Gravity and the Hungarian Railway Network Csaba Gábor Pogonyi

Transcription

1 Statistical Methods in Network Science Gravity and the Hungarian Railway Network Csaba Gábor Pogonyi Table of Contents 1 Introduction Theory The Gravity Model Data Railway network data Population and land area data Results Intuitive analysis Regression analysis building the estimation model Regression analysis - the final estimation model Conclusion References... 17

2 1 INTRODUCTION This paper aims to apply one of the most important models of Economics to a new field. The gravity equation predicts the extent of an economic activity using only two factors: the importance of the actors and their distance from each other. The present paper applies this method on a physical network, the railway network of Hungary. According to the theory, we expect more important links between bigger cities that are close to each other than between smaller cities that are far. This question is important, as if we find the network does not behave according to the theory, we may conclude that economic factors were secondary when the railway network was built, and some modification is advisable. The first part of the paper presents the theory, translating the gravity equation for trade flows into the gravity equation of railway networks. The second part presents the data, whereas the third the testing of the gravity equation. The last part concludes and gives possible solutions. 2 THEORY THE GRAVITY MODEL The gravity model is one of those few Economic models that seem to be working in almost every case and already for a long period of time. Its first form was developed by James Q. Stewart in the 1940 s to predict between two points transportation, migration and communication. He found that these factors are positively related to population and negatively with distance. The theory s big time started in the 1960 s, when Jan Tinbergen used it to predict international trade between countries (Tinbergen, 1962). The core gravity model looks like the following: T ij = AY iy j Dij (1) Where Tij is a value of trade between country i and country j, A is a constant, Yi is the GDP of country i, Yj is the GDP of country j, and Dij is the distance between country i and country j. 2

3 However, in most of the cases they use an other form of the equation that is easier to be predicted: T ij = AY i a b Y j (2) Where a, b and c are allowed to differ from 1. In our research, instead of trade volume, we want to explain the importance of the links between big cities in Hungary. We expect that between bigger cities, the importance of the link (a centrality measure like edge betweenness) is going to be higher on average. We think that the bigger the cities are, the more important they are in the country. As they are more important, also in transport they have to have a more important role, with more important railway links. As the railway network is undirected, there is no difference between going from city A to city B or the other way around: the betweennesses of the edges on the shortest path do no change. Therefore, the model is the following: D ij c EB ij = AY i a c ED (3) ij Where EBij is a centrality measure (edge betweenness) between city i and j, A is a constant, Yi is the GDP of i, and EDij is the sum of edge lengths between the two cities (shortest path in km). It is easier to calculate the logarithms of the variables, this way we are able to measure the percentage changes of the variables of interest: ln (F ij ) = β 0 + β 1 ln(m i ) + β 2 ln(d ij ) + ε ij (4) Where F ij is the edge betweenness of the edges between i and j, M i is the population in i, and D ij is the length of the shortest paths in km edges between the cities. 3

4 3 DATA Our data was put together by using two separate sources: Open Street Map (osm.org, 2013) and Hungarian Statistical Office (ksh.hu, 2013). 3.1 RAILWAY NETWORK DATA The network of the Hungarian railway lines was obtained using relation-data put together by Open Street Map enthusiasts, who listed all the railway routes that together define all Hungarian railway stations and their connections (WikiProject Hungary, 2013). Using the names and length of the links, we created an edge link data frame that was exported into an igraph object. The graph consisted of 5 clusters: one with more than 2000 nodes, and the others with less than 30. The biggest one is the real railway network, the others are small local networks that are connected to the national service. From now on, all measurements refer to the national system. Using the Kamada-Kawai plot from Pajek, the network looks like this (figure 1). Figure 1: the Hungarian railway network graph (own work based on WikiProject Hungary, 2003, graphed with Pajek) The median degree is 2, the average is 2.12 and the maximum is 9 (Ferencváros station in Budapest). The highest node betweenness is also for Ferencváros station. It is not surprising as 4

5 Ferencváros is the main cargo hub and it is situated on the ringrailway that connects all the stations of Budapest. In order to get the shortest paths and average edge betweennesses the data was normalized. The shortest paths among the top9 cities were weighted by the length of the nodes. The average distance is 294 km, the standard deviation is 123. The longest path is between Pécs and Miskolc (571 km), whereas the shortest is between Debrecen and Nyíregyháza. As it can be seen on figure 2, the distribution is not far from normal distribution, skewed somewhat to the right. Figure 2: histogram of shortest paths (own work) Centrality measures can be assessed not only by the average edge betweennesses, but also with median, maximum or minimum measures. See their histograms on figure 3. The average edge betweenness seems to have the distribution closest to normal, whereas the histogram of minimum and maximum look highly uneven. 5

6 Figure 3: histogram of centrality measures (own works) 3.2 POPULATION AND LAND AREA DATA We used the Hungarian Statistical Office s databank to obtain additional data on Hungarian cities (Hungarian Statistical Office, 2013). Sadly, in Hungary there is no data on the GDP of cities; therefore, we had to use population data instead of gross output. We think that this caveat does not cause big differences in the results as output and population is closely related in Hungary. Moreover, in addition to latest population data (2012), we were able to obtain data for 1870 and These years can also be important, as most of the Hungarian railway lines were built between these two years; therefore, the linkages between cities might be even closer related to the situation at the end of the 19 th century, as to the 21 th. 6

7 Table 1: Population, Land area and Density measures by cities (own work, data from Hungarian Statistical Office, 2013) 2010 Population Land area km 2 Density Population/km 2 Budapest Debrecen Szeged Miskolc Pécs Győr Nyíregyháza Kecskemét Székesfehérvár Average If we take a look at table 1 and the various histograms of the variables (figure 4), it can be seen that even among the 9 top cities, there are basically two categories: Budapest, and all the others. Only in the measure of land area is the difference not that huge. It is also interesting to see that this difference has been present already since 1870, and it even grew bigger since. 7

8 Figure 4: histograms of the most important statistics of the cities (own graphs, data from Hungarian Statistical Office, 2013) 4 RESULTS This chapter applies the theory to the data available. The first part assesses only the intuition of the gravity equation, whereas the second part assesses the model itself, testing it with different specifications. The third part presents the final model that fits the gravity equation the most. 4.1 INTUITIVE ANALYSIS The gravity equation s first intuition is that cities with more intensive economic activity will have more important links connected to it. If it is true, the average edge betweenness of the biggest cities in Hungary (Budapest, Debrecen) should be higher than the smaller ones (like Kecskemét or Székesfehérvár). If we take a look at figure 5, we see that this pattern is not straightforward. One of the highest average edge betweennesses belong indeed to Budapest and Debrecen, however, only Székesfehérvár has relatively low value, Kecskemét is close to the average. 8

9 Figure 5: average edge betweennesses by cities (own work) The equation s second intuition tells us that cities that are farther away should have higher average shortest paths from each other. If we take a look at figure 6, we can see this pattern: the two lowest values belong to Budapest and Kecskemét, two cities in the middle of Hungary, whereas Győr and Pécs, two cities in the southern and western part have quite high average shortest path lengths. Figure 6: mean of shortest paths by cities (own work) Another method to check for the validity of the theory is to choose a city and test the equation for it. If we take a look at the example of Szeged, the gravity equation seems to be working. We 9

10 expect that cities that are farther away geographically from Szeged have lower shortest path kilometer on railway. And indeed this is what we can see if we compare figure 7 and 8: Kecskemét is close to Szeged, whereas Győr is the farthest one. Figure 7: map of Hungary (Google, 2013) Figure 8: shortest paths from Szeged (own work) The other part of the gravity equation predicts that the link between Szeged and a big city is more important than it is towards a smaller city. If we take a look at figure 9, we can see that also this part of the equation seems to work: the two highest betweennesses are towards Debrecen and Budapest (the two biggest cities in Hungary), whereas the two lowest are towards Pécs and Székesfehérvár (the 5 th and 9 th biggest cities). Figure 9: average edge betweennesses from Szeged 10

11 Taking into account these findings, the gravity equation seems to explain the railway network of Hungary to some extent. In order to get a more robust answer, we have to estimate the equation using regression analysis. 4.2 REGRESSION ANALYSIS BUILDING THE ESTIMATION MODEL In this chapter, we start with our initial assumption regarding the estimation of the model and try to find out the best specification that both fits the data and fits the gravity equation. In order to estimate the model, we had to face the realities of data availability, therefore, some proxies needed to be found for the theoretical factors. There is no regional GDP calculated on a city level in Hungary, therefore; we used population data instead. An other measure for economic activity could be the land area or the density of cities. Moreover, the density and population variables were also calculated for both 1870 and The regressions were run on 72 observations (9*8 city-pairs) using OLS method with robust standard errors. According to the gravity equation, we expect that sp (shortest path in km) will have a negative effect, whereas economic activity measures will have a positive one. The results of first estimations can be seen on table 2: the first part of the equation seems to work spotlessly; shortest path has a strong and robust negative effect on the centrality measure (average edge betweenness). Regarding the effects of the economic activity measures, the picture is more diverse. The differentiation in time did not change the coefficients significantly. There are however important differences between the three proxies used. Population and density variables achieved low R- squared indicators ( ), and neither were their coefficients significantly different from zero. Population coefficients were at least positive as it was predicted by the model. It seems to be plain that it is not advisable to use the density of cities as a proxy in itself for estimation, even though it would be an easily interpretable result. Interestingly, the variable that behaved in all aspects the most aligned with theory was land area: it achieved the highest R-squared (.21) and also its beta coefficient became positive and significant (.13). 11

12 Table 2: regression output (own work) (1) (2) (3) (4) (5) (6) (7) VARIABLES pop 2012 pop 1910 pop 1870 area dens 2012 dens 1910 dens 1870 pop (0.0207) sp ** ** ** * ** ** ** (0.0428) (0.0429) (0.0427) (0.0418) (0.0403) (0.0402) (0.0379) pop (0.0187) pop (0.0230) land 0.127*** (0.0439) density (0.0269) dens (0.0242) dens (0.0316) constant * (0.538) (0.520) (0.549) (0.529) (0.469) (0.454) (0.441) Observations R-squared Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 However, if we take a look at the predicted values of the models (see figure 10), we can suspect that the reason behind the results is the fact that only in the case of land area is Budapest not hugely different from other cities. In such a case, it is advisable to check if Budapest can be classified as an outlier. Figure 10: predictions by land area and population 2012 (own work) 12

13 If we look at table 3, it can be clearly seen that our suspicions were right: the exclusion of Budapest enhances both the R-squared of the model and its alignment with the gravity equation (strong positive relation with population/land area). In this case, even though we lose 14 observations, it is advisable to declare Budapest as an outlier. Table 3: regression output - testing if Budapest is an outlier (own work) (1) (2) (3) (4) Land area VARIABLES Population Population w/o Budapest Land area w/o Budapest pop *** (0.0207) (0.0758) sp ** *** * ** (0.0428) (0.0407) (0.0418) (0.0439) land 0.127*** 0.178*** (0.0439) (0.0571) constant *** ** * (0.375) (0.940) (0.368) (0.459) Observations R-squared Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 As a centrality measure, the default that we used was the average edge betweenness; however, we tried out also the minimum, maximum and median edge betweennesses between two cities as a robustness check. As it can be seen on table 2, the average, median and minimum measures behaved very similarly to each other. Out of these three, the minimum edge betweenness achieved the highest R-squared, therefore we can conclude that this measure seems to align the best with the gravity theory. 13

14 Table 4: regression output - centrality measures (own work) (1) (2) (3) (4) VARIABLES average median max min pop (0.0207) (0.0143) (0.0398) (0.0172) sp ** ** *** (0.0428) (0.0380) (0.0758) (0.0230) constant (0.538) (0.457) (0.989) (0.335) Observations R-squared Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 Summing up the results from these checks there are three important lessons: first is that using density (population/land area) as a separate variable is not advisable, it is a good idea not to restrict population and area that to a -1 power. Second lesson is that Budapest is an outlier; therefore, it is advisable to leave it out from the regression. Third is that even though average edge betweenneess works well, minimum edge betweenness between cities seems to be able to better estimate the gravity equation. 4.3 REGRESSION ANALYSIS - THE FINAL ESTIMATION MODEL Using the lessons from the previous chapter, we have estimated the best fit for the gravity equation (see table 5). R-squared indices doubled, in spite of the decrease of the number of observations due to the absence of Budapest. The shortest path variable s coefficient seems robust around Among the different proxies for economic activity, we can see some confusion again. The first result is that land area became insignificant in every specification: it seems that the absence of Budapest and the use of the minimum edge betweenness took over its effect. Among the three population measures, the 1910 and 1870 variables were not significant and their R-squared was also lower than the first specification. Therefore, among all the specifications that were tried out, the first one seems to be most reliable: it achieved a.4 R-squared indicator, and both of its coefficients are significant. We might also recognize that both of its coefficients move according to the gravity equation. 14

15 Table 5: final regression output (own work) (1) (2) (3) (4) (5) (6) (7) VARIABLES pop 2012 pop 1910 pop 1870 area dens 2012 dens 1910 dens 1870 pop * * (0.0407) (0.0426) sp *** *** *** *** *** *** *** (0.0227) (0.0235) (0.0233) (0.0245) (0.0251) (0.0256) (0.0239) pop (0.0278) (0.0318) pop (0.0259) (0.0255) land (0.0275) (0.0284) (0.0314) (0.0289) constant * ** (0.477) (0.324) (0.312) (0.238) (0.472) (0.331) (0.320) Observations R-squared Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 If we take a look at the predicted values of the (1) specification (see figure 11), we see that there are significant differences among the cities. Miskolc s presence moves the regression line upwards, whereas Kecskemét downwards. Figure 11: prediction by population 2012 (own work) 15

16 5 CONCLUSION The paper first translated the gravity equation of trade flows into the gravity equation of physical networks, more precisely to railway networks. The intuitive empirical part of the analysis showed that although the application is not straightforward, there are certain patterns that back the model. The regression analysis showed that due to its great size, Budapest has to be taken as an outlier. This finding seems to fit the history of the network, as it was built mainly between 1870 and 1910, to satisfy the needs of a bigger country, the Hungarian part of the Habsburg country. If we take into account the situation back then, Budapest is still the biggest city, however; there are prominent cities that make the difference smaller (Bratislava, Zagreb, Cluj Napoca, Kosice). Another important finding of the regression analysis was that instead of using the average edge betweenness between two cities, the minimum fits the data better. It is in accordance with network economics principles: the capacity of a network (gas pipeline, electronic wire or railway link) is determined by its narrowest part. The final model shows that the gravity equation can be applied to the Hungarian railway network with some modifications. Indeed, distance has a negative effect on the importance of links between two cities, whereas economic activity has a positive relationship with it. This finding means that the network s features makes it suitable for an economically effective usage, no great restructuring is needed. There are three important ways how the analysis could be strengthened. First, a more precise proxy variable for economic activity could be constructed instead of population or land area. Second, the analysis could be extended on the top25 or top50 cities of Hungary. Third, the most important characteristic of the network is seem to be fact that it was built for a bigger country, therefore; testing it for the pre-war borders of Hungary is advisable. 16

17 6 REFERENCES Google. (2013). Map of Hungary. Retrieved August 12, 2013, from = , &sll= , &sspn= , &oq=szeged&t=m&hnear=Szeged,+Hungary&z=7 Hungarian Statistical Office. (2013). Retrieved from OpenStreetMap. (2013). Retrieved from Tinbergen, J. (1962). Shaping the World Economy: Suggestions for an International Economic Policy. New York: Twentieth Century Fund. WikiProject Hungary/Vasút - OpenStreetMap Wiki. (2013). Retrieved from 17