A Topological Interpretation for Mass Transit Network Connectivity July 8, 2006 Chulmin Jun,...

A Topological Interpretation for A Topological Interpretation for Mass Transit Network Mass Transit Network

ConnectivityConnectivity

July 8, 2006July 8, 2006

Chulmin Jun, Seungjae Lee, Hyeyoung Kim & Seungil LeeChulmin Jun, Seungjae Lee, Hyeyoung Kim & Seungil Lee

The University of Seoul, KoreaThe University of Seoul, Korea

ContentsContents

IntroductionIntroduction Hierarchical Network Configuration Hierarchical Network Configuration Applying to Public Transportation Applying to Public Transportation Integrating into GISIntegrating into GIS Generating Paths using GAGenerating Paths using GA Concluding RemarksConcluding Remarks

IntroductionIntroduction

Public-oriented transportation policiesPublic-oriented transportation policies Unbalanced supply due to less Unbalanced supply due to less

systematic route planning and operationssystematic route planning and operations Unbalanced accessibility causes Unbalanced accessibility causes

inequalities in time, expenses and metal inequalities in time, expenses and metal burden of users.burden of users.

Need robust methodology to assess the Need robust methodology to assess the accessibility or serviceability of the accessibility or serviceability of the transport routes.transport routes.

IntroductionIntroduction Space syntax is the technique to Space syntax is the technique to

analyze the connectivity of urban analyze the connectivity of urban or architectural spaces.or architectural spaces.

Has been applied to analyzing Has been applied to analyzing movement in indoor spaces or movement in indoor spaces or pedestrian paths (not in transport pedestrian paths (not in transport network).network).

The study proposes a method to The study proposes a method to evaluate accessibility of public evaluate accessibility of public transport network based on its transport network based on its topological structure.topological structure.

Hierarchical Network Hierarchical Network ConfigurationConfiguration

Movement can be described in an Movement can be described in an abstracted form using its topology.abstracted form using its topology.

Topological description helps focus on Topological description helps focus on the structural relationship among units.the structural relationship among units.• For example, pedestrian movement can be For example, pedestrian movement can be

described using network of simple lines without described using network of simple lines without considering the details such as sizes of forms, considering the details such as sizes of forms, number of people and speed of movement.number of people and speed of movement.


Topological description of streets Topological description of streets networknetwork

1

2

3

6

4

5


Hierarchical structure of a streetHierarchical structure of a street• Representing each component with a node and a Representing each component with a node and a

turn with a link connecting their respective nodes turn with a link connecting their respective nodes

1

2 3

4 5

6

step 1

step 2

step 3


This relationship is described thought a This relationship is described thought a variable called variable called depth.depth.• Depth of one node from another can be directly Depth of one node from another can be directly

measured by counting the number of steps (or measured by counting the number of steps (or turns) between two nodes.turns) between two nodes.


Total Depth(TD)Total Depth(TD)• TDTD11 = 1 x 2 + 2 x 2 + 3 x 1 = 9 = 1 x 2 + 2 x 2 + 3 x 1 = 9

m

ssi NsTD

1

TDi : the total depth of node is : the step from node im : the maximum number of steps extended from node iNs : the number of nodes at step s


Mean Depth(MD) = Mean Depth(MD) = TD / (k-1)TD / (k-1) Normalized Depth(ND)Normalized Depth(ND)

12

3

4

1

2 3 4

1

2

3

4

step k -1

432

1

step 1

b. completely asymmetrical network

a. completely symmetrical network

※ k : the total number of nodes

21

2/)1(

1

)1(...21 k

k

kk

k

kMD

11

1

k

kMD

21

kMD 1

2

)1(20

k

MD

Applying to Public TransportationApplying to Public Transportation

Hierarchical network structure focuses Hierarchical network structure focuses on turns of spaces while the public on turns of spaces while the public transportation entails transfers between transportation entails transfers between vehicles.vehicles.

» In hierarchical network description, the deeper the depth In hierarchical network description, the deeper the depth from a space to others, the more relatively difficult it is to from a space to others, the more relatively difficult it is to move from that space to others.move from that space to others.

» In public transportation, cost generally increases as the In public transportation, cost generally increases as the number of transfers between different modes increases.number of transfers between different modes increases.

Applying to Public TransportationApplying to Public Transportation» One transferOne transfer from a transportation mode to another is the from a transportation mode to another is the

‘spatial transfer’ which becomes ‘spatial transfer’ which becomes one depthone depth between spaces. between spaces.

1 2

5

6

7

3 4

8 9

A B

C

transfer areas1 stops

subway 1subway 2

bus 1bus 2bus 3


• Mapping schematic route connectivity onto a graphMapping schematic route connectivity onto a graph

step 1

step 2

step 3

step 4

2

9

3

5 6 7 8

1

4

step 1

step 2

step 3

step 4

1

9

2

5 6 7 8

4

3

step 1

step 2

step 3

step 4

2

1 4 5 6 73

8

9

3

1 4 6 7 82

5 9

1 3 4

6

5

2

7

9

8

2

9

5

1 4 8

6

3 7

2

9

3

1 45

7

6 8 3

5

7

1 2 4 6

8

9

1 2 4 6

3 7

5

8

9


• Symmetry and asymmetry of the route Symmetry and asymmetry of the route connectivityconnectivity

432

1

step 1

1

2

3

4

step k -1

b. complete asymmetry of the route connectivity

a. complete symmetry of the route connectivity

1 2 3 4

1 2 3 4

※ k : the total number of stops


• Computing depth from each stop

Stop No. TD MD ND ND-1

11 1414 1.7501.750 0.2140.214 4.674.67

22 1111 1.3751.375 0.1070.107 9.339.33

33 1010 1.2501.250 0.0710.071 14.0014.00

44 1414 1.7501.750 0.2140.214 4.674.67

55 1717 2.1252.125 0.3210.321 3.113.11

66 1313 1.6251.625 0.1790.179 5.605.60

77 1212 1.5001.500 0.1430.143 7.007.00

88 1414 1.7501.750 0.2140.214 4.674.67

99 2121 2.6252.625 0.4640.464 2.152.15


Iterative procedure for computing TDIterative procedure for computing TD

1.For i=1~k stops1.For i=1~k stops1.1 For all routes that share stop i1.1 For all routes that share stop i

1.1.1 Step = i1.1.1 Step = i

1.1.2 Find all stops except for stop i and accumulate TD1.1.2 Find all stops except for stop i and accumulate TD

1.1.3 For all transfer areas found1.1.3 For all transfer areas found1.1.3.1 Find all stops in current transfer area1.1.3.1 Find all stops in current transfer area

1.1.3.2 For each stop1.1.3.2 For each stop

1.1.3.2.1 for each route1.1.3.2.1 for each route

Step++ and go to 1.1.2Step++ and go to 1.1.2

Integrating into GIS Integrating into GIS

Typical GIS data structure alone can Typical GIS data structure alone can not capture the complex relationship in not capture the complex relationship in public transportation.public transportation.

The relationship among streets, routes, The relationship among streets, routes, stops and transfer areas can be stops and transfer areas can be abstracted into an entity-relationship abstracted into an entity-relationship model in a relational database.model in a relational database.

Integrating into GISIntegrating into GIS

E-R diagram for public transport E-R diagram for public transport networknetwork

Street

M:N

Transfer area

1:N

Route Stop

1:N

Generating Paths using GAGenerating Paths using GA

Computing depth of a stop requires Computing depth of a stop requires finding paths from that stop to all others, finding paths from that stop to all others, each of which being the minimum-cost each of which being the minimum-cost path.path.

In this study, the minimum-cost path is In this study, the minimum-cost path is the one having the the one having the minimum number of minimum number of transferstransfers between the O-D. between the O-D.

Genetic AlgorithmsGenetic Algorithms

Use the terms borrowed Use the terms borrowed from natural geneticsfrom natural genetics

A global search process A global search process on a certain population on a certain population of chromosomes by of chromosomes by gradually updating the gradually updating the populationpopulation

Exploiting the best Exploiting the best solutions while solutions while exploring the search exploring the search spacespace

A population of chromosomes is created

Search for “better” solutions evaluated by the “fitness”

function

Update the population by the “genetic operators”:

crossover and mutation

Iterate

2 5 9 4 3

Gene

Chromosome


An example network with different types of vehicles.An example network with different types of vehicles. Transfers do not happen at nodes 3 or 7. The rest nodes allow Transfers do not happen at nodes 3 or 7. The rest nodes allow

the traveler to transfer to another mode.the traveler to transfer to another mode.

An example of multi-modal network

1

2

3

4

5 7

8

6

9 : Bus 1

: Bus 2

: Subway 1

: Subway 2


RepresentationRepresentation• Ex: (1, 2, 5, 6, 9)Ex: (1, 2, 5, 6, 9)

InitializationInitialization• C1 = (1, 2, 8, 9)C1 = (1, 2, 8, 9)

• C2 = (1, 4, 5, 6, 9)C2 = (1, 4, 5, 6, 9)

• C3 = (1, 2, 5, 6, 7, 8, 9)C3 = (1, 2, 5, 6, 7, 8, 9)

• ……

EvaluationEvaluation• Rate potential solutions by their fitnessRate potential solutions by their fitness

• Ex) the total time taken from the origin to the destination at Ex) the total time taken from the origin to the destination at chromosome Cchromosome C


SelectionSelection• Good chromosomes are preserved instead of Good chromosomes are preserved instead of

participating in the mutation or crossover.participating in the mutation or crossover.

Genetic OperatorsGenetic Operators• Some members in the population are altered by Some members in the population are altered by

two genetic operators: two genetic operators: crossovercrossover andand mutationmutation..


Genetic Operators (cont’d)Genetic Operators (cont’d)• CrossoverCrossover

» a common node (e.g. Node 5) is selected and the a common node (e.g. Node 5) is selected and the portions of chromosomes after this node are crossed portions of chromosomes after this node are crossed generating new children.generating new children.

• MutationMutation» An arbitrarily selected gene becomes a temporary origin.An arbitrarily selected gene becomes a temporary origin.» The portion after this is generated.The portion after this is generated.

C2’ = (1, 4, 5, 6, 7, 8, 9)C3’ = (1, 2, 5, 6, 9)

C2 = (1, 4, 5, 6, 9)C3 = (1, 2, 5, 6, 7, 8, 9)

C2 = (1, 4, 5, 6, 9) C2’ = (1, 4, 5, 2, 8, 7, 6, 9)

Data Structure in the GISData Structure in the GIS

Need to consider:Need to consider:• A journey can be of A journey can be of different combination of modesdifferent combination of modes..• A transfer takes timeA transfer takes time. (moving time + waiting time). (moving time + waiting time)• Different types of vehicles may Different types of vehicles may share a section of share a section of

networknetwork..• Bus stops or subway Bus stops or subway stops can be located in those stops can be located in those

spots other than crossesspots other than crosses..• Topological relationshipsTopological relationships exist between exist between

nodes(stops) and links(routes).nodes(stops) and links(routes).• A transfer areaA transfer area is where more than one stop are is where more than one stop are

located closely such that the passengers can move located closely such that the passengers can move between the stops on foot.between the stops on foot.


Representation of a transfer area in the GISRepresentation of a transfer area in the GIS

: Subway 1: Bus 1: Bus 2


Entity Relationship DiagramEntity Relationship Diagram

Li nksLI NKI D

FROM_NODETO_NODELengthTi meSTOPI D (FK)MODEI D (FK)

ModesMODEI D

ModeTypeStartTi meEndTi meI ntervalCompanyStartLocat i onEndLocat i on

StopsSTOPI D

StopNameX_CoordY_CoordTRANSI D (FK)

Transf ersTRANSI D

Locat i onNameX_CoordY_Coord

Implementing in the GISImplementing in the GIS Modeling a networkModeling a network

subway-2(rt 2)

subway-3(rt 3)

subway-1(rt 1)

bus20(rt 5)

bus10(rt 4)

2- 1 2- 2 2- 3 2- 4

2- 5

2- 6

2- 7

2- 82- 9

2- 11

2- 10

2- 12

5- 1

2- 13

5- 2

5- 45- 3 5- 5 5- 6

5- 7

5- 8

5- 9

5- 10 5- 18

4- 1

5- 15

5- 16

5- 115- 12

5- 14

5- 13

5-17

4- 2 4- 3 4- 4 4- 5 4- 6 4- 7

6- 8 6- 7 6- 6

6- 4

6- 3

6- 1

6- 2

1- 51- 41- 3

1- 21- 1

3- 1

3- 2

3- 3

3- 4 3- 5

6- 5

bus30(rt 6)

14

1

2

3

46

5

78

910

11

12

13

15

16 17

Implementing in the GISImplementing in the GIS

Implementing in the GISImplementing in the GIS Creating a chromosomeCreating a chromosome

Implementing in the GISImplementing in the GIS Shortest DistanceShortest Distance • No.Transfers: 2No.Transfers: 2

• Dist: 11 kmDist: 11 km

• Travel Time: 46minTravel Time: 46min

Implementing in the GISImplementing in the GIS Min. TransferMin. Transfer

• No.Transfers: 1No.Transfers: 1



Min. Travel TimeMin. Travel Time• No.Transfers: 2No.Transfers: 2



Applying to the CBD of SeoulApplying to the CBD of Seoul

< 그림 8> 지하철 및 버스노선

Subway and Bus Routes

Transfer Areas

Applying to the CBD of SeoulApplying to the CBD of Seoul

NDND-1-1 for bus stops in the CBD of Seoul. for bus stops in the CBD of Seoul.

Concluding RemarksConcluding Remarks

A method to assess accessibility of public A method to assess accessibility of public transport network was proposed by defining transport network was proposed by defining the network relationship onto a graph.the network relationship onto a graph.

An analogy between the concept of depths in An analogy between the concept of depths in pedestrian network and the accessibility of pedestrian network and the accessibility of network of transport routes was used.network of transport routes was used.

An algorithm to automate the computing An algorithm to automate the computing process was developed.process was developed.

Concluding RemarksConcluding Remarks

Each O-D pair must be an optimal path.Each O-D pair must be an optimal path. This is the problem of finding the minimum-This is the problem of finding the minimum-

cost path in the network of multi-modal public cost path in the network of multi-modal public transportation.transportation.

We can’t use shortest path algorithms; we We can’t use shortest path algorithms; we used the GA-based approach.used the GA-based approach.

If the procedure is applied to a city, we can If the procedure is applied to a city, we can quantify the difference in the serviceability of quantify the difference in the serviceability of city areas based on the public transportation.city areas based on the public transportation.

Thank You!Thank You!

A Topological Interpretation for Mass Transit Network Connectivity July 8, 2006 Chulmin Jun,...

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