/hybrid clusters dichotomy. Although this is of course the whole gist of the alternative approach we have taken in the present paper, it induces the problem of the 'right' choice of m. There are no guidelines for an optimal choice of this parameter; a good ad hoc guideline may be exploring different results for different values of m (and C), that this commonsensical guideline is of course by no means an objective operation.
The detailed taxonomy presented in Figure 1 and Table 2 are not 'explanations', neither statistical nor so-called causal. Rather, this classification is merely a tool that allows exploring the patterns that emerge in this global urban network. By discerning the patterns in the complex relational patterns between cities, it can be seen that world cities are tied together in three different ways. First, there is a strong hierarchical dimension to the clusters. Cities with similar levels of overall connectivity tend to be classified together, e.g. New York and London as the dominant nodes are distinct members of a single cluster. Second, there is a strong regional dimension to the clusters. Cities from the same part of the world tend to be classified together. US cities are the exemplary case here, but beyond this clear-cut example there remains an important regional dimension to inter-city relations. Third, there is a tendency for interaction between these two dimensions. Clusters with low average connectivity tend to be more regionally restricted in membership. The theoretical implications of the latter lay in the observation that instantaneous communications and a global network do not imply the 'end of geography' (O'Brien, 1992). Although distance no longer features as a factor in financial transactions, our analysis is picking up the suggestion that cities and their networks do not exist in some sort of abstract service space. Rather, there is a multifaceted geography of world regions through which cities operate as basing points for global capital. Hence, as well as the commonplace notion that individual world cities represent critical local-global nexuses, there are also world regions that represent regional-global nexuses in 'real' geographical space.
We would like to thank the editor and two anonymous referees for useful comments on an earlier draft of this paper. The data on which this research was carried out is part of the ESRC project "World City Network Formation in a Space of Flows" (R000223210).
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1. Obviously, cities with a functional reach beyond the national level are not new. Geddes (1915) and Hall (1966) already identified the existence of cities fulfilling supra-national roles in the world-economy, while Arrighi (1994) has noted that the bias towards historical uniqueness of these world cities can be traced back to the use of discourses and conceptions that are not suited to unveil a capitalist system that is contained by a system of states.
2. Moreover, applying a principal component analysis implies that a significant number of the 123 cities are not classified; the outset of factor analysis techniques is framed upon explaining variance, not classifying all objects.
3. Clustering methods may be divided into two categories based on the cluster structure they produce. Non-hierarchical or partition methods divide a dataset into disjoint clusters, whereas hierarchical methods produce a set of nested clusters in which each pair of objects or clusters is progressively nested in a larger cluster until only one cluster remains. The choice of either of these two techniques in this instance, then, depends primarily on the form of the desired output (Kaufman & Rousseeuw, 1990; Everitt et al., 2001) In the present paper, we apply a non-hierarchical clustering method. The hierarchical element in the clustering result will be represented by ordering the clusters based on the average GNC of the members of that cluster. As such, we will be able to combine the fuzziness of the cluster membership with the hierarchical element in our result.
4. However, one should consider the fact that (i) different variables as constituent components of the classification analysis may be of different relevance for the classification, and (ii) the range of values should be suitably scaled in order to obtain reasonable distance values (Kaufman & Rousseeuw, 1990). Generally, the second problem can be accounted for by using standardized data (z-scores), for this yields a "unit free" measure. Since we use connectivity measures that were derived using a single method and based on real-valued vectors bearing the same meaning, this is of no concern here.
5. It is important to note that the application of our cluster analysis is used as an exploratory rather than a confirmatory research design, which stems from a positive approach towards vagueness: creating many alternative results provides a means for exploring a set of data. Instead of searching for some sort of ideal classification, a multiple-number design allowed for the comparison of results over a range of levels of data reduction (Yates, 1987). In the present paper we will focus on the results for C=20, a pragmatic choice after comparing a number of different 'solutions'.
6. Both the crisp and the fuzzy c-means clustering have been carried out with MATLAB Software.
7. Since its original outset, fuzzy set theory has been employed in many areas to simulate and manage vague information (Höppner et al., 1999). Obviously, these vagueness problems also apply to large geographical databases. MacMillan (1995) has pointed out that fuzzy thinking has been around in geography for as far back as the 1970s. MacMillan himself (1978) and Gale (1972a, 1972b) applied fuzzy set theory with respect to locational decision-making and behavioural geography. However, "at that stage, it did not become fashionable in geographical circles (.)" (MacMillan, 1995, p. 404). More recent examples of applications of the use of fuzzy sets in geography can be found in the domains of spatial analysis (e.g. Leung, 1987; 1988), site selection (e.g. Witlox, 1998), and land-use planning (e.g. Smith, 1992; Xiang et al., 1992). Although there are, then, quite a few examples of the use of fuzzy set theory, research topics and methodology issues relying on the use fuzzy set theory are as yet not a part of mainstream geography. Furthermore, the outset of the basic premises of fuzzy set theory itself was merely the start for a myriad of studies leading to an explosive growth of both the original core ideas and possible extensions, such as research of expert knowledge systems and neural networks.
8. In order to be able to compare this set of results with the crisp classification, we will again focus on the solution for C=20. A more substantive discussion of the patterns that emerge for C=22 is given in Derudder et al. (2002).
Table 1: Results of the crisp c-means algorithm (C = 20)
Table 2: A detailed taxonomy of the world city network (based on the application of the fuzzy c-means algorithm for C=20 and m=1.50)
Figure 1: Service change in world cities, 2000-01
Edited and posted on the web on 26th February 2002; last update 15th March 2003
Note: This Research Bulletin has been published in International Journal of Pattern Recognition and Artificial Intelligence, 21 (3), (2007) , 439-462