STOCKHOLM SVERIGE 2017,
Improving the Robustness of Stockholm’s Metro System
An optimizing algebraic connectivity study
HANNA GUSTAVSSON HAGUSTA@KTH.SE LINNEA THORSTENSSON LINNEATH@KTH.
SE
KTH
Improving the Robustness of Stockholm’s Metro System
-An optimizing algebraic connectivity study Hanna Gustavsson CTFYS14, hagusta@kth.se Linnea Thorstensson CTFYS14, linneath@kth.se Department of Math: Optimization and System Theory
Supervisors: Xiaoming Hu and Han Zhang
May 2017
Abstract
This study is evaluating and improving the robustness of the metro system in Stockholm. It is based on the existing metro system and by interpreting it as a network we can represent it in a mathematical graph.
Furthermore we want to examine how much we can improve the metro system robustness by adding new railway between existing stations for a certain amount of money. More mathematically speaking, which new edges should be added to the existing graph to increase the robustness, where robustness is determined using algebraic connectivity. The problem is solved with di↵erent aspects and using two di↵erent methods, Greedy perturbation heuristic and Semidefinite programming, giving us a result of which new edges that improve the robustness the most. We will eval- uate if it is favorable to build new railway, given the robustness percent increase and the importance of that railway.
I det h¨ar projektet utv¨arderas och f¨orb¨attras Stockholms tunnelbanesys- tems robusthet. Studien baseras p˚a det befintliga tunnelbanesystemet och genom att tyda det som ett n¨atverk kan det representeras i en matema- tisk graf. Vidare vill vi studera hur mycket vi kan f¨orb¨attra robustheten genom att l¨agga till ny j¨arnv¨ag mellan de nuvarande stationerna f¨or en viss summa pengar. Matematiskt beskrivet vill vi allts˚a unders¨oka vilka nya kanter som b¨or l¨aggas till i den befintliga grafen s˚a att robustheten ¨okar.
Robustheten best¨amms genom den algebraiska konnektiviteten. Prob- lemet l¨oses genom olika aspekter och med tv˚a olika metoder, Greedy per- turbation heuristic och Semidefinite programming. Detta ger oss ett re- sultat med vilka nya kanter som ¨okar robustheten mest. Vi unders¨oker om det ¨ar gynnsamt att bygga ny j¨arnv¨ag, givet den procentuella ¨okningen i robusthet samt hur betydelsefull den j¨arnv¨agen kommer att vara.
Contents
1 Introduction 4
1.1 Problem statement . . . 5
1.2 Stockholm metro system . . . 5
1.3 Application Programming Interface . . . 5
2 Mathematical background 6 2.1 Semidefinite Programming (SDP) . . . 6
2.1.1 Semidefinite Cone . . . 6
2.1.2 Eigenvalues and Eigenvectors . . . 6
2.1.3 Symmetric Matrices, some properties . . . 6
2.1.4 SDP . . . 6
2.2 Graph Representation . . . 7
2.3 Laplacian Matrix and Algebraic connectivity . . . 7
2.3.1 A Greedy Perturbation Heuristic . . . 8
2.4 Spherical law of cosines . . . 9
3 Mathematical problem 10 3.1 Entire System Problem . . . 11
3.2 City Center Problem . . . 12
4 Method 15 4.1 Link addition problem . . . 16
4.1.1 A Greedy Perturbation Heuristic . . . 16
4.1.2 SDP relaxation . . . 16
5 Results 18 5.1 Entire Metro System . . . 18
5.1.1 Unweighted . . . 19
5.1.2 Traffic Flow Weight . . . 20
5.1.3 Inverse of Length Weight . . . 20
5.2 City Center Metro Stations . . . 21
5.2.1 Unweighted . . . 23
5.2.2 Traffic Flow Weight . . . 24
5.2.3 Inverse of Length Weight . . . 26
6 Discussion 28 6.1 Entire System . . . 28
6.2 City Center System . . . 29
7 Conclusion 31
8 References 32
9 Appendix A 34
1 Introduction
Graph theory can be used as a method to model ”real-world” networks as math- ematical graphs, [16]. A graph contains vertices, i.e. nodes, and edges which are paths between the vertices. The first occurrence of graph theory is believed to be when Euler in the 18th century was asked to find a path through the two main islands of K¨onigsberg, crossing each of the seven bridges only once, the so called Eulerian path. Nowadays graph theory is used in e.g. GPS systems and to find communities in networks, [19].
It is well known that the Swedish winters are harsh. This is clearly shown every year when the community traffic in Stockholm (SL) and especially the metro, goes out of order. But not only the winters a↵ect the metro system, the rainy spring or the slippery fallen autumn leaves create big challenges for the network. Besides the weather problems there are of course problems with the carriages, signal system failure and power outage every now and then. This leads to a lot of stationary trains and delays. Last year there were as much as 8917 delayed and cancelled trains, which gives an average of 24 deviations every day, [17]. This indicates that the metro system in Stockholm is not very robust. By interpreting the metro system as a network, its robustness can be evaluated. It can be represented mathematically in a graph, where the nodes are the stations and the edges are the railways between them and algebraic connectivity can then be used to analyze the robustness. A more robust and connected metro system has a greater ability to transport passengers between two stations even if there is some railway or station failure.
In graph theory, the connectivity is a measure of how well a graph is con- nected. The algebraic connectivity of a graph is defined as the second smallest eigenvalue of the Laplacian matrix of the graph, [11]. In this work the algebraic connectivity is used as a metric of the network robustness. Researchers have shown that the algebraic connectivity has the closest connection to the network robustness in terms of node and edge connectivity and is therefore the most computational efficient metric of it, [9].
In the last decade, transportation networks and especially air transportation and its robustness has been studied. Guimera and Amaral [13] first studied the scale-free graphical model of the air transportation network. Alexandrov [6] defined that on-demand transportation networks would require robustness in system performance. Bigdeli et al. [7] compared algebraic connectivity to e.g. average degree, average node betweenness and network criticality. Byrne et al. [9] showed that for both small and large sized networks, the algebraic connectivity was the efficient measure for the robustness. In 2010, Vargo et al. [20] introduced the algebraic connectivity to air transportation networks.
They used the algebraic connectivity as a metric of robustness and built the op- timization problem solved by the edge swapping based tabu search algorithm.
There are not as many algebraic connectivity studies on metro systems com-
pared to air transport systems. Derrible and Kennedy [10] adapted network science methodologies to transportation literature and introduced an applica- tion to the robustness of metro systems .
In this study we will examine the robustness of a metro system network by computing the algebraic connectivity. This is, as stated earlier, considered as one of the most efficient methods [9]. To find the most improved algebraic con- nectivity and the new edges providing it, we will use techniques such as greedy perturbation and relaxed semidefinite programming. Together with di↵erent aspects of how to weight the nodes and edges we provide the metro system with a suggestion of new railway routes that improves the robustness of the existing metro system.
1.1 Problem statement
We will study the possibility to improve the robustness of Stockholm’s metro system by maximizing the algebraic connectivity. To improve the connectivity we want to add new routes between already existing stations under a budget constraint. The purpose of this project is then to find the most robust and well connected metro system in Stockholm and determine whether it is profitable to build new railways to increase the robustness of the existing system. This project will have cost as a constraint. Since the cost is nearly proportional to length of railway we will represent the cost constrain with a length constrain in km railway.
1.2 Stockholm metro system
Stockholm metro system is 109 kilometres with 100 stations. The system is divided in three parts called the green-, red- and blue line. They started to build the metro in 1950 and the latest finished extension is from 1994.
Today an extension of the metro system in Stockholm is being built, which is calculated to cost 23 billions Swedish crowns. It will result in 20 kilometers of metro tunnels and ten new stations, [18]. The approximately cost of building one kilometer of metro railway is one billion Swedish crowns (SEK), [14].
1.3 Application Programming Interface
For this project, data for the public transportation is needed and can be re- trieved from Trafiklab.se, which is a website where developers can get data and API:s for public transport in Sweden, [4]. Application programming inter- face, API, is a set of tools for building software applications and specifies how software components should interact. It is also a way for companies to share their data in a controlled way.
2 Mathematical background
2.1 Semidefinite Programming (SDP)
Semidefinite Programming, SDP, is mathematical programming technique with applications in e.g. convex constrained optimization, control theory and com- binatorial optimization.
The idea is to optimize a linear function over the cone of a symmetric semidef- inite matrix subject to a linear constraint, [15].
2.1.1 Semidefinite Cone
A matrix X with dimension (n⇥ n) is a positive semidefinite matrix if X = XT and
vTXv 0 for any v2 Rn
The set of symmetric semidefinite matrices is denoted S+n, also referred to as a semidefinite cone. The symmetric positive semidefinite matrix, X, is denoted X⌫ 0.
2.1.2 Eigenvalues and Eigenvectors
If X is a n⇥ n matrix, the eigenvalue is denoted as and the corresponding eigenvector as x, then:
Xx = x, x6= 0
If Q is the set with eigenvecotrs and the matrix D is a diagonal matrix with the eigenvalues on the diagonal, then:
X = QDQT (1)
2.1.3 Symmetric Matrices, some properties
• X ⌫ 0 $ X = QDQT where the diagonal of D, containing the eigenvalues, is non-negative.
• If X ⌫ 0 then Xii 0, for i = 1, ..., n
• If X ⌫ 0 and Xii = 0, then Xi,j = Xj,i= 0 8j = 1, .., n 2.1.4 SDP
Let X2 Sn, where Sn is the set of symmetric n⇥ n matrices and assume that C(X) is a linear function of X, then
C(X) = C• X = Xn
i=1
Xn j=1
CijXij (2)
If X is a symmetric matrix we can assume that C also is symmetric. This leads to the notation of a SDP problem as:
minimize C• X
subject to Ai• X = bi, i = 1, ..n X ⌫ 0
(3)
C• X is called the objective function, C consists of the data of the objective function and there are n linear functions that X needs to satisfy. X⌫ 0 means that X must lie in the cone of a symmetric positive semidefinte cone, S+n, [12].
A SDP problem can be solved with a standard SDP-solver like S. Boyd’s cvx package for Matlab, [1].
2.2 Graph Representation
We denoted a simple graph by G = (N, E), where the nodes, N = 1, 2, ..., n and the edges, E ✓ {{ij} : ij 2 N, i 6= j}. To bring more information about the graph it is common to introduce weights to the edges. The weight is basically the importance of the edge and is denoted as wij 0, (ij 2 E). If the graph is unweighted, all the weights are equal to one.
The graph G can be represented with an adjacency matrix A with the i:th row and j:th column entry aij and the diagonal are all zeros. The upper and lower diagonal item of A is equal to the weights in the graph G.
aij =
(wij if node i and node j are connected by an edge with weightwij
0 if node i and j are not connected
(4)
2.3 Laplacian Matrix and Algebraic connectivity
From the adjacency matrix A we can create the weighted Laplacian matrix as
lij=
( aij if i6= j Pn
i=1aij if i = j (5)
In the unweighted case we get:
lij= 8>
<
>:
1 if i6= j and (i, j) 2 N Degree of node i if i = j
0 otherwise
(6)
Degree of node is the number of edges the node is connected to, [8].
The Laplacian is a positive semidefinite matrix and its eigenvectors are related
to the graph G’s, structure.
The second smallest eigenvalue of L, 2, is the algebraic connectivity which is a measure of how well-connected the graph G is. The larger value the more connected the graph is, [22]. The normalized eigenvector, v, corresponding to
2 is called the Fiedler vector. Fiedler showed that 2(L) > 0 if and only if G is connected.
Theorem. If G is a general graph with N nodes and G + e is graph G af- ter adding a link between node{ij}, then:
0 = N(G) = N(G + e) N 1(G) N 1(G + e) ... 1(G) 1(G + e) If the multiplicity of N 1(G) is larger than one, N 2(G) = N 1(G), then the algebraic connectivity will stay the same after a link is added. This means that N 2(G) = N 1(G + e), [21].
2.3.1 A Greedy Perturbation Heuristic
Since v is the unit eigenvector corresponding to 2, then vvT is the supergradient of 2(L). If 2 is isolated, an analytic function of L and L is a function of x, we get the supergradient to be:
@
@xl
2(L) = vT @
@xl
v
Since @x@Ll = alaTl, where al is the l:th column in G, (vi vj)2 is the partial derivative of 2(L) with respect to xl. And therefore (vi vj)2gives us the first order approximation of the increase of 2(L) if an edge l is added between node (i, j), [8].
Based on this Gosh and Boyed [8] presented a Greedy Perturbation Heuristic to find which edges to add to increase the algebraic connectivity. The method is a greedy local heuristic method where they added k edges one at a time based on the fact that (vi vj)2is the increase of 2.
Greedy Perturbation Heuristic:
Starting from Gbase, add k edges one at a time:
• Find v, a unit eigenvector corresponding to 2(L), where L is the current Laplacian.
• From the remaining candidate edges, add an edge (i, j) with the largest (vi vj)2.
P. Wei et al.[22] extended the Greedy Heuristic for a weighted problem. The idea is to pick one edge from the candidates with the largest we(vi vj)2, where we is the weight of the edge e. The algorithm that they introduced is called
Modified Greedy Perturbation.
Modified Greedy Perturbation:
• Find v, a unit eigenvector corresponding to 2(L), where L is the current weighted Laplacian.
• From the remaining candidate edges, add an edge (i, j) with the largest we(vi vj)2.
2.4 Spherical law of cosines
Figure 1: Spherical triangle with sides abc, and corners XYZ
In this study we need to determine the distance between metro stations. The position of a metro station is given in geographical coordinates i.e. latitude and longitude. To calculate the distance between two stations the spherical law of cosines (eq. 7) can be used, [3].
In every spherical triangle, see Fig. 1, holds:
cos c = cos a cos b + sin a sin b cos C (7)
3 Mathematical problem
We have an existing network of the metro system which we describe with an undirected graph G(N, V ). The node set, N, represents the stations and the edge set, V, represents the existing links between stations. The aim is to maximize the algebraic connectivity, 2, of the network by adding new routes between existing stations, with cost as a constraint.
The mathematical formulation of our problem becomes:
maximize 2(Lbase+
cand.X
i=1
wixiEi) subject to cTx C
x2 {0, 1}cand.
(8)
Where ci=cost per edge, C is total cost,cand. is the number of added edges and wi is the weight of edge, i. Ei= aiaTi where ai2 Rn is the edge vector between node (i, j) with aii = 1 and aij = 1. All other entries in ai is 0. The cost per edge is calculated by the geographical coordinates of node (i, j) together with the sphercial law of cosines (see section 2.4).
We can change the boolean constraint xi 2 {0, 1} to a linear constraint 0 xi 1. This gives us a relaxation:
maximize 2(Lbase+
cand.X
i=1
wixiEi) subject to cTx C
0 xi 1
(9)
This is a convex optimization problem. Since the linear problem has more possible values for x, its optimal value is an upper bound on the boolean optimal value. The Laplacian, L, is a symmetric positive semidefinite matrix and its smallest eigenvalue is 0 that is associated with the eigenvector 1. The second smallest eigenvalue 2 is given by:
2(L(x)) = min{yTL(x)y | 1Ty = 0, yTy = 1}.
This means that 2(L(x)) is a minimum of linear functions of x. Which means that 2(L(x)) is a concave function of x. This results in a semidefinite program- ming formulation, where 2is the smallest eigenvalue of L + n211T, [8].
The SDP problem can then be formulated as:
maximize 2
subject to 2I 2
n11T L L = Lbase+
cand.X
i=1
wixiEi candX
i
cixi C 0 xi 1
(10)
Here 1 denote a vector with only ones.
Since there are many factors that a↵ect the metro system we will model the problem in di↵erent ways to get the most realistic solution. We will solve the problem both as an unweighted system and a weighted system with di↵erent edge weights. We will not only look at the full system but also solve the same problem for the most important stations in the city center. To find the best solution we will use two di↵erent methods, Greddy Perturbation Heuristic and SDP relaxation, to determine which new routes to be added.
3.1 Entire System Problem
In the full problem we will look at the whole metro system as a graph.
The aim is to find which new routes that improve the robustness of the whole system the most. Our cost constraint is set to 23 billion SEK, the same amount of money as they are building the extension for today. The cost of 23 billion SEK corresponds to 23 km railway. In Fig. 2 the entire metro system in Stockholm is presented. The problem will be divided in three di↵erent sub-problems, an unweighted, a traffic flow weighted and an inverse of length weighted problem.
Figure 2: The entire metro system visualized as a graph. A conversion from the indices n = 1, 2.., 100 to station names can be found in Appendix A.
3.2 City Center Problem
In the city center problem the aim is to find links to add in the inner city metro system. The cost constraint in this problem is set to 10 billion SEK, which corresponds to 10 km of new railway. Fig. 3 shows the graph of the central system containing all the routes. The city center is the most important part of the metro system, there are more people travelling there and it is also where all the lines are connected. The city center problem will be solved as an unweighted problem, a traffic flow weighted problem and an inverse of length weighted problem. These problems will be solved both as a boolean problem, see eq (8), and as a semidefinite programming problem, see eq (10).
Figure 3: The city center metro system visualized as a graph. A conversion from the indices n = 1, 2.., 100 to station names can be found in Appendix A.
• Unweighted problem
In the unweighted problem we say that all edges are equally important.
This means that all edges have the same weight value, wi = 1. The SDP formulation eq. (10) will be revised to:
maximize 2
subject to 2I 2
n11T L L = Lbase+
cand.X
i=1
xiEi candX
i
cixi C 0 xi 1
(11)
• Edge weighted problem
Here every edge has a value wi, not necessarily equal to one depending on the importance of the edge. In the traffic flow case, the value on the weight is given in relation to the number of people travelling that route.
In the inverse of length case the weight is given by the inverse of the length between two nodes. The two weighted problems are suppose to better adapt the mathematical problem to the reality. The most travelled routes are more important since if the routes fail, more people are a↵ected.
Therefore those lines should be weighted with a larger value to prioritize them. The weight given as the inverse of length gives shorter routes more importance. Shorter routes are less expensive to build and also more convenient for the travellers.
4 Method
Our main program is written in Matlab. We formulate the metro system as a graph G(N, V ), where the nodes, N , is the metro stations and the edges, V , is the connection between them. Every station has an index (see Appendix A).
To describe the graph we create an adjacency matrix. In Table 1. we display the adjacency matrix for the first five stations. In the entire system network we have in total 100 stations, hence the adjacency matrix A, will have the dimension 100⇥ 100. In the city center network we have in total 23 stations and therefor the adjacency matrix will have the dimension 23⇥ 23.
1 2 3 4 5
1 0 1 0 0 0
2 1 0 1 0 0
3 0 1 0 1 0
4 0 0 1 0 1
5 0 0 0 1 0
Table 1: Adjacency matrix of the metro system in Stockholm displayed for stations with indices ranging between 1-5.
From the unweighted adjacency matrix we create a graph with Matlab’s built in function G=graph(A). Then we create the Laplacian matrix, both as mentioned in eqns. (5)-(6) and with the built in function L=laplacian(G). For the two weighted problems we create a function that return the Laplacian ma- trix. Both the adjacency matrix and the Laplacian matrix will look di↵erent in the two weighted cases, see eqns. (4)-(6).
The weights representing the number of travellers at each route is based on the number of travellers per day from each station in the metro system in 2013, [2]. In Table 2. a translation from the percentage of travellers from one station to the di↵erent node weights, si, is given. Since we want to have edge weights we converted the node weight in the following fashion, wij = round((si+ sj)/2).
We use Matlab’s built in function round to transformers the weight to an integer.
node weight, si 1 2 3 4 5
percent of total travellers (%) <1 1-4 5-9 10-14 >15
Table 2: Mapping between percentage of travellers from each station and node weights.
When using the inverse of the length as weight we used the spherical law of cosines (see section 2.4) to calculate the distance between node (i, j). Since we did not want the weight to be a too small number, we magnified all the weights with a factor 1000, which is an arbitrarily chosen number.
To calculate the algebraic connectivity of the existing network we use the Mat- lab built in function eig(L) to find the second smallest eigenvalue, 2.
Since we use cost as a constraint in our problem we need to know the cost to build a new route. As mention in Sec. 1.4 the cost of building one kilometer of new rails is 1 billion SEK. We use the limitation of 23 billion SEK since that is the budget that is used when they are extending the metro system today (for the entire system). To calculate the distance between every station we need their geographical coordinates. The data was retrieved from SL:s API SL H˚allplatser och Linjer 2 (Trafiklab.se) by programming a python script that extracts the relevant data. The collected data were station names, station indices and the geographical coordinates for each station. From the data we could pick the station we wanted, and since we got the geographical coordinates we could use spherical law of cosines to calculate the distance.
4.1 Link addition problem
The aim of this thesis is to find a way to add links to the graph to increase the algebraic connectivity. We used two di↵erent methods.
4.1.1 A Greedy Perturbation Heuristic
We use the method presented in Sec. 2.3.1 and the algorithm is as follows:
1. Create the adjacency- and Laplacian matrix for the current network. The Laplacian matrix is called Lbase.
2. Calculate the Fiedler vector, v, from Lbase.
3. Add an edge between node (i, j) with largest (vi vj)2 if edge (i, j) not exists.
4. Calculate the cost of the new edge (i, j).
5. If the total cost, C, does not exceed the maximum cost, begin from step one. Else end.
4.1.2 SDP relaxation
When the problem is formulated as an SDP problem (eq. 10) we can solve it with a standard SDP-solver. We choose to use the Matlab package cvx by Stephen Boyd, [1]. We get a relaxed optimal solution x⇤ where 0 x⇤i 1.
Since we need xi to be either one ore zero in order to know which edge to add, we need a rounding technique. We used a Greedy rounding technique where you choose the k biggest elements from the relaxed solution x⇤. These candidates are rounded to x⇤i = 1 and the rest of the candidates are rounded to x⇤i = 0, [22].
A SDP solver can only be used to solve problems with less than 1000 candidate edges, [8]. In the entire metro system we have approximately 10000 candidate edges and in the city center metro system approximately 529 candidate edges.
Therefore it will only be possible to solve the city center problem with a SDP solver.
5 Results
5.1 Entire Metro System
Figure 4: The entire metro system with added edges. The figure show the results from the unweighted case, the traffic flow weighted case and the inverse of length weighted case.
Figure 5: The increase of 2 when adding k edges in the entire metro system.
Which edges to add is calculated with Greedy Perturbation Heuristic.
In Figure 4 the entire metro system with the calculated new routes is plotted.
As shown in the figure for all the problems, unweighted, weighted with traffic flow and with inverse of length, the new added routes connect the di↵erent lines together. The increase of 2 per added edge for the three problems is plotted in Figure 5.
5.1.1 Unweighted
k Station 1, (index) Station 2, (index) 2% increase 1 Norsborg, (100) H¨asselby Strand, (67) 32.3%
2 Fittja, (97) Hags¨atra, (52) 26.9%
3 Medborgarplatsen, (27) Gamla Stan, (25) 12.1%
Total: 88.1%
Table 3: The increase of 2in percentage when edge k is added. This is for the entire system without any weights.
For the unweighted problem the added edges can be found in Table 3. There we can see how 2increases when new routes are added. The first edge to be added
is also the one that increases 2the most and is the edge between Norsborg and H¨asselby Strand.
5.1.2 Traffic Flow Weight
k Station 1, (index) Station 2, (index) 2% increase 1 Liljeholmen, (81) Fridhemsplan, (4) 34%
2 S¨atra, (92) V¨astertorp, (85) 11.2%
3 Brommaplan, (58) Huvudsta, (13) 7.4%
4 Sk¨armarbrink, (30) Slussen, (26) 16.9%
5 Odenplan, (23) V¨astra Skogen, (6) 4.8%
6 V˚arberg, (94) S¨atra, (92) 12.8%
7 Tensta, (19) Kista, (10) 2.3%
8 R˚acksta, (63) ˚Akeshov, (59) 0%
9 Sockenplan, (46) Gullmarsplan, (29) 2.2%
Total: 131%
Table 4: The increase of 2 in percentage when an edge k is added. The results are for the entire system with traffic flow as edge weights.
When using traffic flow as weight the first edge to be added is between Liljehol- men and Fridhemsplan. All the calculated new edges for this problem can be found in Table 4.
5.1.3 Inverse of Length Weight
k Station 1, (index) Station 2, (index) 2% increase 1 Bred¨ang, (91) Brommaplan, (58) 43.3%
2 Masmo, (96) Hags¨atra, (52) 16.7%
3 Johannelund, (65) Hjulsta, (20) 4.5%
4 Bl˚asut, (31) Stadshagen, (5) 35%
Total: 135.8%
Table 5: The increase of 2 in percentage when an edge k is added. The results are for the entire system with length 1between node (i, j) as edge weight.
When using the inverse of length as weight the route that increases 2the most is Bred¨ang to Brommaplan. This route together with the other new edges are presented in Table 5. The solution with the inverse of length as edge weight results in the largest increase of 2. In Figure 5 we can see that the curve for inverse of length weight has a greater slope then the two others, which corresponds to larger increase of 2.
5.2 City Center Metro Stations
Figure 6: City center metro system with added edges calculated with Perturba- tion Heuristic.
Figure 7: City center metro system with added edges calculated with relaxed SDP.
The city center problem was calculated with two di↵erent methods. Greedy perturbation heuristic, GPH, and Semidefinite programming, SDP. The new systems with the added routes for both the unweighted and the weighted prob- lems are shown in Figure 6 for GPH and in Figure 7 for SDP.
5.2.1 Unweighted
Figure 8: The increase of 2 when k edges are added in the central part of the metro system. The graph is unweighted.
GPH
k Station 1, (index) Station 2, (index) 2% increase 1 Liljeholmen, (81) Alvik, (55) 94.4%
2 Gullmarsplan, (29) Stadshagen, (5) 52.2%
3 Kristineberg, (54) Stadshagen, (5) 12.7%
Total: 233.6%
Table 6: The percentage increase of 2 when edge k is added. This is for the City Central stations without weights.
SDP
k Station 1, (index) Station 2, (index) 2% increase 1 Zinkensdamm, (79) R˚adhuset, (3) 87.1%
2 Hornstull, (80) Thorildsplan, (53) 19.8%
3 Hornstull, (80) Fridhemsplan, (4) 0.4%
Total: 125.2%
Table 7: The increase of 2in percentage when edge k is added. This is for the City Central stations without weights.
For the unweighted problem the first route to add is Liljeholmen to Alvik for greedy perturbation heuristic and Zinkensdamm to R˚adhuset for SDP. These results can be found in Table 6 for GPH and Table 7 for SDP. The greedy perturbation heuristic has a greater total increase in 2 than the SDP. The increase per added edge is plotted in Figure 8.
5.2.2 Traffic Flow Weight
Figure 9: The increase of 2when k edges is added in the central metro system.
The graph has traffic flow as weights.
MGPH
k Station 1, (index) Station 2, (index) 2% increase 1 Liljeholmen, (81) Alvik, (55) 82.6%
2 Gullmarsplan, (29) Fridhemsplan, (4) 22.5%
3 Hornstull, (80) R˚adhuset, (3) 49.8%
Total: 235%
Table 8: The increase of 2in percentage when edge k is added. This is for the City Central stations with the traffic flow between node (i, j) as weight.
SDP
k Station 1, (index) Station 2, (index) 2% increase 1 Hornstull, (80) Fridhemsplan, (4) 73%
2 Skanstull, (28) Fridhemsplan, (4) 42%
3 Alvik, (55) T-Centralen (2) 61.4%
Total: 296.3%
Table 9: The increase of 2in percentage when edge k is added. This is for the City Central stations with the traffic flow between node (i, j) as weight.
For the problem where the traffic flow is used as weight the calculated new routes with MGPH are presented in Table 8. The SDP solution, in Table 9, is quite similar to the MGPH but has a greater total increase in 2. The increase in 2 when adding new edges for the traffic flow weight problem is plotted in Figure 9, both for MGPH and SDP.
5.2.3 Inverse of Length Weight
Figure 10: The increase of 2 when k edges are added in the central metro system. The graph has length 1between node (i, j) as weight.
MGPH
k Station 1, (index) Station 2, (index) 2% increase 1 Hornstull, (80) Thorildsplan (53) 102.1%
2 Gullmarsplan, (29) Stadshagen, (5) 38.6%
3 Kristineberg, (54) Stadshagen, (5) 2.1%
4 Zinkensdamm, (79) R˚adhuset, (3) 16.1%
Total: 232%
Table 10: The increase of 2 in percentage when edge k is added. This is for the City Central stations with the length 1between node (i, j) as weight.
SDP
k Station 1, (index) Station 2, (index) 2% increase 1 Medborgarplatsen, (27) Fridhemsplan, (4) 52.5%
2 Hornstull, (80) Thorildsplan, (53) 87.5%
3 Kristineberg, (54) Skanstull, (28) 3.5%
4 Zinkensdamm, (79) Stadshagen, (5) 9.7%
Total: 224.8%
Table 11: The increase of 2 in percentage when edge k is added. This is for the City Central stations with the length 1between node (i, j) as weight When using the inverse of length as edge weights we got some di↵erents between the two methods. The results for MGPH are presented in Table 10 and the first route that is added is the one between Hornstull and Thorildslpan. The SDP solution has Medborgarplatsen to Fridhemsplan as first added route. All the routes are shown in Table 11. The increase in 2 for MGPH and SDP are very similar, which can be seen in Figure 10.
6 Discussion
6.1 Entire System
As we can see in Fig. 4 all three problems, both the unweighted and the two weighted problems, yields new routes between the existing lines. For the un- weighted case the new edges are edges between end stations or stations near end stations.
The metro system today has a tree structure where all lines are connected at T-centralen and then branches out. A tree structure is not a well connected graph, consequently it is not robust. It is quite obvious that the mathematical solution that will increase the robustness the most is to connect the end sta- tions because if one station goes out of order you can always take the other way around. However this is not the best solution because there are not a lot of people travelling to and from the end stations. In order to get a mathematical solution that a↵ect more people we chose to use the traffic flow as edge weights.
If the inner city edges with more travellers obtain a higher weight than the stations near the end, the mathematical method will prioritize edges near the stations with most travellers. With the traffic flow edge weights we obtained Liljeholmen to Fridhemsplan as the first edge to add, see Table 4. Both Frid- hemsplan and Liljeholmen are quite heavy travelled stations. Fridhemsplan has both the blue and green line connecting and the red line divides in two branches at Liljeholmen. This is a better edge considering peoples travelling routes. A railway between Fridhemsplan and Liljeholmen will be more travelled than a railway between Norsborg and H¨asselby Strand. The increase in robustness, 2
is also greater when adding the edge Liljeholmen to Fridhemsplan, see Table 4, than the edge Norsborg to H¨asselby Strand, see Table 3, so mathematically it is also the better solution. It is interesting to mention that this route is already in the MTR’s 2070 vision plan for Stockholm’s metro system, [5].
The largest increase in 2 can be found in Table 5 and corresponds to the edge between Bred¨ang and Brommaplan. This is the first added edge for the inverse of length weighted problem. All the added edges for this problem are quite long and connects the green line to the red and the blue line compared to the unweighted solution that connected only the red and green lines. Using the inverse of length as weight gives a greater total increase of robustness, 2, than the unweighted and traffic flow weighted problems. Its edges connects all the three di↵erent lines together giving the metro system more of a circle structured system. As mentioned before, a circle structured system increases the robust- ness because you can always take a route past the failed station. The entire system contains a total of 100 nodes which makes it a very large graph and it is also quite large geographically according to the edge length. Because of the size of the graph, the inverse of length weight gives us a pretty good solution.
The distance between the elements in the Fiedler vector is a large number but the weight inverse of length gives us a small number. These two combined gives
a good intermediate solution and that is why the inverse of length weighted problem has the largest increase in 2.
The third edge to be added in the unweighted problem is between Medbor- garplatsen and Gamla Stan which are two stations that are not far away from each other. This edge is added because of the cost constraint. The already chosen two edges are really long and to fulfill the cost constrain only a short third edge can be added. Over all the unweighted problem does not provide us with a good solution, having in mind that most people travel closer to the city center. Accordingly it is more relevant to look at the weighted problems, especially the inverse of length when considering the total solution. If we want to increase the robustness the most while considering a minimal cost, in other words only add one new edge, the traffic flow weighted problem provides us with the best solution.
6.2 City Center System
In the city center system problem we choose to examine di↵erent weights in the graph, the same weights as in the entire system problem. But we also used two di↵erent methods, Greedy Perturbation and a SDP relaxation, to see if we would get di↵erent results. As seen in Tables 6-11 there are some di↵erence in the results comparing the two methods. Both in the 2 increase and which routes to add. But there are also some similarities. In the unweighted problem (Table 6-7) both methods wants to add routes to Hornstull, connect the green line to the blue line and connect the red line to the green line.
When using the traffic flow as weight the similarity between SDP and GPH is that we obtained candidate routes between well-traveled stations. This result is intuitive since routes with more travellers have larger weights. Both methods suggest adding routes to Fridhemsplan and Hornstull. To add routes to Frid- hemsplan feels obvious since it is a well trafficked station and both the green- and the blue line traffic that station. Adding a new route to Hornstull is not what we expected because it is not a well travelled station. On the other hand the station is between two high trafficked stations, Liljeholmen and Slussen, having that in mind it feels like a more obvious route to add.
In the last problem, we used the inverse of the length between two nodes as weights. This is because we wanted to prioritize shorter routes. Since we do not add any new stations it is more realistic to build routes that are not too long.
However, in the city center system problem we believe that using the inverse of length as weight is not very relevant since the system is geographically small and possible routes to add are not too long. The similarities between SDP and Greedy in this case are that both methods suggest to add a route between Horn- stull and Thorildsplan and also they produced candidate routes to Kristineberg and Stadshagen.
The weighted problem that was the most interesting for the city center sys- tem is the one where we used the traffic flow as weight. As mentioned above, having the length as a weight felt unnecessary. The case using no weights is actually fairly similar to the case with weights since basically every station in the city center is well travelled. But we believe that using the traffic flow as weight is a refined mathematical representation of the reality. If we just look at the mathematics, the problem with traffic flow as weights has the highest increase of 2. The method that produces best reslut in this problem was SDP with a total increase of 2with 296.3% compared to 235% with GPH. However, if we are restricted to only add one route, GPH yields a higher increase of 2. The top route with GPH is between Liljeholmen and Alvik, which is similar to the top route to add with SDP, which is Hornstull to Fridhemsplan. This also a solution close to the top-route in the entire system which is the edge between Liljeholmen and Fridhemsplan and is planned to be built in the future.
Comparing the two methods, SDP and GPH, a larger improvement of 2 was obtained with GPH in the unweighted case and in the case with the inverse of lengths as weights. In the problem with traffic flow as weights SDP showed a larger improvement of 2. Both methods were easy to program in Matlab.
The disadvantage of SDP is that it took a while to compute and is not possible to solve for larger system.
7 Conclusion
In this report we show which routes to add in the metro system in Stockholm to increase the robustness of the system the most. The robustness is measured in terms of the algebraic conncectivity represented by 2. The problem is di- vided into two parts, the entire metro system and the city center metro system.
The metro system is represented as a graph where we analyze the increase of
2 when adding edges using di↵erent weights in the graph. We study three di↵erent weight problems, unweighted, traffic flow as weights and the inverse of length between two nodes as weights. By using di↵erent weights we adapt the mathematical model to reality. Our conclusion is that using traffic flow as weights, we get the best solution.
The top-route to add in the entire metro system is between Liljeholmen and Fridhemsplan which yields an increase of the robustness with 34%. The total increase of the robustness, by adding 9 routes that fulfills the cost constraint of 23 billion SEK is 131%.
The top-route to add in the city center metro system is between Liljeholmen and Alvik and yields an increase of 2with 82.6%. This result is obtained with the GPH. The largest total increase of 2 is with the SDP relaxation which produce a total increase of 296.3% by adding three routes. The top-route to add with the SDP relaxation is between Hornstull and Fridhemsplan with an increase of 2 with 73%.
Considering both the entire system and the city center system, our conclusion is that the best increase of the robustness is when adding an edge between the green- and red line. The stations should be nearby Fridhemsplan on the green line and Liljeholmen on the red line.
8 References
[1] cvx research. http://cvxr.com/cvx/. Accessed: 2017-03-14.
[2] Kollektivtrafik, fakta. http://www.sll.se/Global/Verksamhet/
Kollektivtrafik/Fakta%20om%20SL%20och%20l%C3%A4net/SLPR191_
SL_och_lanet_2013_webb.pdf. Accessed: 2017-04-15.
[3] Spherical law of cosines. http://www.math.chalmers.se/Math/
Grundutb/CTH/lnc022/1213/sftrigLNC.pdf. Accessed: 2017-03-01.
[4] Trafiklab. https://www.trafiklab.se/om. Accessed: 2017-03-29.
[5] T¨ank l˚angsiktigt idag. http://www.mtrnordic.se/om-oss/
vision-och-uppdrag/vision-2070/. Accessed: 2017-05-05.
[6] N. Alexandrov. Transportation network topologies. Nasa Langley Research Center, 2004.
[7] A. Bigdeli, A. Tzighadam, and A. Leon-Garcia. Comparison of net- work criticality, algebraic connectivity, and other graph metrics. 1st An- nual Workshop on Simplifying Complex Networks for Practitioners, 2009.
Venice, Italy.
[8] S. Boyd and A. Gosh. Growing well-connected graphs. Stanford University, 2006.
[9] R.H. Byrne, J.T. Feddema, and C.T. Abdallah. Algebraic connectivity and graph robustness. Sandia National Laboratories, Albuquerque, New Mexico, 2009.
[10] S. Derrible and C. Kennedy. The complexity and robustness of metro networks. http://www.sciencedirect.com/science/article/
pii/S0378437110003262, 2010. Accessed: 2017-04-24.
[11] M. Fiedler. Algebraic connectivity og graphs. Czechoslovak Mathematics Journal 23, 1973. page 298-305.
[12] M. Freund. Introduction to semidefinite programming (sdp). MIT open course ware, 2009.
[13] R. Guimera and L. Amaral. Modeling the world-wide airport network.
European Physical Journal B 38, 2004. page 381-385.
[14] L.A. Karlberg. S˚a l˚angt r¨acker en miljard. http://www.nyteknik.se/
bygg/sa-langt-racker-1-miljard-6403890, 2011. Accessed: 2017-03- 11.
[15] S. Reiß. Optimizing extremal eigenvalues of weighted graph laplacians and associated graph realizations. PhD. thesis at Chemnitz University of Technology, 2012.
[16] K. Ruohonen. Graph theory. http://math.tut.fi/~ruohonen/GT_
English.pdf, 2013. Accessed: 2017-03-17.
[17] Slstats. http://slstats.se/#/fardmedel/tunnelbana?period=
tolv-manader. Accessed: 2017-04-02.
[18] A. Sundstr¨om. Notan f¨or t-banan ¨okar 3,5 miljarder. http://www.dn.se/
sthlm/notan-for-t-banan-okar-35-miljarder/, 2016. Accessed: 2017- 03-11.
[19] P. Van Dooren. Graph theory and applications. http://www.hamilton.
ie/ollie/Downloads/Graph.pdf, 2009. Accessed: 2017-03-22.
[20] E. Vargo, R. Kincaid, and N. Alexandrov. Towards optimal transport networks. Systemics, Cybernetics and Informatics 8, 2010. page 59-64.
[21] H. Wang and P. Mieghem. Algebraic connectivity optimization via link addition. Delft University of Technology, 2008.
[22] P. Wei, L. Chen, and D. Sun. Algebraic connectivity maximization of an air transportation network: The flight routes’ addition/deletion problem.
Purdue University, University of Louisville, 2014.
9 Appendix A
Table 12: Appendix A: Every station has an index from n = 1...100.
Station name Index Station Name Index Station Name Index
Kungstr¨adg˚arden 1 Tallkrogen 34 H¨asselby Strand 67
T-centralen 2 Gubb¨angen 35 Ostermalmstorg¨ 68
R˚adhuset 3 H¨okar¨angen 36 Karlaplan 69
Fridhemsplan 4 Farsta 37 G¨ardet 70
Stadshagen 5 Farsta strand 38 Ropsten 71
V¨astra Skogen 6 Hammarbyh¨ojden 39 Stadion 72
Solna Centrum 7 Bj¨orkhagen 40 Tekniska H¨ogskolan 73
N¨ackrosen 8 K¨arrtorp 41 Universitetet 74
Hallonbergen 9 Bagarmossen 42 Bergshamra 75
Kista 10 Skarpn¨ack 43 Danderyds Sjukhus 76
Husby 11 Globen 44 M¨orby Centrum 77
Akalla 12 Enskede g˚ard 45 Mariatorget 78
Huvudsta 13 Sockenplan 46 Zinkensdamm 79
Solna Strand 14 Svedmyra 47 Hornstull 80
Sundbybergs Centru 15 Stureby 48 Liljeholmen 81
Duvbo 16 Bandhagen 49 Midsommarkransen 82
Rissne 17 H¨ogdalen 50 Telefonplan 83
Rinkeby 18 R˚agsved 51 H¨agerstens˚asen 84
Tensta 19 Hags¨atra 52 V¨astertorp 85
Hjulsta 20 Thorildsplan 53 Fru¨angen 86
H¨otorget 21 Kristineberg 54 Aspudden 87
R˚admansgatan 22 Alvik 55 Ornsberg¨ 88
Odenplan 23 Stora Mossen 56 Axelsberg 89
S:t Eriksplan 24 Abrahamsberg 57 M¨alarh¨ojden 90
Gamla Stan 25 Brommaplan 58 Bred¨ang 91
Slussen 26 ˚Akeshov 59 S¨atra 92
Medborgarplatsen 27 Angbyplan¨ 60 Sk¨arholmen 93
Skanstull 28 Islandstorget 61 V˚arberg 94
Gullmarsplan 29 Blackeberg 62 V˚arby g˚ard 95
Sk¨armarbrink 30 R˚acksta 63 Masmo 96
Bl˚asut 31 V¨allingby 64 Fittja 97
Sandsborg 32 Johannelund 65 Alby 98
Skogskyrkog˚arden 33 H¨asselby G˚ard 66 Hallunda 99
Norsborg 100
