This Research Bulletin has been published in Urban Geography, 32 (8), (2011), 1227-1237.
Please refer to the published version when quoting the paper.
Uncertainty in the GaWC global network connectivity ranking
In the Globalization and World Cities research network (GaWC), an interlocking network model has been developed to measure cities’ connectivity into the world city network (Taylor 2001, 2004). The interlocking network model builds upon the aggregated location strategies of leading global service firms across cities. However, the connectivity measurements and consequently the global connectivity ranking rely on the information about the importance of a city within a firm’s office network, (i.e., its service value), which is determined subjectively (Taylor et al 2002, Taylor 2004). Subjective decisions are made in evaluating the importance of a city in a firm’s office network: scores are allocated between 0 and 5, with 0 representing no presence in a city and 5 indicating a firm’s headquarters.This subjectivity exists in allocating scores between 0 and 5 in the data generating process; this implies that individual service values of 1, 2, 3 and 4 in the GaWC dataset are not the only possibilities. This uncertainty requires investigation because it can compromise GaWC's well-known connectivity analyses. However, no existing study has quantitatively analyzed this subjective uncertainty in the interlocking network model. This paper fills this void and analyzes the impact of altering individual service values on city connectivity values and therefore the ranking of cities’ global connectivity, based on the GaWC 2008 dataset. There are three types of error that might be generated by the subjective allocation of service values to cities.
All three possibilities are considered and investigated through simulation of firms as network-makers.
Modeling uncertainty: systematic bias
The GaWC dataset summarizes the location strategies of m firms across n cities, represented by an m-by-n service value matrix V, where v(i,j) represents the importance of city i within firm j's office network. The connectivity measurements of the m cities can be calculated as:
GNC = V*VT*iota –diag(V*VT) (equation 1)
Where GNC is a m-by-1 vector of cities’ global connectivity, GNC(i) represents the global connectivity of city i, VT represents the transposed service value matrix, iota denotes a m-by-1 vector of ones, and the operator diag extracts the main diagonal elements from a matrix.
To ease elaboration, another 1-by-n vector f is created to denote firm size, where f(j) represents the total service value of firm j in the city network. As suggested by equation 1, the global connectivity of city i and k (k≠i) would change by k*f(j) and k*v(k,j) respectively, if the service value of firm j in city i is changed by k. Because the decision making is limited to the boundaries between the middle four scores in the six-point service value scale (Taylor et al 2002), the focus is therefore on the shifts in global connectivity ranking, due to change of one point in the service value of individual firms in individual cities. To begin, the first two types of possible error above are considered: both the under-estimation tendency, by adding one point to service values, as well as over-estimation by taking one point off service values.
Simulations of alternative service values for systematic bias
To evaluate the impact of alternative service values on the global connectivity ranking, one simulation experiment was carried out across all cities and firms.The global connectivity ranking based on the original GaWC dataset served as a benchmark ranking, and the simulation experiment for the case of individual cities and firms involved the following. First, for each of the under-estimation and over-estimation scenarios, a new service value matrix V2 was generated by adding one point to or subtracting one point off the service value of firm j in city i (v(i,j)). Because the measurements of no firms’ presences and firms’ headquarters are rather robust, the service values of 0s and 5s are not changed, and no service value will be changed to 0 or 5. Second, a new global connectivity ranking was produced using the interlocking network model and the new service value matrix V2. Third, the difference between this new ranking and the benchmark ranking, i.e., rank shifts of individual cities, was computed and stored. This process was replicated for all 508 cities and 174 firms, and for both under-estimation and over-estimation cases in the GaWC 2008 dataset, which resulted in 508*174*2 = 176784 simulations (the firm Wachtell, Lipton, Rosen & Katz was excluded from the analysis, as the firm has only one office in the GaWC dataset).
Results for systematic bias scenarios
The global connectivity ranking is relatively robust: there are totally 157754 absolute rank shifts in the 176784 simulations, with 0.9 rank shifts in each simulation on average. We report absolute rank shifts as the summation of signed rank shifts is always zero. This small averaged rank shifts could be ascribed to our constraints that service values of 0s and 5s are not changed and no service value will be changed to 0 or 5. An additional examination of the impacts of altered service values suggests that these altered service values generate 7.1 total rank shifts per simulation. However, relatively large rank shifts could still take place in individual cities. In the following subsections, we discuss the relationship between ranks shifts and firm sizes, the differences among global connectivity values and swap of neighboring ranks, and another simulation on random errors.
Case Studies: Rank Shifts in Individual Cities due to Service Changes in Selected Firms
The absolute rank shifts per service value change of individual firms are presented in Figure 1. It suggests that changes of service values in larger firms would create greater shifts in global connectivity ranking: Service value changes in large firms like Ernst & Young Global and PricewaterhouseCoopers International could change cities’ ranks by 20 and 15 on average in both the under-valuing and over-valuing scenarios respectively. In contrast, service value changes in most firms would create rank shifts less than 5 for individual cities.
Figure 1: Firm Sizes vs (Absolute) Global Rank Shifts per Service Value Change of Individual Firms
(a) The Under-Estimation Scenario, i.e., Alternative Service Values are Generated by Adding One Point
(b) The Over-Estimation Scenario
We should notice that Figure 1 provides a global measure of rank shifts and these shifts do not take place uniformly across cities. Because the average rank shifts are closely related to firm size, four typical firms are selected to explore the local impact of alternative service values on individual cities, i.e., how service value change of firm j in city i alter city i’s connectivity rank. These four firms were Y & R, Euro RSCG Worldwide, Baker Tilly International, and PricewaterhouseCoopers International, with 115, 267, 447, and 629 total service values, respectively. For every city, we plotted its global connectivity rank based on the original GaWC dataset against its rank shift in the new ranking due to the change of its importance in the particular firm's network (Figure 2). The following points can be made:
1) the rankings of the top 130 cities in all four scenarios are relatively robust with few and small rank changes. Although there are around 240 rank changes in the top 130 cities, due to alternative service values in PricewaterhouseCoopers International, rank changes at this scale would not be common, as only 10 and 52 firms in the GaWC dataset have total service values that are larger than 400 and 200, respectively. Changes in most firms would lead to shift patterns similar to that in Figure 2a and 2e. Moreover, what is reported here are rank changes in individual cities generated by alternative service values in these particular cities, and these rank shifts are not taking place simultaneously;
2) Cities with lower ranks have relatively small global connectivity, and increases in service values in these cities would generate boosts to cities’ ranks and there are expected to be more rank shifts at the bottom of rankings, which are most evident in the cases of Baker Tilly International and PricewaterhouseCoopers International (Figure 2); and
3) despite that most connected cities have relatively stable ranks, small changes at the top of the ranking would still give rise to uncertainty in data interpretation. For example, Beijing ranked 10th in the benchmark ranking, and its rank would be increased to 8th, if its service value in PricewaterhouseCoopers International is increased by one. This supports the procedure of grouping cities into tiers of similar global connectivity when interpreting analyses based on GaWC datasets.
Figure 2a: Initial Global Connectivity Ranking vs (Absolute) Rank Shifts in the Under-Estimation Scenario for the Case of Y & R, Euro RSCG Worldwide, Baker Tilly International, and Pricewaterhouse Coopers International
Figure 2b: Initial Global Connectivity Ranking vs (Absolute) Rank Shifts in the Over-Estimation Scenario for the Case of Y & R, Euro RSCG Worldwide, Baker Tilly International, and Pricewaterhouse Coopers International
“Gaps” in GNCs and Swap of Neighboring Ranks
We are interested in the differences in GNC values between cities of neighboring ranks in the original GaWC ranking, i.e., the “gaps” in GNCs, and also the number of firms where changes in service values could lead to swaps of neighboring ranks (Figure 3). We do not report changes in connectivity scores and focus on rank changes, because the changes in connectivity scores are closely related to sizes of firms where service values are changed. The majority of gaps in GNCs are smaller than 500, and the maximum difference is between London and Hong Kong at 15508, though this difference is not rendered in Figure 3a due to scale issue. Although the differences in GNC between city-pairs in the right tail of the distributions are small (Figure 3a), the numbers of firms where alternative service values could generate a swap of neighboring ranks are few (Figure 3b and 3c). These lower-ranked cities, where large rank shifts are likely to happen, usually have a lot of zeros in their service values, whereas we do not allow changes of and changes to zero service values in our simulation. Two additional observations could be made:
(1) there are three and four firms that could create swap of ranks of the top two cities in the under-estimation and over-estimation scenarios respectively. To elaborate this point, the change of service value in Ernst & Young Global’s New York office is used as an example. The difference between the GNCs of London (96267) and New York (95838) is 429, the total service value of Ernst & Young Global is 629, and the service values of Ernst & Young Global’s offices in London and New York are 5 and 3, respectively. By adding one point to the service value of office in New York, New York’s GNC increases to 96464 and surpasses London to become the most connected city.
(2) The highest numbers of firms that could swap neighboring ranks are for the gaps between Milan, Shanghai, and Beijing, which ranked at 8th to 10th in the benchmark ranking. These three cities have very close global connectivity values (65988, 65950, and 65939, respectively), and also host a wide range of middle-size offices that could be assigned alternative service values in the simulation.
Figure 3a: GNC Ranks vs Difference in GNCs Between Neighboring-Ranked Cities
Figure 3b: GNC Ranks vs Number of Firms that could altered Neighboring Ranks in the Under-Estimation Scenario
Figure 3c: GNC Ranks vs Number of Firms that could altered Neighboring Ranks in the Over-Estimation Scenario
Simulation of random errors
As previous simulations focus on the impact of one single error tendency on the global connectivity ranking, the final simulation was developed that introduces random tendencies of alternative service values. The simulation experiment for the case of introducing d random service values involved the following. First, a new service value matrix V2 was produced by generating d alternative service value(s) at random places. For under-valuing and over-valuing scenarios, individual service values were selected randomly and were added one or subtracted by one point accordingly. For the case of random signed errors, an additional uniformly-distributed random indicator was drawn to decide the signs of service value changes: one point was added to the selected service value, if the random indicator is above 0.5, and one point was subtracted otherwise. This random indicator generated approximately d/2 increased and another d/2 decreased service values, and allowed us to test whether the rank shifts due to errors of opposite signs would accumulate or cancel out. Second, a new global connectivity ranking was produced using the interlocking network model and the new service value matrix V2. Third, the difference between this new ranking and the benchmark ranking was computed. The d value ranged from 1 to 200, and the simulation was replicated 100 times for all d values. This experiment thus consisted of another 200*100*3 = 60000 simulations, and the total rank changes in 508 cities were obtained by aggregating the 100 replications for reporting purpose (Figure 4).
Figure 4: Number of Random Service Values Introduced into GaWC Dataset and their (Absolute) Averaged Impacts on GNC Ranks
For the case of random tendencies of alternative service values, greater changes in service values correlate with larger GNC rank changes. However, randomly mixed addition and subtraction of service values generate flatter rank shift distribution than those generated from uni-directional service value changes. This flatter distribution suggests that rank changes due to both processes may partly cancel out, instead of accumulating rank changes. For the case of under-estimation and over-estimation, more alternative service values still lead to larger GNC rank shifts, and they generate similar rank shift patterns, as evidenced by the two overlapping trajectories of increased and decreased service values. The robustness of GNC ranking could also be confirmed by the relatively small averaged rank shifts per city (1.7 rank shifts per city), even when 200 random service values are introduced.
Concluding remarksThis simulation study confirms that the global connectivity ranking for the top 130 cities, which are frequently used in GaWC analyses, is relatively robust. Subjective uncertainty in determining the service values would not create significant rank shifts in these most connected cities, whereas the ranks for cities at the bottom of the ranking are more sensitive to alternative service values. Moreover, small changes at the top of the ranking due to service value changes suggest that we should group cities with close connectivity values in data interpretation. Finally, this simulation study also provides a way to identify specific cities and firms, where alternative service values could lead to relatively large rank shifts and data entries should be double-checked in the collection process.
Taylor, P. (2001) ‘Specification of the world city network’, Geographical Analysis, 33: 181-94.
Taylor, P., Catalano, G. and Walker, D. (2002). ‘Measurement of the world city network’, Urban Studies, 39: 2367-76.
Taylor, P. (2004) World City Network: A Global Urban Analysis, London: Routledge.
This work was supported by the Economic and Social Research Council [grant number RES-000-22-3573].
Note: This Research Bulletin has been published in Urban Geography, 32 (8), (2011), 1227-1237