A Topological Interpretation for Mass Transit Network Connectivity July 8, 2006 Chulmin Jun, Seungjae Lee, Hyeyoung Kim & Seungil Lee The University of Seoul, Korea

Contents Introduction Introduction Hierarchical Network Configuration Hierarchical Network Configuration Applying to Public Transportation Applying to Public Transportation Integrating into GIS Integrating into GIS Generating Paths using GA Generating Paths using GA Concluding Remarks Concluding Remarks

Introduction Public-oriented transportation policies Public-oriented transportation policies Unbalanced supply due to less systematic route planning and operations Unbalanced supply due to less systematic route planning and operations Unbalanced accessibility causes inequalities in time, expenses and metal burden of users. Unbalanced accessibility causes inequalities in time, expenses and metal burden of users. Need robust methodology to assess the accessibility or serviceability of the transport routes. Need robust methodology to assess the accessibility or serviceability of the transport routes.

Introduction Space syntax is the technique to analyze the connectivity of urban or architectural spaces. Space syntax is the technique to analyze the connectivity of urban or architectural spaces. Has been applied to analyzing movement in indoor spaces or pedestrian paths (not in transport network). Has been applied to analyzing movement in indoor spaces or pedestrian paths (not in transport network). The study proposes a method to evaluate accessibility of public transport network based on its topological structure. The study proposes a method to evaluate accessibility of public transport network based on its topological structure.

Hierarchical Network Configuration Movement can be described in an abstracted form using its topology. Movement can be described in an abstracted form using its topology. Topological description helps focus on the structural relationship among units. Topological description helps focus on the structural relationship among units. For example, pedestrian movement can be described using network of simple lines without considering the details such as sizes of forms, number of people and speed of movement. For example, pedestrian movement can be described using network of simple lines without considering the details such as sizes of forms, number of people and speed of movement.

Hierarchical Network Configuration Topological description of streets network Topological description of streets network

Hierarchical Network Configuration Hierarchical structure of a street Hierarchical structure of a street Representing each component with a node and a turn with a link connecting their respective nodes Representing each component with a node and a turn with a link connecting their respective nodes step 1 step 2 step 3

Hierarchical Network Configuration This relationship is described thought a variable called depth. This relationship is described thought a variable called depth. Depth of one node from another can be directly measured by counting the number of steps (or turns) between two nodes. Depth of one node from another can be directly measured by counting the number of steps (or turns) between two nodes.

Hierarchical Network Configuration Total Depth(TD) Total Depth(TD) TD 1 = 1 x x x 1 = 9 TD 1 = 1 x x x 1 = 9 TD i : the total depth of node i s : the step from node i m : the maximum number of steps extended from node i N s : the number of nodes at step s

Hierarchical Network Configuration Mean Depth(MD) = TD / (k-1) Mean Depth(MD) = TD / (k-1) Normalized Depth(ND) Normalized Depth(ND) step k step 1 b. completely asymmetrical network a. completely symmetrical network ※ k : the total number of nodes

Applying to Public Transportation Hierarchical network structure focuses on turns of spaces while the public transportation entails transfers between vehicles. Hierarchical network structure focuses on turns of spaces while the public transportation entails transfers between vehicles. » In hierarchical network description, the deeper the depth from a space to others, the more relatively difficult it is to move from that space to others. » In public transportation, cost generally increases as the number of transfers between different modes increases.

Applying to Public Transportation » One transfer from a transportation mode to another is the ‘spatial transfer’ which becomes one depth between spaces A B C transfer areas1stops subway 1 subway 2 bus 1 bus 2 bus 3

Applying to Public Transportation Mapping schematic route connectivity onto a graph Mapping schematic route connectivity onto a graph step 1 step 2 step 3 step step 1 step 2 step 3 step step 1 step 2 step 3 step

Applying to Public Transportation Symmetry and asymmetry of the route connectivity Symmetry and asymmetry of the route connectivity step step k -1 b. complete asymmetry of the route connectivity a. complete symmetry of the route connectivity ※ k : the total number of stops

Applying to Public Transportation Computing depth from each stop Stop No.TDMD NDND

Applying to Public Transportation Iterative procedure for computing TD Iterative procedure for computing TD 1.For i=1~k stops 1.For i=1~k stops 1.1 For all routes that share stop i Step = i Find all stops except for stop i and accumulate TD For all transfer areas found Find all stops in current transfer area For each stop for each route for each route Step++ and go to 1.1.2

Integrating into GIS Typical GIS data structure alone can not capture the complex relationship in public transportation. Typical GIS data structure alone can not capture the complex relationship in public transportation. The relationship among streets, routes, stops and transfer areas can be abstracted into an entity-relationship model in a relational database. The relationship among streets, routes, stops and transfer areas can be abstracted into an entity-relationship model in a relational database.

Integrating into GIS E-R diagram for public transport network E-R diagram for public transport network Street M:N Transfer area 1:N RouteStop 1:N

Generating Paths using GA Computing depth of a stop requires finding paths from that stop to all others, each of which being the minimum-cost path. Computing depth of a stop requires finding paths from that stop to all others, each of which being the minimum-cost path. In this study, the minimum-cost path is the one having the minimum number of transfers between the O-D. In this study, the minimum-cost path is the one having the minimum number of transfers between the O-D.

Genetic Algorithms Use the terms borrowed from natural genetics Use the terms borrowed from natural genetics A global search process on a certain population of chromosomes by gradually updating the population A global search process on a certain population of chromosomes by gradually updating the population Exploiting the best solutions while exploring the search space Exploiting the best solutions while exploring the search space

Genetic Algorithms 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 the traveler to transfer to another mode. Transfers do not happen at nodes 3 or 7. The rest nodes allow the traveler to transfer to another mode. An example of multi-modal network

Genetic Algorithms Representation Representation Ex: (1, 2, 5, 6, 9) Ex: (1, 2, 5, 6, 9) Initialization Initialization 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) … Evaluation Evaluation Rate potential solutions by their fitness Rate potential solutions by their fitness Ex) the total time taken from the origin to the destination at chromosome C Ex) the total time taken from the origin to the destination at chromosome C

Genetic Algorithms Selection Selection Good chromosomes are preserved instead of participating in the mutation or crossover. Good chromosomes are preserved instead of participating in the mutation or crossover. Genetic Operators Genetic Operators Some members in the population are altered by two genetic operators: crossover and mutation. Some members in the population are altered by two genetic operators: crossover and mutation.

Genetic Algorithms Genetic Operators (cont’d) Genetic Operators (cont’d) Crossover Crossover » a common node (e.g. Node 5) is selected and the portions of chromosomes after this node are crossed generating new children. Mutation Mutation » An arbitrarily selected gene becomes a temporary origin. » 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 GIS Need to consider: Need to consider: A journey can be of different combination of modes. A journey can be of different combination of modes. A transfer takes time. (moving time + waiting time) A transfer takes time. (moving time + waiting time) Different types of vehicles may share a section of network. Different types of vehicles may share a section of network. Bus stops or subway stops can be located in those spots other than crosses. Bus stops or subway stops can be located in those spots other than crosses. Topological relationships exist between nodes(stops) and links(routes). Topological relationships exist between nodes(stops) and links(routes). A transfer area is where more than one stop are located closely such that the passengers can move between the stops on foot. A transfer area is where more than one stop are located closely such that the passengers can move between the stops on foot.

Data Structure in the GIS Representation of a transfer area in the GIS Representation of a transfer area in the GIS

Data Structure in the GIS Entity Relationship Diagram Entity Relationship Diagram

Implementing in the GIS Modeling a network Modeling a network

Implementing in the GIS

Creating a chromosome Creating a chromosome

Implementing in the GIS Shortest Distance Shortest Distance No.Transfers: 2 No.Transfers: 2 Dist: 11 km Dist: 11 km Travel Time: 46min Travel Time: 46min

Implementing in the GIS Min. Transfer Min. Transfer No.Transfers: 1 No.Transfers: 1 Dist: 15 km Dist: 15 km Travel Time: 54min Travel Time: 54min Min. Travel Time Min. Travel Time No.Transfers: 2 No.Transfers: 2 Dist: 12 km Dist: 12 km Travel Time: 44min Travel Time: 44min

Applying to the CBD of Seoul 지하철 및 버스노선 Subway and Bus Routes Transfer Areas

Applying to the CBD of Seoul ND -1 for bus stops in the CBD of Seoul. ND -1 for bus stops in the CBD of Seoul.

Concluding Remarks A method to assess accessibility of public transport network was proposed by defining the network relationship onto a graph. A method to assess accessibility of public transport network was proposed by defining the network relationship onto a graph. An analogy between the concept of depths in pedestrian network and the accessibility of network of transport routes was used. An analogy between the concept of depths in pedestrian network and the accessibility of network of transport routes was used. An algorithm to automate the computing process was developed. An algorithm to automate the computing process was developed.

Concluding 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- cost path in the network of multi-modal public transportation. This is the problem of finding the minimum- cost path in the network of multi-modal public transportation. We can’t use shortest path algorithms; we used the GA-based approach. We can’t use shortest path algorithms; we used the GA-based approach. If the procedure is applied to a city, we can quantify the difference in the serviceability of city areas based on the public transportation. If the procedure is applied to a city, we can quantify the difference in the serviceability of city areas based on the public transportation.

Thank You!