Topological Analysis of the Evolution of Public Transport Networks KTH TSCMT TSCMT

JACOB ROMMEL JACOB Topological Analysis of Analysis Topological the Evolution of Public Networks Transport 2014 ,

Topological Analysis of the Evolution of Public Transport Networks KTH TSCMT TSCMT www.kth.se

Topological Analysis of the Evolution of Public Transport Networks

Jacob Rommel 5/29/2014 Supervisor: Oded Cats

Master Thesis

M.Sc. Transport and Geoinformation Technology Programme

DEPARTMENT OF TRANSPORT AND LOCATION ANALYSIS Abstract

Many studies have been conducted regarding network theory and how it can be applied to public transport network. This has led to knowledge on how network indicators relate to the performance of a network and also to insights of how networks can best be extended. Little is known however on how rail bound public transport networks and their network indicators have evolved over time. This would be interesting to know since many metro and other rail bound public transport networks have evolved over a long period of time with extensions being made at different times by different policy makers and stakeholders. This means that there has not been a unified planning process for many of the networks. It would hence be beneficial to get a better picture of how the networks have evolved, when extending the networks or when creating new ones.

By creating networks for every year in the development of a rail bound public transport network and then calculate the different network indicators, the evolutionary trends could be found. The networks were created in L-space which means that stations were represented as nodes and the rail connection between stations as edges. To every link in the networks, travel time was attached as weights. This was done in order to make the network indicators more realistic. By assigning geographical coordinates to nodes, indicators such as directness and closeness centrality with respect to geographical distance could be derived.

A case study was conducted by applying the methodology to the Stockholm rail bound public transport network. The study period was chosen to be from 1950 up until 2025. 1950 was the year when the Stockholm Metro opened, and the extensions to the network that are decided upon are planned to be completed in 2025. By including the future extensions it was hoped that it could be seen if the future trends are following the trends from the 20th century.

Trends regarding the evolution of the network in Stockholm were found. In general it can be said that indicators were relatively high in the first 15-20 years of the study. This was due to the inner city tram network that existed in these years. The tram network was relatively intra-connected with a relatively high average degree, clustering coefficient and connectivity. When the tram network closed down the indicators drastically decreased, after 1971 many of the indicators started to slowly increase due to the additions of new lines and also extensions of already existing ones. Between the year 2000 and 2025, many of the indicators increased substantially, this was partly due to Tvärbanan that connected many older lines creating nodes with a high degree.

The fact that the future extensions will lead to an increase in many network indicators (and a decrease in average connectivity) was seen as an indication that the future extensions will accentuate trends that have taken place since the early 1970’s. It was also seen that many of the extensions included in this study will help to develop the network in a way that is in line with the overarching planning principles set by the Stockholm council.

The structure of the network consisted of a dense core with branches reaching out to the suburbs in the 1950’s and early 1960’s. In the late 1960’s the network got a radial shape with branches going to the suburbs, no denser core existed in these years. This structure remained relatively unchanged up until the year 2000. After 2000 and up until 2025 a structure emerged in the network with a dense core and also a ring line going around half of the city. This type of structure had been seen in many other rail bound networks around the world.

Acknowledgement

I have always been interested in transportation in general, the interest has been particularly focused on public transport. Whenever I have traveled to a new city, one of the first things I have done has always been to look at public transport maps for the city. Due to my public transport interest it has always been my intent to base my master thesis on public transport. After having discussions with Oded Cats, the subject for this thesis emerged.

I will like to thank Oded, who became my supervisor, for his help throughout the process of writing this thesis. His inputs and guidance have been very helpful.

TABLE OF CONTENT

1 INTRODUCTION ...... 7

1.1 Background ...... 7

1.2 Problem Statement ...... 8

1.3 Objective ...... 8

2 LITERATURE STUDY ...... 9

2.1 Network Theory ...... 9

2.2 Temporal evolution of networks ...... 10

2.3 Public Transport Network ...... 11

2.4 Contribution to the literature ...... 12

3 METHODOLOGY ...... 13

3.1 Network Representation ...... 13

3.2 Creating the Networks ...... 14

3.3 Network Indicators ...... 14 3.3.1 Number of Nodes and Edges ...... 15 3.3.2 Connectivity ...... 15 3.3.3 Clustering Coefficient ...... 15 3.3.4 Degree Centrality ...... 16 3.3.5 Betweeness Centrality ...... 16 3.3.6 Closeness Centrality ...... 17 3.3.7 Network Diameter ...... 17 3.3.8 Directness ...... 17 3.3.9 Assortative ...... 18 3.3.9.1 Average Node Degree ...... 18 3.3.9.2 Pearson Coefficient ...... 18 3.3.10 Summary ...... 19

3.4 Implementation Details ...... 19

4 CASE STUDY ...... 21

4.1 Data ...... 22 4.1.1 Topological Data ...... 22 4.1.1.1 Metro ...... 22 4.1.1.2 The old tram network ...... 24

4.1.1.3 Spårväg City ...... 25 4.1.1.4 Tvärbanan ...... 26 4.1.1.5 Saltsjöbanan ...... 28 4.1.1.6 Roslagsbanan ...... 28 4.1.1.7 Commuter Trains ...... 29 4.1.1.8 Regional Trains ...... 30 4.1.1.9 Summary ...... 30 4.1.2 Time tables ...... 34

5 RESULTS ...... 37

5.1 Number of Nodes and Edges ...... 37

5.2 Connectivity ...... 38

5.3 Clustering Coefficient ...... 39

5.4 Degree Centrality ...... 41 5.4.1 Cumulative Distribution ...... 43 5.4.2 Spatial Analysis of Network Evolution ...... 44 1950 ...... 45 1967 ...... 46 1985 ...... 47 2000 ...... 48 2010 ...... 49 2025 ...... 50 Summary ...... 50

5.5 Betweeness Centrality ...... 52 5.5.1 Cumulative Distribution ...... 52 5.5.2 Spatial Analysis of Network Evolution ...... 53 1950 ...... 53 1967 ...... 54 1985 ...... 55 2000 ...... 56 2010 ...... 57 2025 ...... 58 Summary ...... 59

5.6 Closeness Centrality with respect to travel time ...... 59 5.6.1 Cumulative Distribution ...... 61 5.6.2 Spatial Analysis of Network Evolution ...... 62 1950 ...... 62 1967 ...... 63 1985 ...... 64 2000 ...... 65 2010 ...... 66 2025 ...... 67 Summary ...... 67

5.7 Closeness Centrality with respect to Distance ...... 68 5.7.1 Cumulative Distribution ...... 70 5.7.2 Spatial Analysis of Network Evolution ...... 71 1950 ...... 71 1967 ...... 72 1985 ...... 73 2000 ...... 74 2010 ...... 75 2025 ...... 76 Summary ...... 76

5.8 Network Diameter ...... 77

5.9 Directness ...... 78 5.9.1 Cumulative Distribution ...... 80

5.10 Assortative ...... 81 5.10.1 Neighbor Node Degree ...... 81 5.10.2 Pearson Coefficient ...... 82

5.11 Relations between indicators ...... 83 5.11.1 Clustering Coefficient / Average Degree ...... 83 5.11.2 Closeness Centrality derived from Distance / Closeness Centrality derived from Travel Time...... 85 5.11.3 Directness / Average Degree ...... 86 5.11.4 Network Diameter / Number of Nodes ...... 88 5.11.5 Connectivity / Number of Nodes ...... 89

6 DISCUSSION AND CONCLUSION ...... 91

6.1 Evolutionary Trends ...... 91

6.2 Development according to plan? ...... 93

6.3 Limitations and Future Studies ...... 95

7 REFERENCES ...... 98

8 APPENDIX ...... 101

8.1 Travel time ...... 101

8.2 Geographical Distance ...... 113

1 Introduction

1.1 Background Transportation is one of the major problems for large cities. The road network is often congested and due to lack of available space it is often difficult to build more streets in dense parts of the cities. The congestion creates many problems such as longer travel time which leads to an ineffective transportation network. An important factor that could make transportations more effective in the cities is having good public transport. Public transport is less space consuming than individual transport and it also makes the transport system available to everyone, even those who cannot drive a car.

Recent increase in awareness of the many environmental issues that the world is facing has increased the demand for public transport, this is because public transport is generally a more environmental friendly alternative to other transportation modes. Due to the great demand of modern public transport systems, many systems are being upgraded or plans for upgrading systems are under way.

Metro (Metro is the notation used in this report, other words for metro are subway and underground) is a public transportation system that is a vital part of a many cities’ public transport. Metro consists of trains that run on rail where some parts of the network often are located under street level. A metro system has its own right of way meaning that the rails are not shared with other modes and level crossings do not exist (Fröidh et al 2011). The right of way and also the fact that the lines often run under the ground makes the metro a fast and space efficient transportation mode, metros are often considered as one of the best alternatives in transporting passengers in dense cities. The metro systems usually have other rail bound lines that complements it in areas where the metro has not been built. Together, all the rail bound lines creates a network that serves as a backbone for the public transport system in many major cities around the world.

Over the last century there has been an enormous expansion of rail bound public transport networks around the world. The development of the networks has been incremental meaning that the development has happened step by step where there has been a need for a new line or a station. How the networks have developed over their whole time frame is something that has not been thoroughly studied, and in order to determine if the networks develop according to some preexisting pattern more studies have to be done.

At the end of the 20th century and the beginning of the 21st century there has been a significant expansion of metro networks in developing countries. This is partly because of the large traffic problems that mega cities in mostly developing countries have faced, that in combination with an economic development have lead to a rapid expansion of new metros and other rail bound networks. Examples of cities in developing countries that have built a new metro network in recent years are Shanghai and Seoul. Many of the new systems have also developed in a much faster pace than some of the older rail bound networks. The economic development in many developing countries is expected to continue its increase in the coming years. Due to this it is expected that the world will see more metro systems in the future as well.

1.2 Problem Statement In the literature, a lot of work has been done in investigating public transport networks. This has resulted in knowledge regarding how networks are characterized in terms of network indicators such as betweeness centrality (Derrible 2012), closeness centrality and clustering coefficient (Lin and Ban 2013). The study of public transport networks has also given a greater understanding to how the networks function.

An important point of interest for public transport networks is how they evolve over time. In most cases rail bound public transport networks are built over a long time period, in that way they evolve through a continuous process with new links and stations added where there is a need and some poorly used parts of the network being closed down. Different extensions to a network are also planned and built by many different authorities, there is for example different co-existing political levels that often have different interests. There is also a spatial distribution in political powers, meaning that there might be different municipalities that are affected by an extension and they sometimes have different opinions on how to develop a network. Since a network is built over a long period of time, there is also an issue with successive authorities, this means that different political actors have powers at different points in time during the network’s evolution. Even though the knowledge of how public transport networks and their network indicators have evolved over time might be very useful when new networks are planned or when investments are to be made in already existing networks, few studies have been conducted on the topic.

1.3 Objective This study analyzes how different network indicators change over time in public transport systems consisting of a metro network and also other rail bound public transport modes. A reason why only rail bound modes were looked at was that rail bound lines are high capacity long term investments which are more consistent than road based public transport modes leading to a more unified evolution process.

The network indicators that will be investigated were chosen so that different aspects of the networks can be analyzed. The number of Nodes and Edges and Network Diameter were chosen in order to see the size of the network. To see how well intra-connected the networks are, Connectivity and Clustering Coefficient were used. Four different network centrality measure were also included in order to see how the centrality of the network changes. Another point of interest is if the networks are assortative, to measure this average neighbor node degree and Pearson Coefficient will be used.

It will be investigated if the development of the network indicators follows some pattern and if this pattern change over time. If a pattern can be found, than it could be very useful to use this pattern as a blue print on how future investments in the public transport network could be directed.

To summarize, the purpose of the project is:

Investigate how the above mentioned network characteristics change over time in a public transport system consisting of rail bound modes. Investigate if the observed network characteristics follow a certain pattern over time.

2 Literature Study After several articles had been read, the most relevant information was obtained and it is presented below.

The literature review is divided into three parts. First facts about network theory in general are presented; part two is about temporal development of networks and what effect time has on network indicators. The last part is on public transport networks and how network theory can explain the performance in public transport systems.

At the end of the literature review, this projects contribution to the will be mentioned.

2.1 Network Theory During the 1900’s network theory emerged as a way to represent complex structures. An example was in 1969 when Hagget and Chorley studied spatial structures in channel pattern, river systems and road networks from a network perspective (Lin and Ban 2013).

One major contribution to network theory was given by Watts and Strogatz (1998), up until then networks that were not completely random nor completely regular had not been given much attention; instead most studies had been conducted on these two extreme types of networks. In the article, the networks were shaped as rings. A regular network was defined as a network where every node was connected to its two neighbors and also the two nodes outside the neighbors and in a random network the edges had been added randomly between the nodes. Many real world networks lies somewhere in between a perfectly random and a regular network. In most cases random networks have short path length but low clustering coefficient which means that the average shortest paths between all the pairs of nodes are short and there are few clusters. A regular network, on the other hand, has high clustering coefficient and longer path length. The article showed however that many networks that are in between a random and a regular network had high clustering coefficient and a relatively low average path length. This property was called ‘small world’. Three completely different real life networks that were not random but also not regular were found to posses these small world characteristics. A network was considered to be small world if the average path length of the network was only a little bit greater or similar than the average path length of a random network with the same number of nodes and edges and if the average clustering coefficient was larger than the average clustering coefficient of a random network.

The discovery of the small world properties led to an increasing interest in networks theory since it showed that completely different types of networks showed similar characteristics and could therefore be explained by similar theories (Lin and Ban 2013).

Another property that is shown for many man-made and also complex natural networks is scale free properties, it is discussed by Ash and Newt (2007), Xie and Levinson (2007) and Derrible and Kennedy (2010). A network is considered scale free if its degree distribution P(k) is proportional to k-ϒ meaning that the distribution follows a power law. Since the degree distribution is decaying according to a power law, there will be many nodes having one or two degrees and fewer nodes having more than three or more degrees, this means that most nodes only have one or two connections. The fact that many different networks that have been created more or less randomly have scale free characteristics implies that there is a ‘correct’ way to create a network that some network tends to converge to.

A type of network that will be used in this report is a spatial network called planar network. According to Barthelemy (2010) a spatial network is loosely defined as a network where nodes are located in a space equipped with a metric. A planar network means that no link can intersect each other without creating a node. In real life, an aviation network is a non-planar network since links can intersect without it being an airport. Metro networks on the other hand are generally planar since two rails in most cases do not intersect without there being a station present (there are exceptions to this, when a tunnel or a bridge is built for example).

2.2 Temporal evolution of networks An important aspect in network theory is how network develop over time. This is something that has emerged as a field of study in recent years.

Strano et al (2012) looked at the evolution of a road network in northern Italy. The study looked at a data set with maps showing the evolution of the road network for almost 200 years. Two phases in the evolution of the road network was found; the first was exploration when branches are built into areas not served by roads, after exploration comes densification which is the process where a network gets denser through the addition of links between already existing branches. In the study, there were a lot more of the former phase in the early years and the latter phase was dominant at the end of the study period. The article also found that the main roads that serve as a backbone for the whole network were in most cases the oldest roads in the network. This means that roads added later on in most cases were not as important as the older arterials. The importance of roads was measured through betweeness centrality. Another trend that the article pointed out was that the size and the shape of the cell areas between the roads followed a pattern. The size became more and more homogenous as time went by meaning that the difference in size for the cell areas became smaller in the end than in the early years of the study period. The shape of the cell areas went from being widely distributed in the beginning with many different shapes towards being more homogenously distributed with most cell areas shaped as rectangles. It was also found that the length of links being added to the network got shorter and shorter as time went by in the study period.

Xie and Levinson (2008) discussed the fact that older studies mainly investigated the evolution of top-down planned network, meaning networks that are centrally planned. It is argued however that many transportation networks have been created through a continuous process that involves decision makers, suppliers and users which all have different interests. A simulation model was therefore proposed that looked at how road networks evolve through an agent based process were roads that are not used as much get abandoned and heavily used roads get upgraded into better ones. Through this model it was shown that some networks had similar characteristics even though they had developed from very different types of networks. This was taken as an indication that transportation networks possess similar robust properties that can emerge from the interplay between many different actors.

Roth et al (2012) investigated how metro networks around the world have developed over time. The article described the evolution of 14 major metro networks. It was found that even though the networks had been created at different times and under very different authorities, they still shared some common properties. A trend that could be seen was that metro networks develop into a shape with a denser core surrounded by a circular line and branches going out from the core to suburbs,

10 these branches sometimes had fork stations which are stations where a branch gets divided into two branches. In two of the networks (New York and Chicago), it was harder to detect a core. This is believed to be due to the natural constraints that these cities exhibit with water to the east of Chicago and much of New York being based on an island. Different trends in indicators were found in the article; for example about 60% of nodes in the core have a degree higher than 2, the percentage of stations that are considered to be in the core is about 45% for larger networks. Another trend that was found in the article was that the average degree in the core in the networks tends to go towards 2.35-2.4 when the number of nodes increases.

2.3 Public Transport Network There have been an increasing number of articles written on network theory applied to public transport networks.

One aspect of transit network theory that has been thoroughly investigated is how network should be represented. von Ferber et al (2009) describes four different ways to represent a public transport network; L-space, P-space, C-space and B-space. All of the representations have different advantages and disadvantages depending on what type of network that is being represented. L-space is the most straightforward with stations acting as nodes and the connection between two stations acting as links. This representation is common when presenting a public transport system to the public due to its simplicity. The main disadvantage with L-space is that it does not take the lines into consideration. P-space on the other hand does take the lines into account by representing stations as nodes and then having a link between all stations on the same line. In this way all the stations that can be traveled between without the need of a transfer are connected by a link. P-space is not as straightforward as L-space and it can therefore be more difficult to interpret the network. C-space is represented with each line being a node and the lines that are connected in the public transport system have a link between them, in that way the amount nodes that a journey passes through is the number of transfers. In B-space both lines and stations are represented as nodes and links are connected from the line-nodes to all station-nodes that the line passes by.

Derrible (2012) writes about the importance of centrality measures when examining public transport networks. Betweeness centrality is used as the centrality measure to define the most central and therefore in many cases the most important stations in the networks. Betweeness centrality is defined as the amount of shortest paths that passes through each stations and the more shortest paths that goes through a station the more central it is considered. The article looks at global trends in how betweeness centrality changes in metro networks. There is for example an exponentially increasing trend with higher average betweeness centrality in a network if the number of nodes increases. A cumulative distribution of normalized betweeness centrality was produced for each network, it can be seen that larger networks have a much more distributed CDF than smaller ones pointing towards the fact that central nodes in larger networks have a smaller share of the total betweeness centrality comparing to smaller network. This process is called democratization. Other indicators that help to explain the performance of a public transport network and that are used in this study are mentioned in Barthelemy (2010) and Li et al (2009). These indicators are Directness, Average neighbor node degree, Clustering Coefficient, Gamma Index and Network Diameter. Directness measures the detour that the network is in comparison to the geographical distance, Average neighbor node degree is the average degree of the neighbor nodes for each node which is used when defining if a network is assortative or not, a network is considered assortative if

11 high degree nodes tends to connect to other high degree nodes, clustering coefficient describes how connected the neighbor nodes are with each other, Gamma index is a global indicator of connectivity that gives the ratio between the number of edges in a network and the maximum amount of edges given the number of nodes and diameter is the longest of all the shortest paths in a network.

Another centrality measure that is a good way to estimate the centrality of nodes in public transport networks is closeness centrality (Lin and Ban 2013). It is calculated by taking the ratio of 1 and the average journey length to all other nodes in the network for each node. When measuring the average journey length in a network it can be done in different ways. In this project the journey lengths is measured both in travel time and geographical distance along the network. When investigating if a network has assortative characteristics, average neighbor node degree can be used. Another indicator that measures the assortativeness of a network is Pearson coefficient described by Ash and Newth (2007). It is a global indicator meaning that one pearson coefficient exists for a whole network. It is derived by looking at the number of edges and the degree of the nodes at the ends of each link in the network. If Pearson coefficient is above 0 for a network it is considered assortative and disassortative if it is below 0.

2.4 Contribution to the literature As seen above, many studies exist regarding network theory applied to public transport networks and there are also a few articles regarding temporal evolution of networks. In the literature however there are no articles that do the exact same investigation as this study. Roth et al (2012) did a similar study on metro networks, however that study solely looked at unweighted networks that only consisted of the metro. In this project the travel time for the edges will be applied as weights and also the complete rail bound network, not only the metro, will be looked at. The addition of travel time will make the network more realistic and it will also make it possible to derive more types of network indicators. Also the fact that the whole rail bound network is investigated makes the study more realistic since it then incorporates a larger share of the public transport system than only the metro would. Another interesting aspect of including the whole rail bound network instead of only the metro is that the different rail bound modes have often been built by different actors resulting in a less unified planning process for the rail bound network as a whole. It would therefore be interesting to see how a whole rail bound network has developed.

Another contribution to the literature made by this project is the usage of geographical coordinates when creating the network. This will not only be beneficial when representing the networks but it will also make it possible to derive network indicators that depend on distance such as directness and closeness centrality with respect to distance.

Another aspect that has not been investigated in previous studies is that the future is included in the study. For the future years, extensions that are decided upon are included in the networks. This is beneficial since it can therefore be seen if the future extensions are developing the network in a way that is consistent with the trends up until then. This could be seen as a justification if the proposed extensions are beneficial for the network.

3 Methodology The first step in order to reach the report’s objective was to gain knowledge about network theory through a literature study. The literature was about network theory in general, temporal development of networks and also articles regarding network theory applied to public transport networks.

After the literature study, the different networks were to be built. In order to build the networks information of the network’s topological evolution had to be obtained. This was done through literature studies and also study visits to archives and museums. The information resulted in an inventory where the network for the first year of the study’ was shown and then the incremental changes for each year.

Before the network could be constructed, the appropriate network representation had to be found.

3.1 Network Representation Public transport networks can be represented in many different ways, the different types of network representations have different pros and cons depending on what the researcher is trying to achieve. Below are four different kinds of representations presented.

L-space, P-space, C-space and B-space are all different ways to represent a public transport network (figure 1) (von Ferber et al 2009). All of the representations have different advantages and disadvantages depending on what type of network that is being represented. L-space is the most straightforward with stations acting as nodes and the rail connection connecting two stations acting as links. Due to its simplicity, this representation is common when presenting a public transport system to the public. The main disadvantage is that lines cannot be represented in L-space. P-space on the other hand does take the lines into consideration, it represents stations as nodes and then links are connected to all stations on the same line. In this way all the stations that can be traveled between without the need of a transfer are connected by a link. P-space is not as straightforward as L-space and it can therefore be more difficult to interpret the network. C-space is represented with each line being a node and links are connected between the lines that are connected in the public transport system, in that way the amount nodes that a journey passes through is the number of transfers needed. In B-space both lines and stations are represented as nodes and links are connected from the line-nodes to all station-nodes that the line passes by.

Figure 1. The figure shows the four different network representations described above. From the left, the representations are L-space, P-space, C-space and B-space (von Ferber et al 2009).

For this project, the network was represented in L-space. This way of representing a network is good because it gives a realistic picture of how the network looks like, this will help the visualization of the network evolution. Other advantages are that in this type of study when many different networks are to be compared to each other, L-space is considered favorable since comparisons are the easiest for

L-space. This is because the same stations and rail connections look the same for each year in L-space which makes it easier to follow the evolution of a certain node or link. In P-space however, more links are added if stations are added at the end of a line which can make comparisons between networks more troublesome. A disadvantage with L-space however is that it does not take the lines into account. This is unrealistic since the time it takes to transfer between lines is not included in the total journey time.

By adding travel time and geographical distance to each link and geographical coordinates to the nodes, the L-space representation gets even more realistic.

3.2 Creating the Networks The software Gephi was used to build the networks for every year in the study, it was mainly used for its simplistic inter face which made it easy to create the different networks. It was also used because informative maps can be exported from the software. When creating the networks, the travel time for each link was added as weights (see Appendix for travel time on each link). This was done in order to make the network more realistic and it also allowed for calculating more types of network indicator (closeness centrality with respect to travel time for example). Geographical coordinates were added to each node, these coordinates were obtained from Google Maps. By using the plug in Geo Layout in Gephi, the software could read the coordinates and place the nodes in their correct position. This made the networks look more realistic. The geographical coordinates were also used in order to calculate the distance between nodes. This was done by using a formula that calculates the distance between two points with geographical coordinates; the formula can be seen below (Koordinaten 2014).

d R ARCCOSSINLAT1 SIN LAT2 COS LAT1 COS LAT2 COS LONG2 LONG1 Where: d = Distance between point 1 and point 2 in meters R = The earth’s radius (6 371 000 meters was used) LAT1, LAT2 = Latitude for point 1 respectively point 2 in radians LONG1, LONG2 = Longitude for point 1 respectively point 2 in radians

After the distance was calculated, it was added to each link (see Appendix for distance on each link). The distance was used to calculate more types of network indicators such as directness and closeness centrality with respect to geographical distance. Since the formula only calculates the distance as a crow flies between two nodes and not the actual distance along the rail, there will be some discrepancies to these indicators. It was however decided that the Euclidean distance between stations is sufficient since most stations have a relatively straight connection going between them.

After the network had been created, the next step was to calculate the network indicators. The indicators that were chosen to be part of the study are shown in the section below.

3.3 Network Indicators After reviewing the literature and also analyzing which indicators would be useful for achieving the report’s purpose, the network indicators used in this study were chosen. Other factors that mattered when choosing the indicators were which indicators that were possible to obtain.

The indicators that were chosen are explained below, the section is based on the following articles: Lin and Ban 2013, Li et al 2009, Barthelemy 2010, Derrible 2012 and Ash and Newth 2007.

3.3.1 Number of Nodes and Edges A basic indicator is the number of nodes in a network, it is a simple way to describe the size of a network. An indicator that is similar to the number of nodes is the number of edges. In this project, nodes are defined as stations and edges as the physical rail connection between stations. If two rail connections run parallel between two subsequent stations they are considered as one edge.

The more edges a network have comparing to its number of nodes, the more connected a network is. The indicator connectivity that is described below takes this relation into account.

3.3.2 Connectivity Connectivity is a measure of how well connected a network is. In this project connectivity is measured using the gamma index, it is defined as the ratio between the number of edges and the number of edges in a complete graph.

| E | 3| N | 2

Where: |E| = the number of edges |N |= the number of nodes

The higher the gamma index is the more edges there are in comparison to the number of nodes. This means that the network is considered more connected since there are more routes to choose from in the network.

3.3.3 Clustering Coefficient

A similar term to connectivity is the network indicator Clustering Coefficient. It measures how well intra connected the neighbors of each node are. Two nodes are defined as neighbors if they are connected with a single link.

The indicator is defined as the number of edges connecting the neighbor nodes of each node divided by the maximum amount of edges that could connect the neighbor nodes.

2t C i ( 1) ki ki Where:

Ci = Clustering Coefficient for node i t = the number of edges among node i’s neighbors ki = the number of edges among node i’s neighbors

If there are no edges among the neighbor nodes, the clustering coefficient equals 0. On the other hand if there are as many edges that connect the neighbors as can possibly be, the clustering coefficient equals 1. The more nodes in the network that have a high clustering coefficient the more clusters there are in the network, this is considered as an indication that the network is well connected. This is true since more clusters in general leads to more edges in comparison to the number of nodes in a network, and this relation between edges and nodes is what connectivity measures. Due to this many nodes with a high clustering coefficient in general leads to a more connected network.

In this report, only the clustering coefficient for the closest neighbor is looked at. However clustering coefficient exists for the i:th neighbor as well, for example clustering coefficient for the second neighbor could be obtained. In that case all nodes that are connected with two or less edges are considered neighbors.

3.3.4 Degree Centrality

The number of links connecting a node is called degree. In this paper it is defined as the number of rail connections connecting each station, however if there are two parallel rail services between two stations, they only count as one line. An example of this in Stockholm is that the many metro lines going between T-Centralen and Slussen only counts as one line. Generally the more connections available at a station the more important the station is for the overall performance of the network and the more central the node is considered to be. Degree centrality is considered the most straightforward indicator of centrality.

3.3.5 Betweeness Centrality Betweeness centrality is a centrality measure that indicates the centrality and the importance of a node in a network.

After computing the shortest path between all pairs of nodes in a network, the betweeness centrality is defined as the number of shortest paths that traverse through each node. If one pair of nodes has more than one shortest path between them then the betweeness centrality is divided on the different routes in a way that all nodes along the shortest paths gets an equal share of betweeness centrality. The indicator can be defined as follow.

n (i) B jg Ci B jg n jg

Where: B C i = Betweeness Centrality for node i njg(i) = the number of shortest paths between node j and g that passes through node i ngj = the total number of shortest paths between node j and g

The larger the number of shortest paths that go through a node is, the more important that node is to the network’s performance as a whole and the more central the node is located in the network.

Betweeness centrality can also be defined for edges. It is then calculated in the same way as for nodes and the indicator is then used in order to find the most central edges. In this project however, only betweeness centrality for nodes is used.

Another way to represent betweeness centrality is to use normalized betweeness centrality. It is calculated by dividing each node’s betweeness centrality with the network’s total betweeness centrality. By doing so, each node gets a betweeness between 0 and 1 that represents the percentage betweeness centrality for the whole network. This way of representing the network indicator is favorable because it makes it easier to compare the betweeness centrality for different networks.

3.3.6 Closeness Centrality An important indicator for how central a node is located in a network is its total distance along the network to all other nodes. The network indicator Closeness Centrality is defined as 1 divided by the total distance to all other nodes in the network.

1 C Ci Bdij i j

Where: C C i = Closeness Centrality for node i dij = Distance between node i and j.

The further away from all other nodes a node is located the less central it is in the network.

In this project, both the closeness centrality with respect to travel time and with respect to the geographical distance will be calculated. This is to see if the centrality of nodes varies when considering travel times rather than geographical distances.

3.3.7 Network Diameter In order to measure the size of a network, the network diameter can be used. It is defined as the length of the longest of all the shortest paths in a network. The larger the diameter is the more geographically spread out the network in general is. Another way to measure the diameter is to take the average of all the shortest paths.

In this study, the indicator will be derived by calculating the longest shortest paths in the network.

3.3.8 Directness One important aspect of how well a public transport network functions is how well connected each pair of nodes are. The more connected the nodes in a network are the shorter the shortest distance between each pair of nodes in the network generally becomes. In a perfect network, the distance between every pair of nodes would be the Euclidean distance. In practice however, this cannot be obtained. In order to measure how much the distance in the network differs from the Euclidean distance, the network indicator directness is introduced. For each node it is defined as the ratio between the distance along the network to all other nodes and the Euclidean distance to all other nodes.

BdR,ij i j Qi BdE,ij i j Where:

Qi = Directness for node i dR,ij = The route distance between node i and j dE,ij = The Euclidian distance between node I and j

In an hypothetical network where the distance between all node pairs is equal to the Euclidean distance, directness equals 1, in reality the value is greater than 1 and the higher it gets the longer one has to travel along the network in comparison to the Euclidean distance.

3.3.9 Assortative An assortative network is a network where there is a high possibility that nodes are connected to other nodes with the same degree. In other words, nodes with few degrees are in general connected with other nodes with few degrees and high degree nodes are connected to other high degree nodes. An assortative network has some advantages, for example if a node with many degrees is removed in an assortative network then another nearby high degree node can to some extent compensate for the removed node, making the impact of the removed node not as great, which is improving network robustness.

3.3.9.1 Average Node Degree An indicator used in this project to measure assortativeness is the average degree of every node’s neighbors. It is defined as follows.

1 knn(i) Bk j ki j+ (i)

Where: knn(i) = the average degree of the neighbor’s to node i ki = degree for node i kj = degree for node j

If knn(i) is an increasing function of k, then nodes with higher degree tends to have neighbors that also have higher degree and the network can therefore be considered assortative.

3.3.9.2 Pearson Coefficient Another indicator used to define if a network is assortative is Pearson coefficient. It is derived by looking at the degree for nodes at the edges of every link in the network. If the indicator is above 0, then the network is considered assortative, if the indicator is below 0 the network is considered disassortative.

Pearson coefficient is a global indicator meaning that one number can be derived for a whole network.

C 1 S2 cBBj k Dc ( j k )T i i E 2 i i U ii 1 C 1 S2 cBBj 2 k 2 Dc ( j k )T 2 i i 2 i i iiE U

Where: Γ = Pearson Coefficient c = 1/m (m = the number of edges) ji and ki = degrees of the nodes at the edges of link i

3.3.10 Summary The table below was made in order to summarize the network indicators used in this project.

Table 1. The table shows a summary of the network indicators used in this project. A short description of the information the indicators convey, what level of analysis and in which studies they are mentioned in are showed in the table.

Network Indicator Information the indicator gives Level of Studies Analysis analyzed in Number of Nodes and The size of the network Network Roth et al 2012 Edges Connectivity How intra connected the network is Network Lin and Ban 2013 Clustering Coefficient How clustered nodes are Node Li et al 2009 Degree Centrality The centrality of nodes Node Bartelemy 2010 Betweeness Centrality The centrality of nodes Node Lin and Ban 2013, Derrible 2012 Closeness Centrality The centrality of nodes Node Lin and Ban 2013 Network Diameter The geographical spread of the Network Barthelemy network 2010 Directness How direct journeys the network Node Barthelemy offers in comparison to the 2010 Euclidean distance Average Neighbor node If the network is assortative or not Node Barthelemy degree 2010 Pearson Coefficient If the network is assortative or not Network Ash et al 2007

3.4 Implementation Details To calculate the network indicators, Gephi was planned to be used, however a drawback with Gephi is that it cannot take weights into consideration when calculating indicators. Some of the indicators were therefore calculated using MATLAB instead1. Node lists were exported as csv files from Gephi with the information needed and then MATLAB imported them and calculated the indicators, Dijkstras algorithm was used to calculate the shortest paths between the node pairs in the network. The indicators that were obtained using MATLAB were Betweeness Centrality, Closeness Centrality with respect to travel time and geographic distance, Network Diameter, Directness, average neighbor node degree and Pearson coefficient. The indicators that are not depending on weights, such as clustering coefficient and degree centrality were calculated by Gephi and then exported as csv files to Microsoft Excel.

After the network indicators were obtained from each year in the study, Microsoft Excel was used in order to plot and analyze the data.

The average value of the indicators and the standard deviation were calculated for each network. Cumulative distribution functions were also made in order to see how the indicators were distributed among the nodes. After studying the different graphs, six important years in the study were chosen

1 The MATLAB code used to calculate Dijkstras algorithm and betweeness centrality was downloaded from mathworks.com. The codes were written by David Gleich and are available at http://www.mathworks.com/matlabcentral/fileexchange/10922-matlabbgl. 19 to be the most important ones. In Gephi, maps were created for these years showing how the different centrality measures varied for the nodes, tables were also created that showed top 5 stations for the six years.

Plots were also made showing the relations between some of the indicators, this was done in order to see if any correlation existed. The indicators chosen to be tested for correlation were chosen after their individual plots had been analyzed.

When all plots, maps and tables were done, they were used in order to analyze the different trends that occurred over the study period and if these trends have changed over time.

4 Case Study A case study was done by applying the methodology to the public transport network in Stockholm, Sweden. As mentioned earlier, only rail bound lines will be included in the study and in order to only include mass transit lines not add lines that only have a few departures each day, a threshold of 23 departure a day in each direction was used when including or excluding lines in the network. This was considered to be a reasonable number since 23 departures will result in 2 departures an hour in rush hour and at least one departure an hour in the other times of the day.

The reason why only rail bound lines were included in the study is mainly because rail bound lines demand a large investment cost which reflects a larger development commitment, this means that more work has been put into the process of building a new rail line comparing to the opening of a bus line for example. This results in more long term implications for the rail lines. Due to the low investment costs to open a new bus line, the life time for a bus line could sometimes be very short. A rail line on the other hand have higher investment costs which means that it is not closed down as easily. This makes rail bound lines better for this study since they have more long term effects on networks.

Another reason only rail lines were included is the fact that in most cases the rail network serves as the back bone for the public transport network and it is therefore the most important part.

The case study will look at the network from the opening of the metro in 1950 up until the year 2025.

As of today, the rail bound public transport network in Stockholm has 250 stations, it consists of a metro network, four commuter train lines, four different tram/light rail lines, two sets of local railway networks (Roslagsbanan and Saltsjöbanan)and regional trains going to the outermost suburbs and nearby cities.

The metro is the backbone of the rail bound public transport network for journeys inside the city center and from nearby suburbs to the city. To complement the metro, there are 4 tram/light rail lines. The commuter train network is mainly for journeys to the outermost suburbs. For journeys to suburbs even further away or nearby cities, regional trains are used. Tvärbanan is a tram/light rail line that opened in 2000. It is a transverse line connecting many metro and commuter train lines to each other.

There are also two local railways in the network. They serve as feeder routes from the city center to relatively far away suburbs.

Stockholm is growing in a rapid pace. It is believed that the population in Stockholm will increase with 20 000 inhabitants and 13 000 workplaces each year up until 2020 (SL 2009). The travel patterns with many journeys ending and starting in the city center of Stockholm and also a large share of trips being made locally is believed to be remained unchanged until 2020. These factors are estimated to result in a yearly increase of 3000 travelers in the public transport system during the morning peak between 6 AM and 9 AM. This increase will be difficult for the public transport system to handle. In order to cope with the population increase it is important that the system will be extended in the future.

4.1 Data In order to understand the evolution of the network, data had to be collected.

The data that was gathered could be divided into two main categories; topological data of Stockholm’s rail bound network and time tables. The data was gathered from literature and also from study visits to the following museums and archives: Spårvägsmuseet, Spårvägsmuseet’s archive, Stadsmuseet’s archive, SJK’s archive and the archive at Kungliga Biblioteket.

4.1.1 Topological Data The topological data for the evolution of the rail bound public transport network in Stockholm from 1950 up until 2025 was mainly found in literature studies. Aspenberg (1998), Eriksson (1991) and Lange (1998) have written books explaining the evolution of the tram network in Stockholm. A book describing the evolution of the commuter trains was written by Hällqvist (2008), it describes the topological development from the late 19th century. In order to celebrate Roslagsbanan’s 100th anniversary the book ‘Roslagsbanan 100 år’ (SLJ 1985) was released, it describes the history of Roslagsbanan from 1885 up until 1985. Landgren (1993) wrote about Saltsjöbanan’s evolution in the book ‘Saltsjöbanan’.

Articles regarding the future development were found online, many articles were found on the Stockholm County’s website.

Data regarding the topological evolution of the network was also collected from study visits. The tram museum in Stockholm was visited on two occasions, the 22nd of January and the 7th of February 2014. The main findings were the topological development of the Stockholm Metro. The incremental evolution for each year from 1950 up until today was found.

Some topological data was also gathered from time tables found in archives; this was mainly for the regional train lines.

The evolution of Stockholm’s rail bound public transport network is presented thoroughly below.

4.1.1.1 Metro

Historical Development The metro is the main transportation inside the city center and also between the city and the nearby suburbs. The history of the Stockholm Metro dates back to 1933 when a tunnel for a tram line was constructed (Aspenberg 1998). The tunnel went underneath the island Södermalm from the station Slussen to Ringvägen. Even though trams ran underground from 1933 the Stockholm metro was not officially founded until 1950, the first line used the already existing tram tunnel and also a new line connecting the city center with suburbs south of Stockholm, terminating at Hökarängen. After the first line had opened the system extended gradually over the years, new lines were created and then gradually extended were there was a need.

Cervero (1995) describes the importance of the metro in Stockholm’s transition from being a monocentric city into being a multi-centered metropolis in the later part of the 20th century. The transition was aided by building and extensive rail network with different branches connecting new satellite towns to the city center of Stockholm. The opening of the metro was vital to this process with Vällingby being the first rail served satellite town being built, the green metro line was extended

22 to Vällingby in 1952. In these new rail-severed satellite towns there are a large number of people that use the public transport as their main way of transportation. When comparing to other ‘natural’ suburbs such as Täby, the public transport share is substantially higher in the satellite towns. Other examples of satellite towns that are being served by the metro network are Kista, Rinkeby, Skärholmen and Skarpnäck. Skarpnäck is the newest of these types of suburbs with the metro being extended there in 1994.

Today’s Network Today the metro network is divided into 3 different groups of lines, the red, blue and green lines, the first tram tunnel is now part of the green lines (Lokman.se 2014). The groups are divided into branches, the total number of lines, including all the branches, is seven. As of today the system has 100 stations and the total length of the network is 110 kilometers.

Future Extensions For many years there has been a discussion on how to best expand the metro network. The last time a new metro station was built was 20 years ago (1994) when the green branch to Bagarmossen was extended to Skarpnäck. There have been many suggestions on how to expand the metro, for example extending it to Täby, Nacka and to Karolinska Sjukhuset (Stockholms Handelskammare 2013).

On the 11th of October 2013 the ruling political parties in Sweden presented an agreement to expand the metro system in Stockholm with 9 new stations (DN 2013, Stockholmsförhandling 2013). The proposal included the extension of two lines and the addition of a completely new line with three stations going north from Odenplan to Arenastaden adjacent to the commuter train station Solna Station (Figure 2).

The Blue line will be extended from Akalla in the northwest to Barkarby Station creating a new station in between called Barkarbystaden. Barkarby station is now served by commuter trains but in the future it will also be served by regional trains making it a multimodal hub called Stockholm Väst. The southern end of the blue line will be extended to Nacka, stopping at 4 intermediate stations. By building a connection from Gullmarsplan to Sofia on the new line to Nacka, the branch to Hagsätra, that today is part of the green line, will be part of the new blue line. By making the Hagsätra branch part of the blue line, many passengers that today use the highly used section between Slussen and T- Centralen will instead use the new line going through Kungsträdgården. This will be beneficial for relieving congestion from the section between Slussen and T-Centralen and offering higher frequency on the remaining green line branches.

Similar to how the metro system has developed through the years, the new extensions will not be built in one go but it will be done in an incremental process.

Adolphson (2009) discusses that it has been a goal for the City of Stockholm to achieve a polycentric development since 2001. One of the potential growth areas that have been mentioned by the city is the area around Globen. The fact that the Hagsätra branch will become part of the new blue line is improving the quality of the public transport for the Globen area, since the capacity on the line will increase. Due to this it can be said that the transformation of the Hagsätra line is helping the polycentric development that the City of Stockholm wants to see.

Figure 2. The future extension of the metro. The extensions is planned to be finished in 2025 (http://stockholmsforhandlingen.se/accounts/10965/files/254.pdf).

4.1.1.2 The old tram network At the end of the 19th century a need for efficient transportation arouse in Stockholm. Due to this many tram lines were built in the city (Eriksson 1991). The first two lines were opened in 1877 between Slussen and Roslagstorg respectively Grevbron. In the beginning, the trams were operated by horses. After a few decades however it became clear that it would be better to use the new traction power electricity instead and in 1901 the first electrical tram line opened in Stockholm. The tram network grew steadily over the years into a vast network that covered most parts of the city and also connecting some of the nearby suburbs to the city center.

In 1950 there were 21 tram lines in Stockholm and its nearest suburbs with a total of 212 stations (Eriksson 1991), see figure 3 for a map showing the tram network in 1956 for the city center. However, after the number of automobiles became more abundant and also after the introduction of the motor bus, more people thought that the trams were something of the past. In 1950 the first metro line was also opened and it was seen as the new and modern public transport mode which helped to cement the picture that trams were not part of the future. Because of these circumstances and also other factors, tram lines began to close down in favor of buses and new metro lines in the 1950’s. A decision was made that no new investments should be made in the tram network.

In 1967 Sweden decided to change from driving on the left hand side of the road to right-side traffic (Aspenberg 1998). This was the last blow for the trams in Stockholm since they were an integrated part of the traffic and it would therefore cost a significant amount of money to change the trams into right hand traffic. Due to this all the remaining tram lines in the city center of Stockholm were closed down in 1967. The only remaining lines after this year were the two lines on Lidingö (the lines were shortened to Ropsten in 1967 however), and also the last part of line 12 going from Alvik to Nockeby west of Stockholm. In 1971 the northern of the two tram lines on Lidingö was closed down.

The southern line on Lidingö and the tram line between Alvik and Nockeby are still in operation today.

Figure 3. Map showing the public transport network in 1956 for the city center of Stockholm. The Blue lines on the map are tram lines and the reds are busses (http://media.sparvagsmuseet.se/kartor/Karta_1956_staden/1956_staden.html).

4.1.1.3 Spårväg City In later years it has become obvious that tram and light rail lines have advantages in a densely populated city. This is because that trams have a high capacity and are environmental friendly since they are driven with electricity. Due to this the first leg of a new tram line was opened in the city center of Stockholm in 2010, it is called Spårväg City (SLL 2014).

The first leg is going between Sergels Torg close to the central station and Waldermarsudde on the island Djurgården. The trams are driving on sight and are not fully separated from the rest of the traffic.

Future Extensions Spårväg City will be extended in the future, the line will be prolonged north from the station Djurgårdsbron to the metro station Ropsten, Ropsten is also the terminating station on the old tram line on Lidingö (Lidingöbanan) (SLL 2014). The plan is to connect Spårväg City to Lidingöbanan making it possible to travel from Lidingö to the Stockholm city center without transfers, see Figure 4. Close to Ropsten a new residential area called Norra Djurgårdsstaden is under construction and the plan is to finish the extension of the tram/light rail line until the new area is completed, Norra Djurgårdsstaden is mentioned as one of the growth areas for the polycentric development in the land use development plan from 2010 (Stockholms Stad 2010).

It is also decided that in 2018 Spårväg City will be extended to the west from today’s station Sergels Torg to a new station also called Sergels Torg but which will be located closer to the central station. There are also plans on extending the line west to the island Kungsholmen. However, this extension is not decided upon and is therefore not included in this study.

Figure 4. Map over the future length of Spårväg City including Lidingöbanan (www.sll.se/upload/Trafikf%C3%B6rvaltningen/Bygga%20kollektivtrafik/Sp%C3%A5rv%C3%A4g%20City/Karta_Sergels%20t org-G%C3%A5shaga%20brygga_V3.pdf).

4.1.1.4 Tvärbanan The evolution that was described earlier with the rail network used as feeder routes from satellite towns to the city center (Cervero 1995) has led to a lack of transverse cross-connections between lines outside the city center. This has led to that most transfers between lines have been forced into the city center leading to overcrowding at some stations and also unnecessary long journeys for many travelers (SLL 2014a). In order to some extent relief this problem a new tram/light rail line was opened in 2000. The line is called Tvärbanan and goes between important stations outside the city center of Stockholm. Today the line goes from Solna Centrum in the north to Sickla Udde south-east of the city. On its way the tram/light rail line connects many important stations such as Sundbyberg

(commuter trains, regional trains and metro), Alvik (metro), Liljeholmen (metro) and Gullmarsplan (metro).

On some parts of the line the trains are operated on sight and are integrated with the traffic, on other parts the trains are driven with a signaling system on its own embankment. The line has today 23 stations and the total length is 18 kilometers meters.

Future Extensions In order to create more connections between lines and modes as well as to expand the coverage of the rail bound public transport network, Tvärbanan will be further extended the future (SLL 2014a). In 2014 the line will be extended one stop from Solna C to Solna Station. In 2017 a new branch will be built splitting from the existing line in Norra Ulvsunda and terminate at the commuter train station Helenelund, on its way the new line will go through the metro stations Rissne and Kista, see figure 5 for the map of the line.

Tvärbanan will also be extended in the south from Sickla Udde to Sickla Station on Saltsjöbanan.

Figure 5. Map over the future extension of Tvärbanan to Kista, the new line is colored black in the figure (http://www.sll.se/Handlingar/Trafikn%C3%A4mnden/2011/maj/%C3%84rende%209%20TN%201105- 117%20F%C3%B6rslag%20str%C3%A4ckning%20Tvb%20Norr%20Kistagrenen.pdf).

As mentioned earlier, Adolphson (2009) discussed the fact that the City of Stockholm wants to develop in a polycentric way. Areas like Sundbyberg, Kista and the area around Globen are all areas mentioned in the text as urban nucleus that could grow and aid the polycentric development. All of these three examples have gotten improvements in their public transport accessibility due to extensions of Tvärbanan. The line was built to Globen in 2000, Sundbyberg in 2013 and in 2018 Tvärbanan is to be extended to Kista. Due to this Tvärbanan has helped Stockholm to develop in a polycentric way.

4.1.1.5 Saltsjöbanan Saltsjöbanan was opened as a privately owned railway in 1893 and connected the new recreational suburbs in Saltsjöbaden to Stockholm (Landgren 1993). The railway consisted of one main branch from Slussen in the city center to Saltsjöbaden and one small branch connecting Solsidan to Igelboda on the main branch. There have not been any extensions or closure on Saltsjöbaden during the whole 20th century. The network has 18 stations and the total length is 18.6 kilometers.

4.1.1.6 Roslagsbanan

Historical Development In 1885 the company Stockholm-Rimbo Jernvägsaktiebolag (SRJ) opened a railway line between Stockholm Östra and Rimbo (SLJ 1985). This was the start of a vast network of railway lines that covered most part of the Uppland County, north of Stockholm. Roslagsbanan was a privately owned railway that transported both passengers and freight. In 1937 it grew into its maximum size reaching the city of Uppsala in the west, Hallstavik in the north, Norrtälje in the east and connecting to Engelbrektstorg in the city center of Stockholm in the south. It also had two branches going into the suburb of Djursholm and one branch to Österskär.

Roslagsbanan was built as a narrow gauged railway, with a gauge of 891 millimeters compared with 1492 millimeters which is the standard gauge that was used on most of the railways in Sweden.

After the Second World War, the automobile and motorbus became more popular leading to a decreasing market share for the railway, Roslagsbanan was no exception with some of the lines becoming less profitable (SLJ 1985). This led to the closure of some of the lines. In 1960 the short line between Engelbrektstorg and Östra Station closed, this part of Roslagsbanan went on the city street alongside the regular traffic. More and more lines were closed down and in 1969 all the lines north of Rimbo had been closed, making Rimbo the terminating station to the north. Later (1976) one of the two lines to the suburb Djursholm closed down as well and also a small feeder route to Stocksund was closed during this period. In 1981 the line between Kårsta and Rimbo closed as well and since then the network has remained in its current form.

Between 1967 and 1973, there was no rail bound connection to Roslagsbanan’s terminus Östra Station. In order to avoid getting a disconnected network in the study, bus links were added to Östra Station. The bus links connected Östra Station with the two nearby metro stations Odenplan and Karlaplan. These links were given a relatively high travel time since buses are assumed to have a lower average speed than rail bound modes.

Today’s Network Roslagsbanan is today a narrow gauged network consisting of three lines all terminating at the station Stockholm Östra that also has a connection to the red metro line (SLJ 1985). The three lines

28 terminate at Kårsta, Österskär and Näsbypark. The total length of the network is today 65 kilometers and there are 38 stations.

4.1.1.7 Commuter Trains

Historical Development The state owned railway company SJ has run local trains from Stockholm to its suburbs ever since the end of the 19th century (Hällqvist, 2008). During the last century the demand for train connections grew and the number of train departures a day grew as well. In 1950 SJ had three main lines going form peripheral suburbs in to central Stockholm. In 1968 the newly founded company SL took over the operation of the commuter trains with lines connecting Stockholm with Södertälje and Kungsängen. In 1969 Märsta was introduced in the commuter train network and a few years later, in 1973, commuter trains started to go to Haninge along Nynäsbanan. SL introduced a unified fleet and also a fixed time table, these improvements helped to increase the ridership and made the commuter train network as we know it today.

Today’s Network Today commuter trains are often used to travel to the outermost suburbs. As of today there are four commuter train lines, they all use the conventional railway network, there are often however designated tracks that go next to the main lines that only are being used by the commuter trains (Fröidh et al 2011). When comparing the commuter trains to the metro, the speeds are generally higher and the distances between stations are generally longer.

Future Extensions In order to release capacity on the over saturated railway through Stockholm, a tunnel is being built underneath the city (Trafikverket 2014). Today commuter trains, freight trains and other long distance trains all share the same two tracks going from the central station south through the city. In the future when the tunnel is opened the commuter trains will be removed from the old tracks and only use the tunnel. A part from being beneficial for the freight and long distance trains, it will also be beneficial for the commuter trains since it will be possible to run more commuter trains each hour. When the tunnel is complete the station Karlberg will be removed and replaced by a new station that is being built underneath the metro station Odenplan (Figure 6). This will make transfers easier between the metro network and the commuter trains. The tunnel is planned to open for traffic in 2017.

Figure 6. Map over Citybanan. Citybanan is showed as the dotted line with the future stations showed as blue and white boxes (http://www.trafikverket.se/PageFiles/35906/karta_citybanan_stor1.jpg).

4.1.1.8 Regional Trains There are many trains departing from the central station in Stockholm each day. Many of them travel to nearby cities such as Uppsala, Södertälje and Västerås. The lines that are included in this project are the regional trains to Uppsala stopping in Märsta and Knivsta, to Västerås stopping in Sundbyberg, Bålsta and Enköping and also to Södertälje Syd stopping in Flemingsberg, the reason for excluding the other train lines operating from the central station is the threshold of 23 departures a day that was mentioned earlier.

In 1999 a new railway line was constructed to Arlanda Airport, the main international airport in Stockholm and Sweden (Jarnvag.net 2010). Using the new tracks and also parts of already existing tracks, the company A-Train AB started to run trains between the airport and Stockholm Central Station. The trains stops at 2 different stations at the airport and the travel time between Arlanda and Stockholm is 20 minutes.

4.1.1.9 Summary To summarize the Stockholm rail bound public transport network a map (figure 7) and a table (table 2) are presented below. The map is showing today’s network excluding the regional train lines

30 discussed above and the table shows the evolution of all the lines that will be included in the network between 1950 and 2025.

Figure 7. Stockholm’s rail bound public transport network in 2014 http://sl.se/ficktid/karta/vinter/SL_Sp%C3%A5rtrafik.pdf).

Table 2. The complete topological development of the lines in the Stockholm rail bound public transport network (<1950 means before 1950).

Mode Line Year Extensions Closures Metro Gröna Linjen 1950 Slussen – Hökarängen 1951 Gullmarsplan – Stureby 1952 Hötorget – Vällingby 1954 Stureby – Högdalen 1956 Vällingby – Hässelby Gård 1957 Slussen – Hötorget 1958 Hökarängen – Farsta Skärmarbrink – Bagarmossen Hässelby Gård – Hässelby Strand 1959 Högdalen – Rågsved 1960 Rågsved – Hagsätra 1971 Farsa – Farsta Strand 1994 Bagarmossen - Skarpnäck Röda Linjen 1964 T-Centralen – Fruängen Liljeholmen – Örnsberg 1965 T-Centralen – Östermalmstorg Örnsberg – Sätra 1967 Sätra – Vårberg Östermalmstorg - Ropsten 1972 Vårberg – Fittja 1973 Östermalmstorg – Tekniska Högskolan 1975 Fittja – Norsborg Tekniska Högskolan – Universitetet 1978 Universitetet – Mörby Centrum Blåa Linjen 1975 T-Centralen – Hjulsta 1977 Hallonbergen – Akalla T-Centralen - Kungsträdgården 1985 Västra Skogen - Rinkeby Hallonbergen - Rinkeby 2021 Akalla – Barkarby Station 2025 Kungsträdgården – Nacka C Sofia – Hagsätra Arenastaden 2020 Odenplan - Hagastaden Linjen 2022 Hagastaden - Arenastaden Tram 1 <1950 Norrmalmstorg - Rosenlund 1964 Norrmalmstorg - Rosenlund 2 <1950 Fredhäll – Karlaplan 1952 Rålambsv/V Rydbergsg – Fridhemsplan Rålambsv/V Rydbergsg - (Via Västerbroplan) Fridhemsplan 1963 Fredhäll - Karlaplan 3 <1950 Haga - Heleneborgsgatan 1960 Haga - Heleneborgsgatan 4 <1950 Vitabergsparken – Västerbron – Oxenstiernsg 1952 Ringvägen - Oxenstiernsg Vitabergsparken – Västerbron - Oxenstiernsg 1967 Ringvägen - Oxenstiernsg 5 <1950 Karlberg – Östra Station 1962 Karlberg – Östra Station 6 <1950 Roslagstorg – Sofia 1967 Roslagstorg - Sofia 7 <1950 Vanadisplan - Djurgården 1967 Vanadisplan - Djurgården 9 <1950 Karlberg – Danvikstull 1954 Sofia - Danvikstull 1957 Karlberg - Sofia 10 <1950 Ropsten – Hornstull

1960 Hornsgatan/Varvsgatan – Hornsgatan/Varvsgatan - Heleneborgsgatan Hornstull 1967 Ropsten - Heleneborgsgatan 11 <1950 Tegelbacken - Islandstorget 1952 Tegelbacken - Islandstorget 12 <1950 Tegelbacken - Nockeby 1952 Tegelbacken - Alvik 15 <1950 Norra Bantorget - Sundbyberg 1959 Norra Bantorget - Sundbyberg 16 <1950 Slussen – Mälarhöjden 1964 Slussen - Mälarhöjden 17 <1950 Slussen - Personnevägen 1952 Personnevägen - Västertorp 1956 Västertorp – Fruängen 1964 Slussen - Fruängen 19 <1950 Skärmarbrink – Örby 1951 Skärmarbrink - Örby 20 <1950 Humlegården - Kyrkviken 1967 Humlegården - Ropsten 1971 Ropsten - Kyrkviken 21 <1950 Humlegården - Gåshaga 1967 Humlegården - Ropsten 8 1952 Södersjukhuset - Tessinparken 1967 Södersjukhuset - Tessinparken 13 1954 Fridhemsplan - Mälarhöjden 1964 Fridhemsplan - Mälarhöjden 14 1954 Fridhemsplan - Västertorp 1956 Västertorp – Fruängen 1964 Fridhemsplan - Fruängen Spårväg City 2010 Sergels Torg - Waldermarsudde 2017 Djurgårdsbron – Frihamnen Sergels Torg – Sergels Torg/Centralstationen 2018 Frihamnen – Ropsten Tram/Light Rail Tvärbanan 2000 Gullmarsplan – Alvik 2002 Gullmarsplan – Sickla Udde 2013 Alvik – Solna Centrum 2014 Solna Centrum – Solna Station 2017 Sickla Udde – Sickla Station 2020 Norra Ulvsunda – Helenelund Station Local Railway Saltsjöbanan <1950 Slussen – Saltsjöbanan Igelboda – Solsidan Roslagsbanan <1950 Engelbrektsgatan – Östra Station Östra Station - Lindholmen Djursholms Ösby – Eddavägen Djursholms Ösby - Näsbypark Roslags Näsby - Österskär Stocksund - Långängstorp 1957 Vallentuna - Lindholmen 1960 Engelbrektsgatan – Östra Station 1965 Vallentuna - Lindholmen 1966 Stocksund - Långängstorp 1976 Djursholms Ösby - Eddavägen Regional Train 1950 Stockholm C – Uppsala 1960 Stockholm C – Södertälje S 1999 Stockholm C – Arlanda Central 2004 Stockholm C - Västerås

4.1.2 Time tables In order to find the travel time between the stations each year, timetables for the different lines were looked for. The timetables were found in different archives in Stockholm. The archives that were visited were Spårvägsmuseet’s Archive, Stockholm Stadsmuseum’s Archive, Svenska Järnvägsklubben’s Archive and Kungliga Biblioteket’s Archive. The most timetables were found in the archive in Kungliga Biblioteket. Some troubles occurred however when searching for timetables for the tramlines before 1967, the few that was found only mentioned the departure time at the first station and also the headway. A few time tables that showed the travel time between some of the stations were found however and by using these travel times and also the distance between the stations, the travel time could be estimated between all stations in the network. Due to time limitations, time tables from every year were not looked at. Instead 7-8 years was chosen to be a good interval. Also if it was known that a major improvement had happened on a line, for example a new signaling system or new trains, then time tables for that year were looked for as well. An example of this is the new commuter trains that were introduced in 1968.

Other information that was gained from the time tables was the number of daily departures for the lines. 23 departures a day in each direction was chosen to be a threshold in order to include or exclude lines in the network. The table below (table 3) shows the amount of daily departure in each direction for some of the lines in the network.

Table 3. The daily amount of departures in each direction. If the number was found to be above 23 the line was included in the network. >23 means that there are more departures than 23 and the line is therefore included, - means that there are no information if there are more than 23 departures and the line is then excluded.

Line Year

1950 1954 1957 1960 1965 1970 1972 1977 1981 1987 1990 1992 1994 1995 2000 2002 2003 2007 2014

Commuter Trains Stockholm C – ------>23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 Märsta Stockholm C - 20 - 20 - 22 - - 43 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 Västerhaninge Västerhaninge ------12 19 - - 21 - 26 >23 >23 >23 >23 - Nynäshamn Södertälje C ------17 - - 20 - 21 24 >23 >23 >23 Järna Södertälje C ------17 - - 20 - 18 21 - - 22 Gnesta Local Railways Saltsjöbanan 35 >23 33 >23 32 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 Stockholm Ö – 17 - 18 - 10 9 - 0 0 0 0 >23 0 0 0 0 0 0 Rimbo Stockholm Ö - 23 >23 21 - 24 >23 23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 Lindholmen Stockholm Ö – 37 >23 24 >23 27 >23 24 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 Vallentuna Roslags Näsby 36 20 19 - 24 >23 23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 – Österskär Roslags Näsby >23 25 25 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 – Rydbo Roslags Näsby >23 29 31 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 - Viggbyholm Roslags Näsby >23 >23 >23 >23 37 >23 >28 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23

– Hägernäs Stocksund – 50 >23 42 >23 39 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Långängstorp Djursholms Ö 61 >23 43 >23 41 >23 39 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 – Näsbypark Djursholms Ö 42 >23 40 >23 37 >23 39 0 0 0 0 0 0 0 0 0 0 0 0 – Eddavägen Rimbo – 11 - 12 - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Norrtälje Regional Trains Stockholm C – 28 31 - 32 >23 27 - >23 45 >23 42 41 >23 47 53 >23 52 >23 >23 Uppsala Stockholm C – 17 20 - >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 >23 31 39 >23 Södertälje S Stockholm C – 11 11 - 10 - 11 - - - - 20 14 - 16 21 - 27 25 >23 Västerås Stockholm C - - 5 - - - 6 - - - 11 5 5 - 0 18 22 19 - - Eskilstuna

5 Results The results in this study are presented in the section below. First the result for every network indicator is presented one by one and then the correlations between some of the indicators are shown.

5.1 Number of Nodes and Edges The number of nodes is a basic measure of a network’s size. As seen in figure 8, the number of nodes changes substantially over the years.

Number of Nodes 350 330 310 290 270 250 230 No of Nodes 210 190 170 150 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 8. Number of nodes each year.

In the first 10 years the amount of nodes remained relatively stable, then the closure of the many tram lines led to a sharp decline in the number of stations. The tram lines had a high density of stations which led to the sharp decline. After 1971, the number of nodes slowly started to grow again. This was mainly due to the opening of new metro and commuter train lines and also after the year 2000 the addition of new tram / light rail lines such as Spårväg City and Tvärbanan.

Even though the number of stations has grown since 1971, there were more stations in 1950 than it will be in the year 2025.

A similar indicator of a network’s size to the number of nodes is the number of edges; the number of edges each year is shown in figure 9.

Number of Edges 350 330 310 290 270 250 230 No of Edges 210 190 170 150 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 9. The number of edges each year.

The evolution follows a similar trend as the number of nodes. It can be observed however that in relation to the number of nodes, the number of edges increases more in the last decades of the study. This relation between the number of edges and number of nodes is something that connectivity is measuring.

5.2 Connectivity Connectivity is a measure of how well intra-connected a network is. In order to measure connectivity in this study, the gamma index was used. In figure 10 the gamma index for each year is presented.

Connectivity 0.375

0.37

0.365

0.36

0.355

0.35 Gamma Index

0.345

0.34

0.335 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 10. The development of Gamma Index.

It can be seen that the evolution of the gamma index to some extent follow the evolution of the number of nodes and edges. This is especially true in the 20th century. When the tram network was closed down the connectivity declined as well. This is because many stations in the tram network were high degree nodes in comparison to the other nodes in the network.

When the small but steady increase of nodes began at the beginning of the 1970’s, the connectivity increased as well. This meant that the nodes that were added in general had a relatively high degree.

During the last years when Tvärbanan was built the connectivity increased more comparing to the number of nodes. This is because Tvärbanan connected many different metro and commuter train lines resulting in a high number of high degree stations.

An increase in the number of nodes is not automatically equal to an increase in gamma index. This can for example be seen the year 2010 when the first leg of Spårväg City was completed from Sergels Torg to Waldermarsudde. This meant an increase in the number of stations but since most of the new stations had one or two degrees, the addition of these nodes resulted in a decreased gamma index.

5.3 Clustering Coefficient The clustering coefficient measures how well connected each node’s neighbors are. The evolution of the average clustering coefficient can be seen in figure 11.

Average Clustering Coefficent

0.014

0.012

0.01

ng Coefficient 0.008

0.006

0.004

Average Clusteri 0.002

0 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 11. Average Clustering Coefficient for each year.

As seen in figure 11 the average clustering coefficient is very low, in 39 of the 76 years every node in the network have a clustering coefficient of 0. In other years only a few stations in the network have a clustering coefficient greater than 0. This was somewhat expected since a majority of stations in the networks are located along branches with a degree centrality of 2. A high percentage of stations having two degrees mean that the possibility of a node’s neighbors to be connected is relatively low. Due to this correlation between the average clustering coefficient and the average degree, the two

39 indicators somewhat follows the evolution of each other with peaks in the first and last decades of the study period.

From the year 2020 up until 2025, the average clustering coefficient increases substantially. This is because of the extended Tvärbanan and the new metro lines. These extensions result in a more intra-connected network in some parts which increases the clustering coefficient for some nodes.

A table was made for six important years in the study period. The years chosen were 1950, 1967, 1985, 2000, 2010 and 2025. 1950 was chosen because it’s the first year, 1967 is the year when the tram network was closed down which led to dramatic changes in network indicators, 1985 is when the blue metro line was completed, 2000 is the year when the first leg of Tvärbanan opened, Spårväg City opened in 2010 and 2025 is the last year of the study. In the table below the five stations with the highest clustering coefficient are presented for the six years, the value of the clustering coefficient is also shown for the top five stations.

Table 4. The table shows the five stations with the highest clustering coefficient for six different years in the study period. The value of the clustering coefficient is shown in brackets after the station’s name. The average clustering coefficient and standard deviation are shown for the 6 years as well.

Clustering Coefficient Year 1950 1967 1985 2000 2010 2025 Top 5 1 Hantverkargat - - - Karlberg Nacka (1.0) stations an/ (0.333) St Eriksgatan (clustering (0.333) coefficient 2 St Eriksg - - - Sundbyberg Spånga (1.0) in /Drottninghol (0.0667) brackets) msv. (0.333) 3 Fridhemsplan - - - Centralstation Hagastaden (0.167) en (0.0152) (1.0) 4 - - - - - Solna Station (0.167) 5 - - - - - Saltsjö-Järla (0.167) Average 0.002723 0 0 0 0.001694 0.014057

Standard 0.0285 0 0 0 0.021717 0.107008 Deviation

As seen for three of the years, the clustering coefficient is equal to 0 for all nodes in the network. In most of the other years only a few stations have a clustering coefficient above 0. The exception is 2025 where there are more stations with a relatively high clustering coefficient. This is due to the new metro lines and also the extension of Tvärbanan. Even though more stations get higher clustering coefficient in 2025, 257 of 267 still have a clustering coefficient of 0 in 2025.

5.4 Degree Centrality The average degree and standard deviation for each year can be seen in figure 12 and figure 13.

Average Degree with Standard Deviation 3.3 3.1 2.9 2.7 2.5 2.3 2.1 Degree 1.9 1.7 1.5 1.3 1.1 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 12. The evolution of average Degree Centrality with standard deviation.

Average Degree 2.22

2.18

2.14

2.1 Degree 2.06

2.02

1.98 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 13. The evolution of Average Degree Centrality.

The trend that can be observed is that the average degree was relatively stable in the first 15 years of the study, it then decreased when the tram network was closed down in the late 1960’s. This was due to the fact that the tram network had relatively high degree nodes, and when these were closed down the average degree decreased. After 1970, the average degree in the network slowly increased. In most cases the addition of new lines led to an increase in the average degree of the network, an exception to this was in 2010 when the opening of Spårväg City led to a decrease in the average degree. Since 2000 the average degree has increased substantially, this can mainly be traced to Tvärbanan that connects many different lines resulting in more high degree stations. The increasing trend is even larger in the last 10-15 years of the study, this is due to the extension of Tvärbanan to Kista and also other additions to the network such as new metro lines and the extension of Spårväg City.

The standard deviation tends to grow when the average degree grows. This is something that was expected since most nodes have 2 degrees and if more high degree nodes are added, the more spread out the entire sample gets.

5.4.1 Cumulative Distribution In order to see how the nodes’ degree for the different years are distributed in the network a cumulative distribution for each year is presented in figure 14.

Cumulative distribution of Degree 1

0.9

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0.5 Percentage 0.4

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0 1 2 3 4 5 6 7 8 9 10 11 12 Degree 1950 1951 1952 1954 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1975 1976 1977 1978 1982 1985 1986 1987 1993 1994 1995 1996 1999 2000 2001 2002 2003 2006 2008 2009 2010 2012 2013 2014 2017 2018 2020 2021 2022 2025

Figure 14. Cumulative Distribution of Degree Centrality for every year.

The trend that can be seen in the plot is that for the first decade in the study period the distribution was relatively widely distributed. This means that there was a relatively large share of nodes having more than 2 degrees, in the first decade. In the 1960’s when the tram network closed down the distribution changed into being not as widely distributed with fewer stations having more than 2 degrees.

After 1971, this trend reversed into a wider distribution with more stations having more than 2 degrees. This trend is accentuated after the year 2000 when Tvärbanan was built and with the addition of the new lines in the last years of the study period the distribution gets even wider than in the 1950’s. Another difference between the last years’ distribution and the first years’ is that the last years’ have a higher maximum node degree. Tvärbanan’s large impact on the degree distribution was

43 due to the fact that it connects many important stations and lines to each other which created high degree nodes. These nodes include Liljeholmen, Gullmarsplan, Alvik and Sundbyberg.

The transition for degree centrality from being more widely distributed in the first decades to being less distributed after 1967 and then being widely distributed again in the final years can also be seen in figure 12 were standard deviation is following this trend.

5.4.2 Spatial Analysis of Network Evolution Maps were made in order to see how degree centrality was distributed among the nodes in the network. Maps are presented for the six years showed in table 4, the years are 1950, 1967, 1985, 2000, 2010 and 2025.

For each year, one map showing the whole network and one map zoomed in to the central parts is presented. The zoomed in maps are included because they make it easier to see the different colors of the nodes.

1950 The maps showing degree centrality for nodes in 1950 are shown below (figure 15).

Slussen

Figure 15. Maps that show degree centrality for every node in the network in 1950. The more yellow a node is the higher its degree is.

In the map from 1950 it can be seen that there were many stations located in the central parts of the network. Most of these nodes were stations in the tram network. In comparison to other modes, many tram stations had high degree.

The node with the highest degree in 1950 was Slussen with 6 degrees. Slussen worked as the main hub in the tram network and was also served by the metro and Saltsjöbanan. Other high degree nodes in 1950 were Centralstationen, which was the main train station in Stockholm, Tegelbacken and Norrmalmstorg. Tegelbacken and Norrmalmstorg both served as important hubs in the tram network. Another node with high degree in the network was Djursholms Ösby, it was a transfer station located on Roslagsbanan.

1967 The maps below (figure 16) are from the network in 1967.

Centralstationen

Figure 16. Maps that show degree centrality for every node in the network in 1967. The more yellow a node is the higher its degree is.

As seen, a majority of the centrally located nodes were gone. This was due to the closure of the tram network earlier the same year. The old hubs in the tram network Norrmalmstorg and Tegelbacken were gone and the most centrally located node from 1950, Slussen, had decreased its degree to 4. In 1967, Centralstationen was the main hub in the metro network with both the red and green metro lines going through the station, it also still served as the main regional and commuter train station. This meant that Centralstationen was the highest degree node in 1967 with 7 degrees. Apart from Centralstationen other high degree nodes in the network were Slussen and Djursholms Ösby which was an important station on Roslagsbanan.

1985 Figure 17 shows degree centrality for the nodes in the network in 1985.

Centralstationen

Figure 17. Maps that show degree centrality for every node in the network in 1985. The more yellow a node is the higher its degree is.

In 1985, apart from one station, the metro network as we know it today (2014) was completed. The network was a radial network with different branches going out from the city center. Many lines reached the central part of the network in Centralstationen making it an even more important station than in 1967. Centralstationen’s degree was 9 in 1985. Apart from Centralstationen, Slussen was also an important hub with many metro lines and it was also the terminus for Saltsjöbanan. With the creation of the blue metro line, Fridhemsplan also emerged as a hub in the network. With its 4 degrees, transfers between the green and blue metro lines were possible there. Another node that got higher degree due to the blue metro line was Sundbyberg, in 1985 it had 4 degrees with connections to both commuter trains and the blue metro line.

2000 In the year 2000, Tvärbanan opened from Gullmarsplan to Alvik. The network showing degree centrality for the nodes can be seen in figure 18.

Centralstationen

Figure 18. Maps that show degree centrality for every node in the network in 2000. The more yellow a node is the higher its degree is.

Centralstationen, with 10 degrees, was still the station with the highest degree. The increase in Centralstationen’s degree was due to the opening of the direct connection between Arlanda and Centralstationen. Other important hubs in 2010 were Liljeholmen with 5 degrees and Alvik with 4 degrees. Both of these stations were located along the newly opened cross radial line Tvärbanan.

2010 In figure 19 the degree for the network in 2010 can be seen.

Centralstationen

Figure 19. Maps that show degree centrality for every node for the network in 2010. The more yellow a node is the higher its degree is.

In 2010, Centralstationen’s degree centrality had increased even more to 12. This was due to the new regional train line to Västerås and also the opening of Spårväg City. In 2010 Tvärbanan was also extended to Sickla Udde increasing Gullmarsplan’s degree to 4.

2025 The maps from the last year of the study period are shown in figure 20.

Centralstationen

Figure 20. Maps that show degree centrality for every node in the network in 2025. The more yellow a node is the higher its degree is.

In 2025, the highest degree node will still be Centralstationen with 12 degrees. The extension of Tvärbanan to Helenelund and Solna Station will create more hubs outside the city center of Stockholm. These hubs include Helenelund (3 Degrees), Kista (4 degrees), Rissne (4 degrees), Solna Station (4 degrees), Solna Centrum (4 degrees) and the hubs on the older part of Tvärbanan. One station that will increase its degree from 2010 is Odenplan. This is because the commuter trains will stop there in 2025 and also a new metro line will have its terminus at Odenplan. In 2025 Odenplan will have 5 degrees.

Summary To summarize important information from the maps, the 5 stations with the highest degree centrality for every map are presented in table 5, the degree for the top five stations is shown in brackets. Average degree and standard deviation are also shown in the table.

Table 5. The table shows the five stations with the highest degree centrality from the six years showing in the maps, the degree for the top five stations are shown in brackets. Average degree centrality and standard deviation are also presented.

Degree Centrality Year 1950 1967 1985 2000 2010 2025 Top 5 1 Slussen Centralstatione Centralstation Centralstation Centralstation Centralstatio stations (6) n (7) en (9) en (10) en (12) nen (12) 2 Centralstation Slussen Slussen Liljeholmen Sundbyberg Sundbyberg en (5) (4) (4) (5) (6) (8) 3 Östra Station Djursholms Fridhemsplan Slussen Liljeholmen Gullmarsplan (5) Ösby (4) (4) (4) (5) (6) 4 Skanstull Gullmarsplan Sundbyberg Fridhemsplan Slussen (4) Odenplan (6) (4) (3) (4) (4) 5 Engelbrektspl Skärmarbrink Gullmarsplan Alvik Gullmarsplan Alvik an (4) (3) (3) (4) (4) (5) Average 2.085 2.011 2.039 2.053 2.082 2.202 Standard 0.647 0.586 0.683 0.751 0.879 1.013 Deviation

5.5 Betweeness Centrality

5.5.1 Cumulative Distribution In order to see how betweeness centrality was distributed in the networks, a cumulative distribution for betweeness centrality was made, it can be seen in figure 21. The plot shows the distribution for each year in the study. Normalized Betweeness centrality was used in order to make the comparison between years easier.

Cumulative Distribution over Betweeness Centrality 1

0.9

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Percentage 0.4

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0.2

0.1

0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Percentage Betweeness Centrality

1950 1951 1952 1954 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1975 1976 1977 1978 1982 1985 1986 1993 1994 1995 1996 1999 2000 2001 2002 2003 2006 2008 2009 2010 2012 2013 2014 2017 2018 2020 2021 2022 2025

Figure 21. Cumulative Distribution of Betweeness Centrality for each year.

It can be observed that during the first years in the study, the node’s betweeness centrality is not so widely distributed, this means that relatively few nodes had a high betweeness centrality and many nodes had a relatively low betweeness centrality. When fewer nodes are having a high betweeness

52 centrality in a network, there are a few nodes that are the most central and important to the networks overall function.

During the 1960’s and 1970’s the distribution got more distributed. This means that a higher percentage of nodes got high betweeness centrality and were therefore more important to the networks performance. This trend reversed during the last decades of the study, resulting in that the last years have a similar distribution to the first years with a smaller percentage of nodes having a relatively high betweeness centrality.

A reason for the trend discussed above is democratization that was mentioned by Derrible (2012). It means that normalized betweeness centrality becomes lower for all nodes in a network when the number of nodes increases. Democratization can be seen in figure 21, since normalized betweeness centrality was relatively low for most of the nodes in the 1950’s, it then increased for the nodes in the end of the 1960’s and after 1970 it slowly decreased up until 2025. The opposite trend can be seen for the number of nodes and there is therefore an inverse correlation between the overall value of normalized betweeness centrality and the number of nodes.

5.5.2 Spatial Analysis of Network Evolution Maps are presented below that show betweeness centrality for each node in the network. Maps were made for the same six year as for degree centrality; 1950, 1967, 1985, 2000, 2010 and 2025. The more yellow a node is in the maps the higher that node’s betweeness centrality is.

Two maps are presented for each year, first a map showing the whole network and also a zoomed in map over the central parts in order to better see the difference in the node’s betweeness centrality.

1950 In figure 22 the betweeness centrality for the different nodes in 1950 is shown.

Slussen

Figure 22. Maps that show betweeness centrality for every node in the network in 1950. The more yellow a node is the higher its betweeness centrality is.

As seen, the nodes with the highest betweeness centrality were located in the in the central part of the network. The node having the highest betweeness centrality was Slussen (2.27% of the network’s total betweeness centrality), other nodes with high betweeness centrality in 1950 were Norrmalmstorg and Karl XII Torg.

In 1950 there were three routes connecting the northern and southern part of the network. There were the western connection between Hornstull and Kungsholmen along Västerbron and two lines going on each side of Gamla Stan. By inspecting the three routes in figure 22 it is clear that the two routes along Gamla Stan had higher betweeness centrality than the western route. This means that the stations at Gamla Stan were highly utilized in 1950 and due to that, they had high betweeness centrality.

1967 The betweeness centrality for the nodes in 1967 is shown below in figure 23.

Centralstationen

Figure 23. Maps that show betweeness centrality for every node in the network in 1967. The more yellow a node is the higher its betweeness centrality is.

The station with the highest betweeness centrality in 1967 was Centralstationen with 4.31% of the total network’s betweeness centrality. Other nodes with high betweeness centrality were stations located in the central parts of the network such as Slussen (3.21%), Karlaplan (2.90%) and Gamla Stan (2.82%). The reason for Karlaplan’s high betweeness centrality was due to its connection to Roslagsbanan and most of the journeys to and from Roslagsbanan therefore went through Karlaplan. Since the tram network had closed down, every journey that went between the southern and

54 northern part of the network had to go through Centralstationen, this is one of the reasons why its betweeness centrality had increased.

1985 In 1985 the blue metro line as it is known today was completed, the maps showing betweeness centrality for the nodes in 1985 can be seen in figure 24.

Centralstationen

Figure 24. Maps that show betweeness centrality for every node in the network in 1985. The more yellow a node is the higher its betweeness centrality is.

It can be seen that Centralstationen still was the station with the highest betweeness centrality in 1985 (5.0% of the total network’s betweeness centrality). After Centralstationen the two centrally located stations Gamla Stan and Slussen also had high betweeness centrality.

In comparison to 1985, the metro connection to Östra Station and Roslagsbanan was now completed. This meant that there only was one route to go from all other stations to Roslagsbanan, this meant that the most centrally located stations on Roslagsbanan had a high betweeness centrality. In order to go to any destination on Roslagsbanan all passengers had to travel through Östermalmstorg. Due to this, Östermalmstorg had the third highest betweeness centrality in the whole network.

In 1985, Fridhemsplan worked as an important hub in the network since it was connected to both the green and blue metro lines, due to this Fridhemsplan had a high betweeness centrality as well.

2000 In 2000 the first leg of Tvärbanan was completed. The maps showing betweeness centrality from the year 2000 can be seen in figure 25.

Centralstationen

Figure 25. Maps that show betweeness centrality for every node in the network in 2000. The more yellow a node is the higher its betweeness centrality is.

Even though Centralstationen still was the station with the highest betweeness centrality, its betweeness centrality had decreased due to the opening of Tvärbanan, from 5.0% to 4.8%. The reason for the relatively small decrease is partly because of the opening of Arlanda Express that connected Centralstationen and Arlanda leading to an increase in betweeness centrality for Centralstationen. Apart from the decrease in Centralstationen’s betweeness and the increase in the hubs along Tvärbanan (Gullmarsplan, Alvik and Liljeholmen), the betweeness centrality distribution was similar to the one in 1985.

2010 As seen in figure 26, betweeness centrality was quite similar distributed in the network in 2010 as it was in 2000.

Centralstationen

Figure 26. Maps that show betweeness centrality for every node in the network in 2010. The more yellow a node is the higher its betweeness centrality is.

With the addition of the new regional train line to Västerås and Spårväg City, Centralstationen’s betweeness centrality increased from 4.8% to 5.0 %. The extension of Tvärbanan to Sickla Udde had an increasing effect on Gullmarsplan’s betweeness centrality.

In 2006 the station Årstaberg on Tvärbanan opened for commuter trains as well. This made it one of the nodes with the highest betweeness centrality in the whole network.

The stations that followed after Centralstationen in descending order were Östermalmstorg (Due to its connection to Roslagsbanan), Slussen and Stadion (Stadion also had a connection to Roslagsbanan).

2025 In 2025 Tvärbanan will be extended creating more short cuts for travelers, figure 27 shows how betweeness centrality will be distributed among the stations in 2025.

Centralstationen

Figure 27. Maps that shows betweeness centrality for every node in the network in 2025. The more yellow a node is the higher its betweeness centrality is.

Tvärbanan results in a decrease of travelers needing to go through Centralstationen. This will have a decreasing effect on Centralstationen’s betweeness centrality, however the addition of a commuter train line to Uppsala and also more stations on Spårväg City leads to an increasing effect on Centralstationen’s betweeness centrality. All together, Centralstationen’s betweeness increases in 2025 from 5.0% to 5.1%.

Since Sundbyberg has a fast connection to Centralstationen through a regional train line and also the fact that Tvärbanan was extended there makes it a station that many travelers go through when travelling from the north-west to other areas in the network. Due to this, Sundbyberg is the node with the highest betweeness centrality north west of the city center.

In 2025, Odenplan is an important station in the network, this is due to the new commuter train line that passes through the station and also the new metro line going north from Odenplan to Solna Station. This will increase Odenplan’s betweeness centrality from 0.0050% in 2010 to 0.51% of the network’s total betweeness centrality in 2025.

By looking at the map, it can be seen that Roslagsbanan is more vulnerable than many other lines in the network. This is because many stations on Roslagsbanan have a high betweeness centrality in comparison to other lines. This means that a failure on one of these stations would have large negative effects to the network’s performance. The reason why Roslagsbanan’s most central stations have such high betweeness centrality is because there is only one connection from Roslagsbanan to other parts of the network, this connection is the red metro line through Östermalmstorg. Other lines in the network such as metro and commuter train lines often have more stations were it is possible to change to other modes. This allows for alternative routes in case of failure.

Summary Important facts from the maps are summarized in the table below (table 6). The 5 stations with the highest betweeness centrality for each map are presented with the normalized betweeness centrality for each station shown in brackets.

Table 6. The table shows the five stations with the highest betweeness centrality from the six years represented in the maps, the value for each station is shown in brackets.

Betweeness Centrality Year 1950 1967 1985 2000 2010 2025 Top 5 1 Slussen Centralstation Centralstation Centralstation Centralstation Centralstatio stations (2.27%) en (4.31%) en (4.99%) en (4.84%) en (4.96%) nen (5.15%) 2 Norrmalmstor Slussen Slussen Slussen Östermalmsto Östermalmst g (2.0%) (3.21%) (3.20%) (2.90%) rg (2.71%) org (2.77%) 3 Karl XII Torg Karlaplan Östermalmsto Östermalmsto Slussen Stadion (1.86%) (2.90%) rg (3.00%) rg (2.75%) (2.13%) (1.93%) 4 Tegelbacken Gamla Stan Gamla Stan Gamla Stan Stadion Östra Station (1.75%) (2.82%) (2.79%) (2.59%) (1.92%) (1.89%) 5 Stureplan Östermalmsto Östra Station Stadion Östra Station Universitetet (1.65%) rg (2.54%) (2.10%) (2.02%) (1.88%) (1.89%)

5.6 Closeness Centrality with respect to travel time The average closeness centrality with respect to travel time can be seen in figure 28; the values for each node in the networks were derived by dividing 1 and the average journey length to all other nodes in minutes and then an average for each network was calculated. Average Closeness with respect to Travel Time 0.038

0.036

0.034

0.032

0.03

0.028

0.026

0.024

Closenesstravel with respectto time 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 28. Average Closeness Centrality with respect to travel time for each year.

As seen the average closeness was relatively high in the beginning and it then decreased when the tram network was closed down. This was expected since the tram network had relatively high connectivity and closing it down meant longer average journeys in the network. Also the fact that the tram network had many stations located close to each other resulted in a relatively high closeness centrality, this is because stations located close to each other means that travelling between these stations have a decreasing effect on the average closeness centrality. Another reason for the relatively high average closeness centrality in the 1950’s and the early 1960’s was that many of the stations that are located the furthest out in the network were not part of the network at that time, for example Bålsta, Järna, Arlanda and Nynäshamn. Naturally the average journey in the network became longer when these stations were added.

After the sharp decline at the end of the 1960’s the average closeness centrality with respect to travel time slowly started to increase during the following years. This increase got smaller but continued all the way up until 2025, there were a few years when the average closeness decreased however, for example between 1999 and 2001. The main reasons for this were the addition of Arlanda Express in 1999, the commuter train line between Västerhaninge and Nynäshamn in 2000 and the extension of the commuter trains from Kungsängen to Bålsta in 2001. The addition of these lines in the periphery of the network made the average journey time longer and the closeness therefore decreased, it is believed however that the opening of Tvärbanan between Gullmarsplan and Alvik in 2000 to some extent halted this decrease.

5.6.1 Cumulative Distribution In order to see how closeness centrality with respect to travel time was distributed, a cumulative distribution was made for each year. Values for the cumulative distribution were derived by taking the ratio of 1 and the total travel time in minutes to all other nodes in the network. Below in figure 29, the CDF for closeness centrality with respect to travel time is presented.

CDF for Closeness Centrality with respect to travel time 1 0.9 0.8 0.7 0.6 0.5 0.4 Percentage 0.3 0.2 0.1 0 4E-05 6.5E-05 9E-05 0.000115 0.00014 0.000165 0.00019 0.000215 0.00024 0.000265 0.00029 Closeness with respect to Travel Time

1950 1951 1952 1954 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1975 1976 1977 1978 1982 1985 1986 1987 1993 1994 1995 1996 1999 2000 2001 2002 2003 2006 2008 2009 2010 2012 2013 2014 2017 2018 2020 2021 2022 2025

Figure 29. Cumulative Distribution of Closeness Centrality with respect to travel time for each year.

An evolution pattern can be seen in the graph above, in the first years of the study, the distribution is not as widely distributed relative to the other years, this means that the difference in closeness between the nodes at this time was not as large as in other years. Later on in the study period, the distribution of closeness centrality in the network got more spread out meaning that more nodes had a higher closeness centrality and were therefore considered to be more central. This trend is then reversed and in the last decades the majority of nodes are at the same level as in the 1950’s. The main difference however is that during the later years it can be seen that the distribution has a fatter tail than the first years meaning that there are some stations that have a higher value of closeness centrality compared to the 1950’s. These nodes are centrally located stations in the network.

Due to the fact that closeness centrality was derived by taking the total travel time to all other nodes, it is depending on the number of nodes in the network. This is because the total travel time is likely to increase when new nodes are added to a network since there will be more nodes to travel to. When analyzing figure 31, it can be seen that this is the case since the graphs get shifted to the right when the number of nodes decrease and shifted to the left when the number of nodes increase. This means that the average closeness centrality with respect to travel time is dependent on the number of nodes in the network which is explaining the evolutionary trends observed in figure 29.

5.6.2 Spatial Analysis of Network Evolution Maps showing closeness centrality with respect to travel time are presented below. As before, maps are presented from the six years 1950, 1967, 1985, 2000, 2010 and 2025.

For each year one map showing the entire network and one zoomed in to the more central parts are presented. For each map, the more yellow a node is the higher the closeness centrality with respect to travel time is for that node.

1950 Figure 30 illustrates the network in 1950 with closeness centrality with respect to travel time shown for the nodes.

Karl XII Torg

Figure 30. Maps that show closeness centrality with respect to travel time for every node in the network in 1950. The more yellow a node is the higher its closeness centrality is.

It can be seen that stations located in the city center had the highest closeness centrality meaning that the travel time to other nodes generally was shorter. One result that can be obtained from the map is that Alvik had a high closeness centrality even though it was located outside the jurisdictional city center of Stockholm. This is due to its good connections to the central part. The island

Djurgården on the other hand was located in the city center but had a relatively low closeness centrality.

In 1950, the station having the highest closeness centrality was Karl XII Torg close to Kungsträdgården in the central parts of town. This station was a centrally located station in the tram network. Its closeness centrality with respect to travel time was 0.000186. After Karl XII Torg, Tegelbacken and Norrmalmstorg were the most centrally located nodes according to closeness centrality. These stations were also important stations in the tram network.

1967 When the tram network closed down in 1967 the closeness centrality with respect to travel time changed in the network. Figure 31 shows the closeness centrality for the different nodes in 1967.

Centralstationen

Figure 31. Maps that show closeness centrality with respect to travel time for every node in the network in 1967. The more yellow a node is the higher its closeness centrality is.

As seen above, the most centrally located station according to closeness centrality in 1967 was Centralstationen, its closeness centrality was 0.000239. The reason why the most centrally located node had changed to Centralstationen was because the inner city tram lines were gone and therefore most journeys in the network went through Centralstationen. After Centralstationen the most centrally located stations were Hötorget and Östermalmstorg.

The stations along Roslagsbanan had relatively low closeness centrality with respect to travel time even though some stations were located near the central parts of the network. The reason for this was that in 1967 there were no rail bound connection to Roslagsbanan’s terminus Östra Station, and

63 therefore in order to reach Östra Station a bus link had to be used. In comparison to other links, the travel time on the bus link was high making journeys to Roslagsbanan relatively time consuming.

1985 Apart from one station, the metro network as we know it today (2014) was completed in 1985. Maps showing how closeness centrality was distrusted in the network are shown in figure 32.

Centralstationen

Figure 32. Maps that show closeness centrality with respect to travel time for every node in the network in 1985. The more yellow a node is the higher its closeness centrality is.

In 1985, the station with the highest closeness centrality was Centralstationen with a closeness centrality of 0.000264. Other stations that had high closeness centrality were Rådhuset on the blue metro line, Gamla Stan and Östermalmstorg. All of these stations were centrally located stations in the metro network.

In 1985 Östra Station was connected by metro instead of the slower bus links. This meant that traveling to Roslagsbanan was not as time consuming as in 1967. Therefore the stations on Roslagsbanan had higher closeness centrality with respect to travel time than what they had in 1967.

2000 In the year 2000 the first leg of Tvärbanan was completed, below are two maps that show closeness centrality with respect to travel time in 2000.

Centralstationen

Figure 33. Maps that show closeness centrality with respect to travel time for every node in the network in 2000. The more yellow a node is the higher its closeness centrality is.

The station having the highest closeness centrality with respect to travel time in 2000 was Centralstationen with closeness centrality 0.000241. After Centralstationen the most centrally located stations were Gamla Stan, Rådhuset and Kungsträdgården.

It can be seen in the map that stations located on branches that had a connection to Tvärbanan had a higher closeness centrality than in 1985. This was because Tvärbanan made it possible to travel between these lines without having to go in to the city center for transfer.

2010 The map showing closeness centrality with respect to travel time in 2010 can be seen below in figure 34.

Centralstationen

Figure 34. Maps that show closeness centrality with respect to travel time for every node in the network in 2010. The more yellow a node is the higher its closeness centrality is.

The station having the highest closeness centrality in 2010 was Centralstationen with 0.00226. The second, third and fourth stations were Gamla Stan, Kungsträdgården/Hamngatan on Spårväg City and Kungsträdgården on the blue metro line.

It can be seen in the map that the new tram line Spårväg City was geographically located in the center of the network; however its closeness centrality was relatively high. This was due to its relatively slow speed resulting in a high travel time.

2025 The last year in the study is 2025. Among other additions, Tvärbanan is extended to Helenelund. Figure 35 shows closeness centrality with respect to travel time in 2025.

Centralstationen

Figure 35. Maps that show closeness centrality with respect to travel time for every node in the network in 2025. The more yellow a node is the higher its closeness centrality is.

In 2025 Centralstationen will still be the station with the highest closeness centrality (0.000211) followed by Gamla Stan, Hötorget and Kungsträdgården/Hamngatan.

It can be seen that the new links that have been added up until 2025 have made closeness centrality higher for many nodes surrounding the city center. This is especially true for stations that lie in proximity to the extended Tvärbanan.

In 2025 Citybanan is completed making Odenplan a station on the commuter train lines going north from Centralstationen. A new metro line connecting Odenplan to Solna Station is also completed this year. These actions will increase Odenplan’s centrality in the network. It can be seen in figure 37 that Odenplan’s closeness centrality increases between 2010 and 2025 (from 0.000188 to 0.00193).

Summary The five stations with the highest closeness centrality with respect to travel time for each map are presented in the table below (table 7), the value for each station is showed in brackets. The average closeness centrality for the six years is shown as well. Observe that the average closeness centrality is derived by taking the average journey length while the closeness centrality for the individual stations are derived by using the total travel time to all other nodes, thereof the difference.

Table 7. The table shows the five stations with the highest closeness centrality with respect to travel time from the six years showing in the maps, the value for each station is shown in brackets. Average closeness centrality is also shown.

Closeness Centrality with respect to travel time Year 1950 1967 1985 2000 2010 2025 Top 5 1 Karl XII Torg Centralstation Centralstation Centralstation Centralstation Centralstatione stations (1.86E-04) en (0.239E- en (2.64E-04) en (2.41E-04) en (2.26E-04) n (2.11E-04) 04) 2 Tegelbacken Hötorget Rådhuset Gamla Stan Gamla Stan Gamla Stan (1.84E-04) (2.34E-04) (2.55E-04) (2.32E-04) (2.15E-04) (2.03E-04) 3 Norrmalmstor Östermalmsto Gamla Stan Rådhuset Kungsträdgår Hötorget g (1.83E-04) rg (2.33E-04) (2.53E-04) (2.29E-04) den/Hamngat (2.01E-04) an (2.15E-04) 4 Gustav Adolfs Gamla Stan Östermalmsto Kungsträdgår Kungsträdgår Kungsträdgård Torg (1.83E- (2.32E-04) rg (2.52E-04) den (2.29E- den (2.14E- en/Hamngatan 04) 04) 04) (2.01E-04) 5 Skeppsbron / Rådmansgata Hötorget Östermalmsto Rådhuset Kungsträdgårde Slottet (1.82E- n (2.32E-04) (2.50E-04) rg (2.29E-04) (2.14E-04) n (2.00E-04) 04) Average 0.033642 0.02586 0.03006 0.030425 0.030671 0.0316

5.7 Closeness Centrality with respect to Distance The development of the average closeness centrality with respect to geographical distance can be seen in figure 36; the values for each node in the networks were derived by dividing 1 and the average journey length to all other nodes in meters and then the average for each network was calculated.

Average Closeness with respect to Distance 9E-05 8.5E-05 8E-05 7.5E-05 7E-05 6.5E-05 6E-05 5.5E-05 5E-05 4.5E-05 Closenessdistance with respectto 4E-05 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 36. Average Closeness Centrality with respect to distance for each year.

As seen above, the closeness centrality was relatively high during the 1950’s. This was partly due to the extensive tram network that existed at this time. In comparison to other modes the distance between the stations was short for the trams, which made some of the journeys short. The fact that the tram network had many stations with a high degree also meant that the network offered more routes to chose from for travelers and the overall length of a journey could therefore be shorter. Another aspect why closeness centrality was high during the first decade was because many of the

68 most peripheral destinations such as Lindholmen, Arlanda and Järna were not part of the network at this time.

After the tram network started to close down in the 1960’s, closeness centrality decreased. This was again due to the short distance between stations in the tram network which made more trips at that time longer than during the tram era. Also the extension of Roslagsbanan to Österskär and Lindholmen in 1965 enhanced the decrease. During the three decades after 1970, closeness centrality remained relatively stable, this is believed to be due to two different phenomena. First of all the slow improvement of average degree in the network during this time was having a positive effect on the development of closeness centrality. This is because more high degree nodes, in general, leads to shorter journeys in the network. The other phenomena that is believed to have had a negative effect on closeness centrality was the fact that new stations were added at the edges of the network, these new stations meant that the average journey length in the system got longer. Examples of these extensions were Märsta in 1969 and the branch to Västerhaninge in 1973.

In 1999 the average closeness centrality decreased, this was due to the opening of Arlanda Express which increased the average travel distance. In 2000 Tvärbanan opened which had a positive effect on closeness centrality since many travelers now could use Tvärbanan as a short cut. At the same time however the branch between Västerhaninge and Nynäshamn in the far south of the network opened, this had a great negative effect on closeness centrality and due to this the average closeness centrality decreased. The decrease continued in the first decade of the 21st century due the additions of more branches at the edge of the network, these included the commuter train line to Bålsta in 2001, the commuter train line between Södertälje and Järna in 2002 and Regional trains to Västerås in 2003.

In the last 15 years of the study, closeness centrality with respect to distance will increase. This is due to the extension of Tvärbanan and also the opening of new metro lines. These actions will help to increase the average connectivity in the network and due to this more routes will be created and travelers will therefore be able to choose routes that are shorter.

In general it can be said that the main reason why average closeness centrality with respect to geographical distance is the lowest in the last decades of study is because the area covered by the network becomes larger at the end of the study period comparing to the first decades. This has an increasing effect on the average journey length in the network which decreases closeness centrality.

5.7.1 Cumulative Distribution Values for the cumulative distribution of closeness centrality with respect to geographical distance were derived by taking the ratio of 1 and the total distance along the network to all other nodes in the network. The CDF of closeness centrality with respect to distance for each year can be seen in figure 37.

CDF for Closeness Centrality with respect to geographical distance 1 0.9 0.8 0.7 0.6 0.5 0.4 Percentage 0.3 0.2 0.1 0 0 0.0000001 0.0000002 0.0000003 0.0000004 0.0000005 Closeness with respect to Distance

Figure 37. Cumulative distribution of closeness centrality with respect to distance for each year.

It was seen in figure 36 that in the first 10-15 years, the average closeness centrality with respect to geographical distance was high. This can also be observed in figure 37 where it is seen that many nodes had a relatively high closeness in the 1950’s and early 1960’s because the first years’ cumulative distributions are shifted to the right.

Another trend is that the CDF for the networks in the 1950’s have a fatter tail since they had more nodes that were located in the central part, this means that more nodes had a high closeness centrality. The majority of these centrally located nodes were in the 1950’s part of the tram network. After the tram network was closed it can clearly be seen that many nodes with high closeness centrality disappeared in the cumulative distribution.

In the last decades of the study period, there are many nodes with a low closeness centrality. This was also seen in figure 37.

Considering that closeness centrality for the cumulative distribution was derived by using the total journey length to all other nodes, closeness centrality is depending on the number of nodes. This is the reason why the last years’ graphs are shifted to the left which means that closeness centrality with respect to distance was low for many nodes. Even though it can be seen in figure 37 that closeness centrality was low in the last 20 years, relatively it is even lower for many nodes in figure 36. This can be explained by the increasing number of nodes in the last years which have a decreasing effect on closeness centrality.

5.7.2 Spatial Analysis of Network Evolution Maps showing closeness centrality with respect to geographical distance are presented below. As before, maps are presented from the six years 1950, 1967, 1985, 2000, 2010 and 2025. For each year one map showing the entire network and one zoomed in map to the more central parts are presented. For each map, the more yellow a node is the higher the closeness centrality with respect to geographical distance is for that node.

1950 Maps showing how closeness centrality in 1950 was distributed in the network can be seen in figure 38.

Karl XII Torg

Figure 38. Maps that show closeness centrality with respect to distance for every node in the network in 1950. The more yellow a node is the higher its closeness centrality is.

The station with the highest closeness centrality with respect to geographical distance was Karl XII Torg with a closeness centrality of 4.94E-7. Other stations with high closeness centrality were

Tegelbacken and Norrmalmstorg. These were all centrally located stations in the tram network. The top three stations were the same as for closeness centrality with respect to travel time which is shown in figure 30.

Generally in the network, there were many stations with a relatively high closeness centrality, most of these stations were located in the tram network.

In comparison to figure 30 that is showing closeness centrality with respect to travel time, it can be seen that stations along the commuter train lines were less central when closeness was calculated with respect to distance. This is because the commuter trains were faster than most of the other modes in the network.

1967 In 1967, the tram network closed down. Figure 39 shows closeness centrality with respect to distance in the network.

Centralstationen

Figure 39. Maps that show closeness centrality with respect to distance for every node in the network in 1967. The more yellow a node is the higher its closeness centrality is.

The station with the highest closeness centrality with respect to distance in 1967 was Centralstationen with closeness centrality of 4.82E-7. The three nodes that followed were Hötorget, Rådmansgatan and Odenplan. All of these stations were located on the green metro line. One difference between figure 39 and figure 31 showing closeness centrality with respect to travel time is that the stations on Roslagsbanan had higher closeness centrality in the map above. This is because Roslagsbanan’s terminus Östra Station was connected with the rest of the network by bus links with a relatively high travel time. When calculating closeness centrality with respect to travel time these bus links became quite long even though their distance was relatively short. In the map

72 above closeness centrality was derived by looking at distance and Östra Station and other stations along Roslagsbanan therefore got a higher closeness centrality than in figure 31.

1985 Figure 40 shows maps that present the closeness centrality with respect to distance for the network in 1985.

Centralstationen

Figure 40. Maps that shows closeness centrality with respect to distance for every node for the network in 1985. The more yellow a node is the higher its closeness centrality is.

Comparing to 1967, the closeness centrality was similarly distributed in the network. Centralstationen was still the station with the highest closeness centrality (4.288E-7). The stations that followed were Gamla Stan, Slussen and Rådhuset. These were all centrally located metro stations.

2000 Below are the maps that show closeness centrality with respect to distance for the network in 2000.

Centralstationen

Figure 41. Maps that show closeness centrality with respect to distance for every node in the network in 2000. The more yellow a node is the higher its closeness centrality is.

The station that is considered the most central according to closeness centrality in 2000 was Centralstationen with a closeness centrality of 3.848E-7. The three nodes that followed were Gamla Stan, Slussen and Rådhuset. The same phenomenon that was seen in for closeness centrality with respect to travel time in figure 33 can be seen above as well, that is that the stations on branches that are connected to Tvärbanan got an increased closeness centrality. This was because the transverse connections that Tvärbanan offered made many of the trips in the network shorter.

2010 In figure 42 the maps showing closeness centrality with respect to distance for the network in the year 2010 can be seen.

Centralstationen

Figure 42. Maps that show closeness centrality with respect to distance for every node in the network in 2010. The more yellow a node is the higher its closeness centrality is.

The node having the highest closeness centrality in 2010 was Centralstationen with a closeness centrality of 3.14E-7. The following stations in descending order were Gamla Stan, Slussen and Rådhuset.

One difference when comparing figure 42 and figure 34 showing closeness centrality with respect to travel time is that Spårväg City was more centrally located when closeness centrality was derived from distance instead of travel time. This can be explained with the fact that Spårväg City had a relatively slow speed in comparison to other modes which meant that closeness centrality with respect to travel time was lower than closeness centrality with respect to distance for the tram line.

2025 The maps showing closeness centrality with respect to distance for the last year of the study can be seen in figure 43.

Centralstationen

Figure 43. Maps that show closeness centrality with respect to distance for every node in the network in 2025. The more yellow a node is the higher its closeness centrality is.

The station that will have the highest closeness centrality in 2025 is Centralstationen with closeness centrality of 3.00E-7. The stations that follow in descending order are Gamla Stan, Slussen and Hötorget. All of these stations are metro stations located in central parts of the network. It can be seen that nodes that are located close to a connection to Tvärbanan, which in 2025 is extended to Helenelund, gets a higher closeness centrality than in earlier years. This is due to the short cuts that Tvärbanan offers.

In general it can be seen in figure 43 that the different extensions of the network up until 2025 will result in more nodes with relatively high closeness centrality. This means that the central part of the network has extended from earlier only being in central Stockholm to now include some stations further away from the city center.

Summary The five stations with the highest closeness centrality with respect to distance for each map are presented in the table below (table 8), the values for the stations are shown in brackets. The average closeness centrality for the six years is shown as well. Observe that the average closeness centrality with respect to distance is derived by taking the average journey length while the closeness centrality for the individual stations are derived by using the total distance to all other nodes in the network, thereof the difference.

Table 8. The table shows the five stations with the highest closeness centrality with respect to distance from the six years showing in the maps, the value for each station is shown in brackets. Average closeness centrality is also shown.

Closeness centrality with respect to distance Year 1950 1967 1985 2000 2010 2025 Top 5 1 Karl XII Torg Centralstation Centralstation Centralstatione Centralstatione Centralstatione stations (4.49E-07) en (4.82E-07) en (4.29E-07) n (3.59E-07) n (3.14E-07) n (3.00E-07) 2 Tegelbacken( Hötorget Gamla Stan Gamla Stan Gamla Stan Gamla Stan 4.49E-01) (4.74E-07) (4.19E-07) (3.43E-07) (3.08E-07) (2.94E-07) 3 Norrmalmstor Rådmansgata Slussen Slussen Slussen Slussen g (4.47E-07) n (4.69E-07) (4.13E-07) (3.41E-07) (3.06E-07) (2.91E-07) 4 Strömgatan/ Odenplan Rådhuset Rådhuset Rådhuset Hötorget Drottninggat (4.69E-07) (4.09E-7) (3.34E-07) (3.00E-07) (2.88E-07) an (4.45E-07) 5 Gustav Adolfs Gamla Stan Östermalmsto Hötorget Hötorget Rådhuset Torg (4.68E-07) rg (3.33E-07) (3.00E-7) (2.88E-07) (4.44E-07) (4.08E-07) Average 7.87E-05 5.09E-05 4.85E-05 4.37E-05 4.24E-05 4.52E-05

5.8 Network Diameter In order to see how geographically spread out the network is, network diameter was calculated for each year. It was derived as the longest of all the shortest paths for each year.

The graph showing network diameter for each year is presented below.

Network Diameter 160000

150000

140000

130000

120000

110000

Network Diameter (meters) 100000

90000 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 44. The figure is showing the network diameter for each year.

As seen in figure 44, the network diameter is kept relatively low during the first 40-50 years of the study. At the turn of the century the network diameter increased substantially making it a lot higher in the end in comparison to the beginning of the century. The reason for the increase in network diameter is believed to be due to the many extensions of stations at the edges of the network, meaning that the network covered a larger area than before. These extensions include Arlanda, Järna, Bålsta and Västerås.

5.9 Directness It is believed that directness is strongly correlated to connectivity since the more connected a network is the more routes there is to chose from and generally the less of a detour the network is when comparing to the Euclidean distance.

The average directness for each year with the standard deviation is presented in figure 45 and figure 46.

Average Directness with Standard deviation 1.55

1.5

1.45

1.4

1.35 Directness 1.3

1.25

1.2 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 45. The development of average directness and standard deviation.

Average Directness 1.44 1.42 1.4 1.38 1.36

Directness 1.34 1.32 1.3 1.28 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 Year

Figure 46. The development of average directness.

When analyzing the charts above it becomes clear that there exists and inverse relation between connectivity and average directness over the time period. This means that when connectivity is decreasing, directness is increasing and when connectivity is increasing directness is decreasing. At the time of the closure of the many tram lines directness increased substantially, this was because when the many relatively high degree tram stations closed down, many travelers had to take a longer detour than before. Between 1967 and 2000 directness remained relatively stable with a few exceptions in the early 1970’s. In 2000 the transverse connection Tvärbanan was built, this made it possible to change between many branches without entering the city center of Stockholm. This shortened many trips in the network and therefore the average directness decreased at this year. From the year 2000 and onwards, the average directness kept on decreasing due to the opening and extensions of new lines. Examples of these are the extension of Tvärbanan to Sickla and Kista, the new multi modal station Årstaberg that was opened in 2006 and the opening of Citybanan in 2017. One main difference from the inverse relation between average directness and connectivity is that the decrease in the last decades is smaller in comparison to the increase during the same time for connectivity. This means that the early years’ directness is at the same level as the last years’, for connectivity the last years are higher than the first years which would not have been the case if there would have been a perfect inverse correlation.

The standard deviation for the average directness increases when directness is increasing, this means that when the average directness is higher the total sample of directness is more spread out than otherwise.

5.9.1 Cumulative Distribution In order to see how directness is distributed each year, a cumulative distribution plot for each year was made (figure 47).

Cumulative Distribution of Directness 1

0.9

0.8

0.7

0.6

0.5

Percentage 0.4

0.3

0.2

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0 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 Directness 1950 1951 1952 1954 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1975 1976 1977 1978 1982 1985 1986 1987 1993 1994 1995 1996 1999 2000 2001 2002 2003 2006 2008 2009 2010 2012 2013 2014 2017 2018 2020 2021 2022 2025

Figure 47. Cumulative distribution of directness for each year.

The trend that was observed is that the first years’ distributions are not so widely distributed meaning that the node’s directness is not so spread out comparing to other years. This can also be seen in figure 45 where the standard deviation is smaller in the first years comparing to the period between 1970 and 1990. The fact that the graphs are shifted to the left for the first years also implies that many nodes had a relatively low directness in the first decade, this can also be seen in figure 46.

At the end of the 1960’s the distribution of directness got more spread out meaning that the difference between the node’s with high directness and low directness nodes got larger, this resulted in some nodes having a relatively high directness while other still had a low directness. In general however, more nodes had a higher directness after the 1960’s, this can also be seen in figure 46.

After the year 2000 the trend is that the distribution of directness returns to the shape of the 1950’s with a distribution not as spread out. This trend can also be observed in figure 45 were it is seen that the standard deviation gets smaller and returns to the 1950’s level of standard deviation in the last decades.

5.10 Assortative In order to measure if the network is an assortative or a disassortative network, the neighbor node degree and pearson coefficient was used.

5.10.1 Neighbor Node Degree When plotting the average degree of the neighbor nodes for each node degree a network is considered assortative if the function is increasing.

In figure 48 the average neighbor node degree is plotted versus degree.

5.5

4.5

3.5

3 Average Degree of Neighbor Nodes

2.5

2 1 2 3 4 5 6 7 8 9 10 11 12 Degree

Figure 48. Average degree of neighbor nodes for each node degree.

It can be seen that for the first decades, the function is increasing for most years, at least for degree 1 to 3. When degrees gets higher than 3 a trend cannot be seen in the graph. This is partly believed to be due to the fact that there are not too many nodes with a degree of 4 or higher meaning that the average neighbor degree can be easily affected by variation in one or only a few nodes. For the last two decades in the study period the function is decreasing from 1 to 2 degrees and then, as for the early years, increasing between 2 and 3 degrees. One of the reasons behind the decrease is that after 1977 the one degree node Kungsträdgården is built and its only neighbor is the station with the highest degree in the network, Centralstationen. This raises the average node degree for nodes having one degree substantially.

In summary it can be said that there is a weak trend that the plots are increasing between 1 and 3 degrees. Above 3 degrees no trend can be seen, this however is believed to be due to the lack of nodes having more than 3 degrees which makes the result less significant.

Even though there is a weak trend that the average node degree function is increasing it cannot be seen if the network is assortative or not. Due to this, the pearson coefficient was also derived in order to further investigate the assortativeness of the network.

5.10.2 Pearson Coefficient If the pearson coefficient is positive then a network is considered assortative.

In figure 49 the pearson coefficient for each year is showed.

Pearson Coefficient 0.12

0.08

0.04

0 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025

Pearson Coefficient -0.04

-0.08

-0.12 Year

Figure 49. Pearson Coefficient for each year.

As seen, according to the pearson coefficient all the networks are considered assortative. This means that in general in these networks high degree nodes are connected to other high degree nodes and many nodes with few degrees are connected to other nodes with few degrees.

In the first two decades the coefficient varies back and forth even though it stays above 0, in 1977 the coefficient decreased, this was partly because the station Kungsträdgården opened that year. Kungsträdgården had one degree but its only neighbor was the highest degree node in the network, Centralstationen, this had a substantial decreasing effect on assortativeness. From 1977 up until 1994 there was little change in the coefficient, after 2000 an increasing trend can be detected. This is believed to have many explanations, the opening of Citybanan is one of them since it connects the high degree nodes Centralstationen and Odenplan to each other. Another explanation is the extension of the blue metro line from Kungsträdgården since it increases Kungsträdgården’s degree by one and therefore removes the connection between a one degree node and the highest degree node in the network, the extension also connects the two high degree nodes Sofia and Gullmarsplan to each other.

5.11 Relations between indicators In order to get a greater understanding on the reason for some of the indicators’ development, relations between some of the indicators were investigated. This was done in order to see in what way some indicators are affecting others. This would be interesting to know since an increase in one indicator could have an effect on another indicator that might be important for the network’s function. The relations that are investigated below were chosen due to similarities in their individual graphs, which implied that a relation existed. The relations were all investigated at a network level, this means that global indicators and the average value for node specific indicators were used.

Plots showing the investigated relations are presented below.

5.11.1 Clustering Coefficient / Average Degree As seen in figure 13 and figure 11, when average degree increases the average clustering coefficient often increases as well. In a more connected network the probability of having a higher clustering coefficient is higher, because of this it is expected that clustering coefficient will increase as average degree in the network is increasing.

The relation between the average degree and clustering coefficient was investigated and the correlation can be seen in figure 50.

increasing when new nodes are added. It can be seen that connectivity decreased steadily when the number of nodes decreased between 1950 and 1971. After 1971, the number of nodes increased more or less steadily throughout the study period. This is reflected by an increasing connectivity for these years. However it can be seen that the increase in the last years is steeper in comparison to the rest of the study period. The increase gets accentuated after the year 2012 when commuter trains started to run to Uppsala and in 2013-2014 Tvärbanan got extended to Solna Centrum and Solna Station. During the remaining years in the study period the connectivity is increasing along the accentuated trend due to the extensions that in many cases connect already existing stations, such as Tvärbanan to Kista and the new the metro lines to Nacka and Solna Station.

For the whole period, there is an exponential relation between the number of nodes and connectivity. Due to the difference in increase between the regimes the R2 value for the relation is only 0.5227 which means that 52.27% of changes in connectivity can be explained by variation in the number of nodes. This is not a very strong relation but as mentioned before, there is strong evidence that an increasing number of nodes lead to an increase in connectivity for the network. One exception to the relation is the opening of the first leg of Spårväg City in 2010. The tram line resulted in an increasing number of nodes by 10 but the connectivity decreased from 0.351 to 0.350.

To summarize this relation it can be said that an increasing number of nodes has lead to an increased connectivity throughout the study period and this trend has been accentuated after the year 2012.

6 Discussion and Conclusion

6.1 Evolutionary Trends When analyzing the results obtained in the previous sections it becomes clear that most of the average values of the indicators follow a distinctive trend during the study period. Many indicators are relatively high in the beginning, then they decrease up until the closing of the tram network in 1967 and then they slowly start to increase again. In the final years, many of the indicators surpass its value from the first decades. This is true for connectivity, average clustering coefficient and average degree centrality. For the number of nodes and edges, the same trend can be seen but the last years have a smaller value than the first decade. Comparing to the above mentioned indicators, directness is different since a higher value indicates a less functioning network and a smaller value is a sign of a better network. Because of this directness’ trend over the study period is inverted to the indicators mentioned above meaning that the average directness is relatively low in the beginning, it then increases up until 1967 and then it decreases for the rest of the study period.

The above mentioned trend is explained by the fact that in the first 17 years of the study the old tram network still existed. In general, the nodes in the network had a relatively high degree meaning that they contributed to a high average degree, average clustering coefficient and connectivity and also a low average directness. The fact that the tram network had many nodes also made the number of nodes high. When the tram network closed down all of these indicators decreased, and directness increased. When the network size slowly started to increase again in the 1970’s the average degree and connectivity increased as well. This was because many of the stations that were added had a relatively high average degree making the network more intra-connected. This trend accentuated after the year 2000, this is especially true for average degree and connectivity. One of the reasons behind this is that the first leg of Tvärbanan was built in 2000. Tvärbanan connected many different metro and commuter train lines creating many high degree stations. Another reason behind the accentuated trends in the last year is that many of the new extensions connected already existing stations, for example the extension of Tvärbanan to Solna in 2013/2014 and to Kista in 2020, the addition of the new metro lines to Nacka, Solna Station and Barkarby Station and also the opening of Citybanan in 2017. There is one exception to this however, it is the completion of the first leg of Spårväg City in 2010. The new line led to a decrease in both connectivity and average degree.

Another trend that is seen is that average closeness centrality with respect to geographical distance is at its lowest and network diameter is at its highest during the last decades of the study period. This implies that the geographical spread of the network is larger in the last decades in comparison to the rest of the study period. The fact that the networks in the latter years are both the most intra- connected and also has the largest geographical spread implies that the extensions added to the network in the last 20-30 years are beneficial for the network. This is because the latter networks are not only the most intra-connected but they also cover the largest area meaning that more people are served by the networks.

Network theory can be used when analyzing if a planned extension is beneficial for the overall performance of a network. This can be done by investigating how different network indicators change when new additions are added to a network, if these additions are beneficial for the evolution of different indicators, then it can be said that the extensions are improving the network. Network theory can also be used to evaluate different extension plans to each other. This can be

91 done by creating different networks with the different extension plans and then analyze how they affect the network indicators, due to time limitations different extension plans were not investigated in this study but only the extensions that have been decided upon. In this study it is seen that the future extensions accentuates many of the positive trends seen throughout the study period and in this way network theory can justify that these extensions are beneficial for the network.

When Strano et al (2012) investigated the evolution of a road network in northern Italy, they found two different evolutionary phases, exploration and densification. These two phases can be seen in the evolution in this study as well. From the 1970’s up until 2000, most of the extensions that were built were branches that went into areas not before served by the network. This is typical of exploration, after the year 2000 however most of the extensions made the network denser by connecting already existing stations. Examples of these are Tvärbanan, the extension of the blue metro line to Barkarby and also the new metro line between Odenplan and Solna Station. These extensions are in line with the phase densification and it can therefore be said that in the last two decades of the study period, the trend has changed from exploration to densification.

In the article written by Roth et al (2012) a clear trend was seen that metro networks have developed into similar shapes even though they had been build at different times, over different time frames and by very different authorities. The shape was a dense core with a ring line going around it. Outside the core, the networks consisted of branches reaching out from the core to peripheral parts of the network, along the branches fork stations sometimes existed. Because this shape was seen in nearly all of the major metro networks, it was considered to be a good and efficient way to build a rail bound public transport network.

When analyzing the shape of the network in this study, it is clear that in the early years, a denser core existed with a ring line circling around most of the core. From the core, branches reached out to suburbs and there were also fork stations along some of those lines. The core and ring line was mainly made up of the old tram network. After the tram network closed down, the shape was more of a radial shape with branches going in and out of the central part, no real core existed in these years. This shape was more or less the same up until the year 2000 when Tvärbanan was built. Tvärbanan can be seen as the start of a new ring connection going outside the city center. With the extensions of Tvärbanan and also the addition of more stations in the central parts of the network, it can be said that the evolution from the year 2000 up until 2025 goes towards the shape discussed by Roth et al (2012). In 2025, Tvärbanan is creating half a circle from Sickla in the south east to Solna Station in the north, and the part of the network inside Tvärbanan is denser than the rest of the network. Even though the trend is towards the metro shape seen in Roth et al (2012) it is not a complete core with a ring line going all the way around and branches going in every direction. This is due to the natural constraints that Stockholm has, the constraints are that the city is built on islands and to the east the Baltic Ocean and the archipelagoes are located meaning that rail lines cannot be built everywhere in the city. In spite of the natural constraint, the future extensions are definitely developing the network towards the ‘standard’ metro network shape discussed by Roth et al (2012). The fact that systems tends to converge into a similar shape even though they have been constructed at different times and under different authorities implies that this shape is beneficial for the network. This should be of interest when new rail bound public transport networks are being planned around the world.

The development in Roth et al (2012) also has some differences in comparisons to the development in this study. By looking at the major metro networks in the world, Roth et al analyzed how average degree in the dense parts of the network changes with respect to the number of nodes in the network, it can be seen that the average degree in the core is increasing with network size up until 200 nodes. When the network size increases up until 300, average degree starts to vary up and down with a decreasing trend. This trend is not seen in this report where the average degree is steadily increasing even when the number of nodes in the network increase above 200, it has to be mentioned however that only the average degree in the core was looked at by Roth et al, but it is believed that the average degree for the core in this study is increasing as well. The core is approximated as being the part of the network inside Tvärbanan.

6.2 Development according to plan? As mentioned earlier, the future extensions to the network result in accentuating many of the trends that had occurred since the late 1960’s.

A point of interest is if these extensions are developing the network in a way that follows the guidelines for the development of the transportation system in Stockholm put down by the city council.

In Framkomlighetsstrategin 2030, the city council of Stockholm has written down in what way they want the overall transportation in the city to develop up until 2030 (Stockholms Stad 2012). An improved public transport system is mentioned as vital part in this development.

Four main goals are pinpointed in the report:

A, More people and gods should be transported through high capacity modes such as public transport, bicycles, walking and freight vehicles with high occupancy.

B, The accessibility in the road network should be improved through an increase of the travel speeds for high capacity modes and the travel time reliability for all travelers.

C, The streets’ role as attractive meeting places should be strengthens through improving the accessibility for pedestrians.

D, The negative effects of the road network should be minimized through controlling car usage to only include car journeys that are the most beneficial for the society.

In Framkomlighetsstrategin it is mentioned that 100 000 new housing units will be built in Stockholm up until 2030, for the street network in the inner city to handle this increase, the car usage need to decrease. One major contribution to decrease car usage is to improve the public transport system in order to make the system more attractive. The extensions of the public transport system that are included in this report are improving the system to the better which hopefully will lead to a decrease in car usage in the city. In this way it can be said that the evolution of the public transport network is helping the city to develop according to goal D in framkomlighetsstrategin.

Today there are 570 000 workplaces in Stockholm, 54% of these 570 000 workers are living inside the city of Stockholm, 22 % live in the 10 surrounding cities, 16 percent live in other parts of the Stockholm County while only 8 percent are commuting from another county (Stockholms Stad 2012).

It is believed that these travel patterns will remain relatively stable up until 2030. This means that the demand for public transport journeys in the city center and the nearby suburbs will be high in the future. Many of the planned extensions will improve the quality in the central parts of the network, examples of these are Spårväg City, the new metro lines and also the extensions of Tvärbanan. Tvärbanan’s extensions are believed to be the most beneficial for this process since it gives travelers the option to transfer outside of the city center. This will release capacity in the central parts of the network that is needed for the increasing demand induced by the population increase.

There have been discussions in Stockholm to make the buses along the trunk lines in the city center into tram lines (DN 2011). Most discussions have been over bus number 4, which is the most frequently used bus line. The reasons for turning the bus into a tram line are because it would increase the capacity on the line since a tram can accommodate more travelers than a bus, it would also increase the speed on the line since trams often gets its own right of way meaning that it does not have to adjust its speed to the traffic, it would also be beneficial since trams are more environmentally friendly than buses due to the fact that they are driven on electricity. As of today, no decisions have been made to turn the bus lines into trams and they are therefore not included in this report as one of the future extensions. Goal B in framkomlighetsstrategin says that the speed and capacity should increase for the high capacity lines in the network. The bus line number 4 is definitely one of the high capacity lines and it would therefore be beneficial for framkomlighetsstrategin to turn the bus line into a tram line since it would increase both the speed and capacity. Therefore it can be said that the lack of a decision to build tram lines along the trunk bus lines is not a development according to how the city council wants Stockholm to develop.

Goal C in framkomlighetsstrategin says that the attractiveness of the streets should be improved by increasing the accessibility for pedestrians. In this way the streets can be attractive meeting places for people and not only a way to transport vehicles. The public transport system’s role in achieving this goal is substantial since an increased public transport ridership would get more people not to use their car which would release space on the streets that can be claimed by the pedestrians. The trends that have been seen in this study with a more connected and larger public transport network will have positive effects on the overall quality of the public transport system in the city. This is believed to make more people use public transport instead of their cars and in this way, the trend that can be seen in this study is helping the city to achieve goal C.

One of the actions mentioned in framkomlighetsstrategin in order to achieve goal B is to increase the capacity for critical sections of the network. A critical section in today’s network is the two tracks going from the central station south through the city, these tracks are used by mainline trains and also the commuter train lines to Södertälje and Nynäshamn. This is one of the most critical sections in the entire Swedish rail network and there have been talks on increasing the capacity on this section for many years (Trafikverket 2014a). Citybanan, which is a new train tunnel that will go underneath the city will be opened in 2017. Two tracks will be built in the tunnel that will be used by the commuter trains, this will release capacity on the old critical section. Another critical part of today’s network is the metro section between Slussen and T-Centralen were three green lines are sharing one double track and two red lines share another double tracks. The extension of the blue metro line through Sofia to Gullmarsplan will offer an alternative route other than the busy section between Slussen and T-Centralen. This will release capacity on the old metro tracks. As seen, both of these new constructions will offer relief to two of the most critical sections in the

94 rail bound network. It can therefore be said that these two future extensions will help to achieve goal B in framkomlighetsstrategin.

Adolphson (2009) mentions that there is a goal for the City of Stockholm to develop in a polycentric manner, this can also be seen in Stockholm’s land use development plan (Stockholms Stad 2010).When analyzing the results in this report it can be seen that the future extensions are aiding the polycentric development. In 2025, the degree for nodes in urban nuclei such as Alvik, Sundbyberg, Solna and Kista are all increasing when comparing to 2010. This means that more lines are connecting these stations and it that way the possibility for the urban nuclei to grow is increasing. The same trend can be seen for the maps showing betweeness centrality for the different nodes. When comparing the maps from 2010 and 2025 it is seen that more stations have a relatively high betweeness centrality in 2025 than in 2010. This means that these stations have increased its importance in the network, and the conditions for the surrounding developments to grow therefore increases. Many of these stations are situated along the extended Tvärbanan which has increased the centrality of these nodes.

When looking at the land use development plan certain areas are mentioned as being potential growth areas in order to aid the polycentric development, many of these areas’ public transport is improved by the future extensions included in this report which will help the areas to grow. Examples of these areas are, Norra Djurgårdsstaden that will be served by Spårväg City in 2018, Ulvsunda and Kista that both will be served by the extension of Tvärbanan, Hammarby Sjöstad which will be a station on the metro line to Nacka and also the area around Karolinska Sjukhuset and Norra Station that is located in proximity to the new metro station Hagastaden on the new metro line going north from Odenplan. An urban area that is not mentioned in the land use development plan as being a potential growth area for the polycentric development is Barkarby. It can be seen however that the centrality of Barkarby in the system will increase up until 2025. This due to the extension of the blue metro line to Barkarby and also the opening of the regional train station Stockholm Väst. Barkarby’s degree centrality will increase from 2 to 5 degrees between today (2014) and 2025 and its share of the network’s total betweeness centrality will also increase, from 0.20% to 0.48%. Even though Barkarby is not one of growth areas in the land use development plan, the relatively nearby area Kista is mentioned in the plan. Kista will benefit from the public transport improvement at Barkarby since the extensions of the blue metro line will make it possible to change from regional and commuter trains at Barkarby station and then take the metro to Kista.

In summary, it can be said that the future extensions are aiding the polycentric development in Stockholm, many of the areas that are mentioned in the land use development plan as potential growth areas will benefit from the future public transport extensions.

As seen in the above section, many of the future extensions to the rail bound public transport network in Stockholm will develop the network in a way that is according to the development sought after by the city council.

6.3 Limitations and Future Studies There are some limitations in this project which means that there are discrepancies to some of the calculated indicators.

The fact that the network representation does not account for the lines in the network means that the extra time a transfer results in is not included in the total travel time. This means that when calculating the shortest paths between the node pairs, some of the shortest paths are not going through the stations they would in reality. This has effects on betweeness centrality, closeness centrality with respect to travel time and also directness. A solution to this, which is not done in this study, would be to include a transfer penalty when transfers are made between lines.

Also the fact that the distance between nodes was calculated as a straight line is a limitation for the study. This is because the distance between two stations in most cases is longer since most of the rail connections are not straight lines. This means that when calculating the network indicators that are based on distance such as directness and closeness centrality with respect to geographical distance there will be some small errors associated to the results. It was decided however that the distance as a crow flies between nodes were sufficient since most rail connections are relatively straight between stations.

Another limitation is that time tables were not found for every year in the study period. This means that the travel time for many of the networks are not from the actual year the network is representing. It has been observed however that travel time is not changing drastically from one year to another and it was therefore considered sufficient to only include the travel time from every 7-8 years.

Another aspect of the study that is a limitation is that time tables for the old tram network in the 1950’s were difficult to find. Only a few time tables were obtained and these only gave the travel time between some of the stations. By using these travel times and also the distance between stations the travel time for every link in tram network could be estimated. Even though the correct travel time was not found for every link it is still considered to be adequate since the approximation is believed to be close to reality.

For future studies it would be good to replicate this project without some of the limitations. An example is adding a transfer penalty at stations with different lines which would, if implemented correctly make the study more realistic. Another way to increase the reliability of the study would be to include travel demand in the network. In this report demand is ignored all together in order to make calculations easier. Introducing travel demand would make the results even more realistic.

Another way to improve the study would be to include the capacity for lines and stations. In this way travelers would use alternative routes if capacity is reached on parts of the network which would make the results more realistic. For future studies, the network representation could also be improved by including headway for links. This would make journey times more according to reality since how long one has to wait at a station is an important aspect when choosing which route to use in the network.

In order to incorporate a larger share of the public transport network it would have been a good idea to add certain trunk bus lines to the study as well. This would replicate journeys along the networks in a better way since many journeys include both bus and rail bound modes. This is something that could be added to the analysis in future studies.

Another aspect to for future studies would be to replicate this study on other networks around the world as well. This is to see if the trends that have been seen in this study also occur in other networks. It would be especially interesting to investigate networks that have been built at a different time period and also by different types of authorities.

When analyzing the indicators, it became clear that many of the indicators are dependent on the number of nodes in the network. This is true for betweeness centrality and the two closeness centralities. Since those indicators are not a measure of the number of nodes in the network it would have been good to compensate for the number of nodes and in that way calculate what the indicators were used for (centrality of nodes in these cases). One thing that future studies could do would be to calculate closeness centrality by using the average journey length in a network instead of the total journey length that was used in this report when creating the maps and the cumulative distributions. If the average journey length was used, the comparison between years would not have been dependent on the number of nodes and the evolution of centrality could therefore easier have been seen.

Another thing that could be useful for future studies would be to add planned future extensions to the study as well. In this way the different planned extensions could be compared to one another and it could be seen how the different extensions affect the indicators and that might be useful when choosing among the different planned extensions. Another thing that could be done would be to test the extensions that are included in this report one by one and see how each extension affects the indicators in the network. This could be used do decide which of the extensions that are decided upon that will be the most beneficial for the network’s performance.

7 References Adolphson, M., 2009. Estimating a Polycentric Urban Structure. Case Study: Urban Changes in the Stockholm Region 1991–2004, Stockholm: Royal Institute of Technology.

Aspenberg, N., 1998. The Tramways in Stockholm: The history of the tramways, interurbans, trolleybuses and underground. Oslo: Banerforlaget.

Ash, J., Newth, D., 2007. Optimizing complex networks for resilience against cascading failure, Albury: Charles Sturt University.

Barthélemy, M., 2010. Spatial Networks, Paris: Ecole des Hautes Etudes en Sciences Sociales.

Cervero, R., 1995. Sustainable new towns: Stockholm’s rail-served satellites, Berkley: University of California.

Derrible, S., 2012. Network Centrality of Metro Systems, Singapore: Singapore-MIT Alliance for Research and Technology.

Derrible, S., Kennedy, C., 2010. The complexity and robustness of metro networks, Toronto: University of Toronto.

DN, 2011. Sex spårvägslinjer ska ersätta de blå bussarna, http://www.dn.se/sthlm/sex- sparvagslinjer-ska-ersatta-de-bla-bussarna/, accessed on: 2014-05-09.

DN, 2013. Så ska vi bygga ut t-banan och bostäderna i Stockholm, http://www.dn.se/debatt/sa-ska- vi-bygga-ut-t-banan-och-bostaderna-i-stockholm/, accessed on: 2014-01-18.

Eriksson, G., 1991. Alla Stockholms Spårvagnar – en spårvagnsresa i Stockholm under nittio år. Stockholm: Jan Jangö AB.

Fröidh, O., Kordnejad, B., Kottenhoff, K., Lindahl, A., Nelldal, B., Troche, G., 2011. Kompendium i Tågtrafikplanering, Stockholm: Royal Institute of Technology.

Hällqvist, A., 2008. Pendeltåg i Stockholmsområdet. Stockholm: Trafiknostalgiska Förlaget.

Jarnvag.net, 2010. Stockholm-Märsta/Arlanda-Uppsala, http://www.jarnvag.net/index.php/banguide/banor-svealand/stockholm-uppsala, accessed on: 2014-04-04.

Koordinaten, 2014. Formula for the distance calculation between two coordinates on earth, http://www.koordinaten.com/informations/formula.shtml, accessed on: 2014-02-20.

Landgren, K. ,1993. Saltsjöbanan. Stockholm: Svenska Järnvägsklubben.

Lange, T., 1998. Stockholm på spåret. Från hästspårvagn till snabbspårväg. Stockholm: Svenska Spårvägssällskapet.

Li, P., Zhu, Z., Zhou, J., Ding, J., Wang, H., Wei, S., 2009. Performance Analysis of Public Transport Systems in Nanjing Based on Network Topology, Nanjing: Nanjing Institute of Technology.

Lin, J., Ban, Y., 2013. Complex Network Topology of Transportation System, Stockholm: Royal Institute of Technology.

Lokman.se, 2014. Tunnelbanan, http://www.lokman.se/tagbilder/Svenska_tag/SL_Spartrafik/TUB.htm, accessed on: 2014-04-28.

Roth, C., Kang, S., Batty, M., Barthelemy, M., 2012. A long-time limit for world subway networks, Paris: Institut des Systemes Complexes.

SL, 2009. Trafikplan 2020, http://www1.sl.se/pdf/Trafikplan%202020_klickbar.pdf, accessed on: 2014-04-23.

SLJ, 1985. Roslagsbanan 100 år, Malmö: Frank Stenvalls Förlag.

SLL, 2014. Spårväg City, http://www.sll.se/verksamhet/kollektivtrafik/aktuella-projekt/sparvag-city/, accessed on: 2014-02-23.

SLL, 2014a. Tvärbanan, http://www.sll.se/verksamhet/kollektivtrafik/aktuella-projekt/tvarbanan/, accessed on: 2014-02-23.

SLL, 2014b. Tvärbanan till Sickla, http://www.sll.se/verksamhet/kollektivtrafik/aktuella- projekt/tvarbanan/tvarbanan-sickla/, accessed on: 2014-03-03.

Stockholmförhandling, 2013. UTBYGGD TUNNELBANAFÖR FLER BOSTÄDER, http://stockholmsforhandlingen.se/accounts/10965/files/276.pdf, accessed on: 2014-01-18.

Stockholms Handelskammare, 2013. Tunnelbana möjliggör 350 000 nya bostäder, Stockholm: Stockholms Handelskammare.

Stockholms Stad, 2010. PROMENADSTADEN Översiktsplan för Stockholm, Stockholm: Stadsbyggnadskontoret.

Stockholms Stad, 2012. Framkomlighetsstrategin för Stockholm 2030, http://www.stockholm.se/PageFiles/227129/Framkomlighet/Framkomlighetsstrategi_%20bilaga%20 1.pdf, accessed on: 2014-05-10.

Strano, E., Nicosia, V., Latora, V., Porta, S., Barthelemy, M., 2012. Elementary processes governing the evolution of road networks, Lausanne: Ecole Polytechnique Federale de Lausanne.

Trafikverket, 2014. Om Citybanan, http://www.trafikverket.se/Privat/Projekt/Stockholm/Citybanan/Om-projektet/, accessed on: 2014- 04-02.

Trafikverket, 2014a. Citybanans Bakgrund, http://www.trafikverket.se/Privat/Projekt/Stockholm/Citybanan/Bakgrund/, accessed on: 2014-04- 02.

Watts, D., Strogatz, S., 1998. Collective dynamics of ‘small-world’ networks, Ithaca: Cornell University. von Ferber, C., Holovatch, T., Holovatch, Y., Palchykov, V., 2009. Public transport networks: empirical analysis and modeling, Coventry: Coventry University.

XIe, F., Levinson, D., 2007. Modeling the Growth of Transportation Networks: A Comprehensive Review, Minneapolis: University of Minnesota.

Xie, F., Levinson, D., 2008. Topological evolution of surface transportation networks, Minneapolis: University of Minnesota.

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8 Appendix Maps are presented below that show travel time and geographical distance for each link. For travel time, maps for every 15th year are presented and for distance only the first and last year are presented. Zoomed in maps are included in order to make it easier to see the different numbers.

8.1 Travel time

Figure A. The map is showing the travel time for each link in 1950.

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Figure B. The map is showing the travel time for each link in 1950.

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Figure C. The map is showing the travel time for each link in 1965.

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Figure D. The map is showing the travel time for each link in 1965.

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Figure E. The map is showing the travel time for each link in 1980.

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Figure F. The map is showing the travel time for each link in 1980.

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Figure G. The map is showing the travel time for each link in 1995.

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Figure H. The map is showing the travel time for each link in 1995.

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Figure I. The map is showing the travel time for each link in 2010.

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Figure J. The map is showing the travel time for each link in 2010.

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Figure K. The map is showing the travel time for each link in 2025.

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Figure L. The map is showing the travel time for each link in 2025.

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8.2 Geographical Distance

Figure M. The map is showing the geographical distance for each link in 1950.

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Figure N. The map is showing the geographical distance for each link in 1950.

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Figure O. The map is showing the geographical distance for each link in 1950.

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Figure P. The map is showing the geographical distance for each link in 1950.

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Figure Q. The map is showing the geographical distance for each link in 1950.

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Figure R. The map is showing the geographical distance for each link in 2025.

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Figure S. The map is showing the geographical distance for each link in 2025.

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Figure T. The map is showing the geographical distance for each link in 2025.

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JACOB ROMMEL JACOB 2014 , of Analysis Topological the Evolution of Public Networks Transport

Topological Analysis of the Evolution of Public Transport Networks KTH TSCMT TSCMT www.kth.se

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