This Research Bulletin has been published in Transportation Planning and Technology, 33 (4), (2010), 343-366.
Please refer to the published version when quoting the paper.
In their paper in Urban Studies, Limtanakool et al. (2007) introduce a number of spatial interaction indices and apply them to data on business, holiday, and leisure journeys with the aim of examining the pattern of interaction between Functional Urban Areas (FURs) in France and Germany . On the basis of the values of the indices, the urban network configuration in the two countries can be located on the continuum between the archetypal fully monocentric and fully polycentric networks.
In this paper, the analytical framework of Limtanakool et al. will be modified by calculating two additional overall measures and by normalizing the indices through a comparison with the rank size distribution. We interpret the spatial interaction indices as measures of connectivity and dominance and investigate their usefulness in the context of the European airline network by applying them to a dataset on air passenger flows. Connectivity and dominance are two aspects of hierarchical differentiation, a concept we borrowed from Pumain (2006). Hierarchical differentiation involves a ranking of elements from large to small (e.g. the rank size rule for cities). It differs from hierarchical organisation, which indicates different levels, with new properties emerging at each level (e.g. a Christaller pattern of central places).
The paper is organized as follows. Section 2 explains the rationale for our research, i.e. the use of air passenger flows to measure inter-city relations. In section 3, our dataset is introduced and the transformations that were needed to make the data suitable for our analysis are described. Next, we present the four spatial interaction indices (section 4) and explain the way in which they were normalized (section 5). Section 6 focuses on the main results of the research. The final section discusses the main findings of this paper.
2. Rationale: the use of air passenger data
Within research on contemporary globalization, studies on transnational urban networks have taken on a prominent place. Since the 1990s, it has therefore become commonplace to emphasize the importance of border-crossing relations between cities. M ost urban researchers nowadays acknowledge that major cities are increasingly produced and reproduced by what flows through them, rather than by what remains fixed within them. In this context, numerous authors have pointed to severe data deficiencies as regards these inter-city relations. Taylor (1999, p. 1901) has argued that there are two main reasons for this empirical deficit: “First, statistics in general have their origin in servicing the information needs of states and this has resulted in our contemporary world being measured through state-centric data. [...] Second, statistics in general have developed a critical bias towards measuring attributes at the expense of connections.” In recent years, two different approaches have been adopted in averting this empirical deficit (Derudder, 2006), i.e. (i) the corporate organization solution, analysing organization networks created by firms that pursue global strategies, and (ii) the infrastructure solution, describing the telecommunication and transport networks that have enabled organisations to go global. The research presented in this paper is part of the latter approach, and makes use of air passenger data in measuring the relation between world cities and globalization. 1
Air passenger data have become popular for analysing flows between major cities because they are publicly available, because transport is mainly about connections and flows, and because they allow for a relatively straightforward analysis of the spatial patterning of transnational inter-city relations. (Derudder and Witlox, 2005; Taylor et al., 2007)
Besides being popular for research on inter-city relations, airline data are also pre-eminently suitable for examining the effects of recent developments in the airline industry. In the US , the deregulation of the passenger aviation market in 1978 has resulted in a radical reorganization of the airline network. More specifically, agreements between airports and airlines have tended to result in hub-and-spoke configurations in which a small number of key airports (hubs) serve as transfer points where passengers change planes. From these hubs, the spoke flights then take passengers to their final destinations. (Burghouwt et al., 2003)
The process of deregulation also took place in Europe , albeit in a more gradual way. Three packages of deregulation measures (in 1987, 1989 and 1992) have led to a shifting of power from governments towards the European airlines. (Button et al., 1998; Hakfoort, 1999). Because European carriers already showed a very high traffic concentration rate before deregulation, the deregulation process did not result in a restructuring as radical as in the US (Burghouwt et al., 2003). However, Burghouwt et al. (2003, p. 320) notice that “at a smaller scale, radial strategies can be observed among regional carriers”. The advantages of such a radial hub-and-spoke configuration, as compared to a point-to-point configuration (Figure 1), are obvious: for the same number of destinations, there are fewer routes to serve, which in turn yields the possibility of higher flight frequencies and bigger aircrafts (Burghouwt and Hakfoort, 2001). There is however an important countertendency to this evolution, viz. the mounting success of low-cost carriers. These low-cost carriers tend to prefer a point-to-point organization (Alderighi et al., 2005) and in this way challenge the shift towards hub-and-spoke models.
Figure 1: Point-to-point network (on the left) and hub-and-spoke network (on the right)
Numerous studies have tried to measure the spatial configuration of the airline network, thereby comparing the real network configurations with ideal hub-and-spoke and point-to-point structures (for an overview, see Alderighi et al., 2007). However, in view of the larger research project of which our analysis is part (cf. footnote 1), the focus in this paper is somewhat different. Rather than measuring the airline network as such, air passenger data are used to examine the hierarchical differentiation in the European urban network. The formative aim of this paper is to investigate the empirical merits of four spatial interaction indices, adapted from Limtanakool et al. (2007).
3.1. Description of the AEA-database
The analysis in this paper makes use of AEA-data. AEA stands for Association of European Airlines, a non-profit-making organisation that brings together 31 major European airlines2. The AEA represents its member airlines in dialogue with all the relevant European and international organisations in the aviation value chain, thus ensuring the sustainable growth of the European airline industry in a global context. (http://www.aea.be)
Through the cooperation of an airline, we were able to obtain a AEA-database that contains, for each connection, information on the carrier, origin and destination (airport, city, country and region), number of passengers (subdivided into first class, business class and economy class), freight, mail, number of flights (subdivided into passenger flights and freight flights) and distance between origin and destination. The data is summarized on a monthly basis for the period January 2001 – December 2005, which allows a detailed analysis on recent data. Because of some difficulties with the homogeneity of the data for different years3, in this paper we will only discuss the results for 2005.
Obviously, the AEA-data is a very rich source of information. There are, however, two drawbacks that should be taken into account when interpreting the results. The first problem is the fact that no low cost carriers are member of the AEA. According to the European Low Fares Airline Association (www.elfaa.com), the low cost carrier sector accounted for approximately 30% of intra-European traffic in 2006. As mentioned above, low-cost carriers tend to prefer a point-to-point organisation of air traffic. Not including them will therefore have some influence on the results.4 A second and more important disadvantage of the AEA-data, and a potential source of distortions and misinterpretations, is the lack of real origin-destination data: the database records the individual legs of a trip rather than the trip as a whole. For example, a flight from Oslo to Madrid via London will be recorded as two separate flights, one from Oslo to London and one from London to Madrid . Any possible stopovers are not registered as such, which implies that the connectivity of cities with an important hub function, like London and Paris , will be overestimated.
3.2. Transformation of the Data
The AEA-database includes flights within Europe , as well as flights between Europe and other regions. For our research purposes, we only selected those flights where both the origin city and destination city are European cities. Since our analysis is part of a research on flows between cities, we converted the airport-to-airport database into a city-to-city database by summing the number of passengers over all the airports for a given city. Finally, given that we do not know the home-based location of the travellers, we summed the passengers travelling from city A to city B with those travelling in the opposite direction, and grouped the same connections, resulting in a database of non-directional flows. After these transformations, our database contains 130,663,329 passengers (of which 90% economy class and 10% business and first class), divided over 22 carriers, 35 countries and 183 cities.
3.3 Various Analyses
The AEA data offers the possibility for various analyses and comparisons: the evolution in time (2001-2005), the difference between business class and economy class, and separate analyses per carrier or country. With regard to the latter, we choose the three carriers with the largest number of passengers flying within Europe , namely Air France , British Airways and Lufthansa, and their home base countries France , the United Kingdom and Germany . The evolution in time will not be dealt with in this paper, because of the above mentioned homogeneity problem.
Our expectations regarding these comparisons can be summarized in four hypotheses. First, since not all cities are business centres, we expect business class to be more hierarchically differentiated than economy class. Second, we expect a difference in hierarchical differentiation between France and the United Kingdom on the one hand, and Germany on the other, because of their different urban configuration (primacy of London and Paris , versus a more polycentric structure in Germany , see e.g. Krätke, 2001; Taylor et al. 2006). Third, the carriers will probably show more dominance and less connectivity than their respective home countries, because of the hub-and-spoke strategy applied by most full service airlines nowadays (cf. supra). Fourth, since each carrier has a home base city, frequented by a majority of its passengers, we expect the links to be less hierarchically differentiated than the cities. It is on the basis of these four hypotheses that we will assess the empirical merits of the spatial interaction indices.
4. Spatial interaction indices
As mentioned in the introduction, hierarchical differentiation has two aspects, namely dominance and connectivity.5 Since the same degree of connectivity in a network can be associated with different levels of dominance of the cities in this network, we had to use more than one index. Four indices were selected, two that measure dominance and two that give an indication of the connectivity within the network. One of the dominance indices (the overall distribution index based on cities ODI c) is measured at the level of the overall network, while the other (the non-directional dominance index DIT i) is measured at the level of the individual cities. Of the connectivity measures, again one is calculated at the level of the overall network (the overall distribution index based on links ODI l), and one at the level of the individual connections or links (the relative strength index RSI ij). The latter three indices are taken from Limtanakool et al. (2007), who use them, as mentioned in the introduction, to locate urban network configurations on the continuum between the archetypal fully monocentric and fully polycentric networks. Since ‘polycentricity’ has no meaning within the context of our research, and since our database provides no real origin-destination data, we prefer to use the more neutral term ‘connectivity’.
The first index, the overall distribution index based on cities ODI c, is an entropy measure that measures the extent to which the total interaction is distributed evenly across all cities in the network:
ODI c = - (1)
where Z i is the share of passengers associated with city i in the total number of passengers, and I is the number of cities in the network. A value of 1 indicates an equal distribution over the I cities.
The second index is the non-directional dominance index DIT i, calculated as the ratio between the sum of the interactions associated with city i and the average size of the interactions associated with the other cities in the network:
DIT i = (2)
where T i is the total number of passengers associated with city i and i ≠ j. Cities with a DIT i value above 1 are considered dominant cities because they are more important than the average of the other cities in the network. Large differences between DIT i values for different cities indicate a high degree of hierarchical differentiation.
The third index, the overall distribution index based on links ODI l, is again an entropy index, measuring the extent to which the total interaction is distributed evenly across all links (city-pairs) in the network:
ODI l = - (3)
where Z l is the share of passengers travelling on link l in the total number of passengers, and L is the potential number of links in the network. The maximum ODI l value of 1 indicates a fully connected structure.
Finally, the fourth index is the relative strength index RSI ij, which is simply the proportion of interaction on a single link between two cities relative to the total interaction in the network:
RSI ij = (4)
where T ij is the total number of passengers travelling between city i and city j, and i ≠ j. The RSI ij values for all links in the network sum to unity, while individual values range from 0 to 1. Similar to the DIT i measure, large differences between RSI ij values are an indication of hierarchical differentiation.
Before applying the analytical framework of Limtanakool et al. (2007) to airline data, we modified it in two ways. First, the DIT i and RSI ij indices offer values for each individual city/link. This poses no problem when only a small number of nodes is analysed, as is the case in the paper of Limtanakool et al. (8 FURs in Germany , 6 in France). However, when the number of nodes is large, as is the case in our research, conclusions on the hierarchical differentiation are not always straightforward to make. We therefore calculated the standard deviations of the values of both indices as a second overall measure of hierarchical differentiation that may be helpful in interpretation. High standard deviations reflect large differences in the values of the indices and thus point to more dominance and less connectivity. In other words: t he higher the standard deviations, the less equally divided passengers are between cities and links.
Second, the measures are sensitive to the number of cities and links. This can be shown by calculating them for rank size distributions. A rank size distribution can be defined as:
= (for cities) and (for links) (5)
where Ti (resp. T1) is the number of passengers associated with city i (resp. link l), I (resp. L) is the total number of cities (resp. links), T’1 is the virtual number of passengers of the largest city/link, and r is the rank of the city/link. One would expect the values of the entropy indices and the standard deviations to be the same for each rank size distribution, irrespective of the number of cities or links.
The values of the indices and standard deviations for rank size distributions with the same number of cities/links as the real distributions are given in Table 1. If the number of cities/links would not matter at all, then all values in each column would be the same, which is clearly not the case. This implies that comparisons between countries, carriers or service classes cannot be made without normalizing the results first. The following section explains how the indices are normalized by comparing them to the values for the corresponding rank size distribution.
Table 1: Results for rank size distributions
5. Normalization of the indices
The sensitivity of the indices to the number of cities/links not only implies that statements about individual values6 cannot be substantiated, since there is no reference point to judge them by, but also that the results of analyses that are based on different numbers of cities/links cannot be compared. Whereas for the judgment of individual values a simple comparison to the rank size value is sufficient7, matters are slightly more complicated when comparing results of different analyses. Imagine two countries X and Y, with x and y cities respectively, where x > y. If ODI c for country X is smaller than ODI c for country Y, this does not necessarily imply that country X is more hierarchically differentiated than country Y. We can however obtain comparable results if we use the rank size distribution (as defined above) as a reference point. However, simply dividing the real values of the entropy indices and standard deviations by their corresponding rank size distribution values does not solve the incomparability problem. What we need is a normalization that, indifferent of the number of cities/links in the network analysed, results in a value between 0 and 1, where 0 indicates a completely even distribution, 1 points to maximum hierarchical differentiation (all passengers concentrated in one city/on one link) and 0.5 reflects a rank size distribution. For the entropy values (ODI c and ODI l), this can be achieved by applying the following formula’s (the subscript N refers to the normalized value of the index or standard deviation, while the subscript RS indicates the value of the index or standard deviation for a rank size distribution):
ODI N = when ODI ≥ ODI RS (6)
ODI N = when ODI ≤ ODI RS (7)
The normalized value of the standard deviation (SD) of RSI ij is computed as follows:
SD N(RSI ij) = when SD ≤ SD RS (8)
SD N(RSI ij) = when SD ≥ SD RS (9)
In a situation of complete absence of hierarchical differentiation, all links have the same RSI ij value (i.e., 1/L) and the standard deviation is 0, resulting in a normalized standard deviation of 0 (formula (8)). When there is maximal hierarchical differentiation (all passengers concentrated on one link), the standard deviation reaches its maximum value of (1000/L)√(L-1) (proof in Appendix), resulting in a normalized standard deviation of 1 (formula (9)).
For the standard deviation of DIT i, the normalization formula’s are:
SD N(DIT i) = when SD ≤ SD RS (10)
SD N(DIT i) = when SD ≥ SD RS (11)
When there is no hierarchical differentiation at all, all cities have a DIT i value of 1 and the standard deviation is 0, resulting in a normalized standard deviation of 0 (formula (10)). When all passengers are concentrated in one city (maximal hierarchical differentiation), the standard deviation becomes infinite, resulting in a normalized standard deviation of 1 (formula (11)).
Normalizing the indices by comparing them to the corresponding indices for a r ank size distribution with the same number of cities/links allows for comparisons between the various analyses. In the example given above, if ODI c for country X is larger than ODI c for a rank size distribution with x cities, while ODI c for country Y is smaller than ODI c for a rank size distribution with y cities8, then the conclusion (after applying the above formula’s) would be that country Y is more hierarchically differentiated then country X, which will be reflected in a larger normalized value for country Y.
Before focusing on the results, a final remark should be made on the way the different indices deal with differences from the mean. The entropy measures ODI c and ODI l are not very sensitive to changes in the values of the largest cities/links, because the proportions of passengers are multiplied by their logarithm. On the other hand, although the standard deviations treat positive and negative deviations from the mean in the same way, they are more sensitive to higher deviations because of the squaring. Therefore, in interpretations, it is best to combine entropy values and standard deviations.
6.1. City and Link Level
Table 2 lists the top-5 of the most important cities and links in 2005, subdivided into business class and economy class. From the table, it can be seen that while Frankfurt holds the fourth place for economy class flows, it comes second for business class flows. This is no surprise, given the importance of this city for (international) business. The analyses with regard to the links show that the fourth and fifth most important links for economy class are the connections of Madrid to Paris and to London . For business class flows on the other hand, places 4 and 5 are taken by Scandinavian capital cities: Oslo-Stockholm and Copenhagen-Stockholm. The latter is, however, not a reflection of the economic importance of these Scandinavian cities. Business class was first introduced in Europe by Scandinavian Airlines (SAS) and has since then been very popular in the Scandinavian countries. The cumulated RSI ij values (multiplied by 1000) of the five most important connections amount to 61.01 for economy class and to 110.39 for business class, reflecting the more hierarchically differentiated nature of the business class network.
Table 2: Top-5 DIT i and RSI ij values in 2005
Table 3 shows the DIT i values of all dominant cities in 2005. In other words, the DIT i values larger than 1. Dominant cities within the same country are listed together. As can be seen from the table, only seven countries have more than one dominant city: the UK , France , Germany , Italy , Spain , Sweden and Switzerland . The three countries for which we did a separate intra-country analysis are marked in bold. Germany has six dominant cities, while the UK and France only have three. Moreover, the range of DIT i values of the six German cities is smaller than that of the British and French cities. As mentioned above, we therefore expect domestic flights in Germany to show more connectivity, or less hierarchical differentiation, than domestic flights in France and the United Kingdom . Of the other two countries, the UK and France , the former will probably be more hierarchically differentiated, given the large DIT i value for London .
Table 3: Dominant cities in 2005
6.2. Network Level
Table 4 shows the normalized values of the entropy values and the standard deviations, calculated by formula’s (6)-(11). Remember that a figure larger than 0.5 indicates a distribution that is more hierarchically differentiated than the rank size distribution, while a value smaller than 0.5 indicates a less hierarchically differentiated distribution.
Table 4: Normalized results for 2005
To visualise the results, some graphs were made of the distributions of the number of passengers for the different analyses (Figures 2 to 9). The bold line indicates the real values, the dashed thin line the values for the corresponding rank size distribution. The x-axis is made logarithmic to ease interpretation. The corresponding normalized index values are indicated beside the graphs.
Figures 2 and 3 show the distributions for the three carriers (Air France, British Airways and Lufthansa). At the city level (Figure 3), all three have a more hierarchically differentiated distribution of passengers than the rank size distribution, which is reflected in values for the indices above 0.50. Of the three carriers, Lufthansa has the least hierarchically differentiated network. At the link level however, the bold lines are less steep than the dashed lines, which means that the passengers are more evenly distributed than would be the case for a rank size distribution. Here again, Lufthansa’s network is the least hierarchically distributed.
When we look at the three selected countries ( France , United Kingdom and Germany ), the distribution of the passengers at the link level (Figure 5) is again less hierarchically differentiated than the distribution at the city level (Figure 4), but the difference in hierarchical differentiation between cities and links is less pronounced than was the case for the carriers. At the link level, France and the United Kingdom approach the rank size distribution, Germany is clearly less hierarchically differentiated. At the city level, a clear distinction can be made between the graphs for France and the United Kingdom on the one hand, and the graph for Germany on the other. The first two are steeper than the rank size distribution, while the graph for Germany approaches the rank size distribution, although it seems somewhat less steep. The index values for France and the United Kingdom are above 0.50, while the values of the indices for Germany seem to contradict each other: the entropy value is above 0.50, the standard deviation is below 0.50. In other words: the entropy value points to more hierarchical differentiation than the rank size distribution, the standard deviation to less hierarchical differentiation.
The same is true for both economy class and business class passengers at the city level (Figure 6), and the explanation for this seeming contradiction is very simple. When the y-axis is also made logarithmic (Figure 7), the graph clearly shows how the real distribution turns downwards at a relatively early stage. This means that there are a lot of small values and these small values are overrepresented in the entropy value (because of the multiplication by the logarithm). At the link level, the real distribution also goes downwards (Figure 8), but at a much later stage, resulting in an entropy value below 0.50. The indices for the links (Figure 9) are substantially smaller than those for the cities, which indicates that the links are much less hierarchically differentiated than the cities.
Figure 2: Carriers, city level
Figure 3: Carriers, link level
Figure 4: Countries, city level
Figure 5: Countries, link level
Figure 6: Service classes, city level
Figure 7: Service classes, city level, log-log graph
Figure 8: Service classes, link level, log-log graph
Figure 9: Service classes, link level
Both the growing attention within urban studies for inter-city relations, and the changes in the spatial configuration of the airline network resulting from the deregulation of the European airline market have given rise to a search for adequate data and adequate measurement tools. In this paper, we examined the empirical merits of four spatial interaction indices for measuring the hierarchical differentiation within the European urban network. The standard deviations of two of the indices, DIT i and RSI ij, were calculated as a second overall measure of hierarchical differentiation. Because of their sensitivity to the number of cities or links, these standard deviations and the entropy indices ODI c and ODI l should be normalized. This can be done by comparing them to their corresponding values for a rank size distribution with the same number of cities or links. The four normalized measures, when interpreted together, appear to give a good idea of the tendency to hierarchical differentiation. This appreciation is based on the fact that our four hypotheses (cf. 3.3) are confirmed by the results: business class is more hierarchically differentiated than economy class, Germany is less hierarchically differentiated than France and the United Kingdom , the carriers are more hierarchically differentiated than their home countries, and the links are less hierarchically differentiated than the cities.
The methodology presented in this paper was applied to air passenger data between European cities, but there seems no reason to assume that the procedure discussed here cannot be extended to other kinds of flow data and/or networks at other scales.
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The standard deviation (SD) for a population is given by:
with x i the value for element i, X the average value and N the number of elements.
The maximum SD for RSI ij is reached when all passengers are concentrated on one link. The RSI ij value for this link is 1, while the RSI ij values for all other links are 0. If L is the number of links in the network, then the maximum SD for RSI ij is given by:
SD max =
The use of 1000 in the formula reflects the fact that we multiplied the RSI ij values by 1000 to ease their interpretation. If the original RSI ij values are used, the 1000 should self-evidently be changed to 1.
The formula can be further worked out as follows:
SD max =
SD max =
SD max =
SD max =
SD max =
SD max =
This research was conducted in the context of a large scale project on globalized urbanization, funded by the Flemish Fund for Scientific Research (research project G.0214.04 of the ‘Fonds voor Wetenschappelijk Onderzoek’, www.fwo.be)). The overall goal of this research project is to investigate the possible ways in which transnational urban networks can be empirically investigated.
1. The research presented in this paper is part of a larger research project on the quantification of the world city network. This research is funded by the Fund for Scientific Research Flanders and carried out at the Department of Geography of Ghent University in co-operation with the Globalisation and World Cities (GaWC) Study Group, a research group founded by Peter Taylor that focuses on the external relations of world cities. Its principal aim is a thorough investigation of the specification, measurement and analysis of the relation between world cities and globalization.
2. Situation on 19th September 2007 .
3. Between 2003 and 2004, the number of passengers shows a major increase, mainly caused by a growth in domestic passengers (passengers flying within one country), which is very probably due to a change in the registration procedure.
4. However, the distortion will probably not be very large. The results of a recent empirical analysis by Alderighi et al. (2007) indicate that the spatial network configuration of full service carriers and low cost carriers is very similar. On the other hand, their temporal configuration is very different: by adopting a wave-system structure in the airline flight schedule, full service carriers show high temporal concentration, while low cost carriers have almost a zero temporal concentration.
5. There is in fact a third aspect of hierarchical differentiation, namely symmetry, but this could not be investigated in our research: since we do not know the home-based location of the travellers (cf. 3.2.), the directional flow information of our AEA-database cannot be straightforwardly interpreted in terms of actual origins and destinations. For the sake of convenience, we therefore assume that the general level of dominance also applies to each pair of cities/links.
6. E.g. the statement of Limtanakool et al. (2007) that an entropy value of 0.81 points to a fairly even distribution of interaction.
7. In the case of the entropy values, a value smaller than the value for the rank size distribution with the same number of cities/links is interpreted as a tendency towards hierarchical differentiation, a value larger then the corresponding rank size value as a tendency towards an equal distribution of the number of passengers. For the standard deviations, the interpretation is the other way round.
8. Cf. Table 1. Notice that ODI c for a rank size distribution with x cities will be smaller than ODI c for a rank size distribution with y cities, since x > y (in the hypothetical case that the number of cities in country Y is larger than the number of cities in country X, the example given in the text would be mathematically impossible). The same holds for ODI l and for RSI ij: the fewer the number of links, the larger the index value for the rank size distribution. For DIT i on the other hand, fewer cities will result in a smaller standard deviation when a rank size distribution is applied.
Edited and posted on the web on 2nd November 2007
Note: This Research Bulletin has been published in Transportation Planning and Technology, 33 (4), (2010), 343-366