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Modeling Transportation Network Dynamics as a Complex System David Levinson Bhanu Yerra Objectives Why do networks expand and contract? Do networks self-organize into hierarchies? Roads - an emergent property? Can investment rules predict location of network improvement?

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Modeling Transportation Network Dynamics as a Complex System


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objectives
Objectives
  • Why do networks expand and contract?
  • Do networks self-organize into hierarchies?
  • Roads - an emergent property?
  • Can investment rules predict location of network improvement?
  • How can this improved knowledge help in planning transportation networks?
how networks change with time
How networks change with time?
  • State of a network node changes
  • Travel time of a link changes
  • Capacity of a link changes
  • Flow on a link changes
  • New links and nodes are added
  • Existing links are removed
  • System properties, like congestion, change over time
how to model a network as a complex system
How to model a network as a complex system?
  • Links and nodes are agents
  • Agent properties
  • Rules of interaction that determine the state of agents in the next time step
  • Spatial pattern of interaction between agents
  • External forces and variables
  • Initial states
transportation network as a complex system
Transportation network as a complex system
  • Network is split into two layers
    • Network layer
    • Land use layer
  • Network is modeled as a directed graph
  • Land Use Layer has small land blocks as agents that determine the populations and land use
slide6

Network Layer

Land Use Layer

Figure 1: Splitting the system into network layer and land use layer

models required
Models Required
  • Land use and population model
  • Travel demand model
  • Revenue model
  • Cost model
  • Network investment model
network
Network
  • Grid network
  • Random networks will also be tested
  • Networks under modeling stage
    • Cylindrical network
    • Torus network
  • Initial speed distribution
    • Every link with same initial speed
    • Uniformly distributed speeds
land use and demography
Land Use and Demography
  • Small Land Blocks are agents
  • Population, business activity, and geographical features are attributes
  • Uniformly distributed and bell shaped land use are modeled
  • Land use is assumed to be static
  • Dynamics of land use will be added later
travel demand model
Travel Demand Model
  • Trip Generation
    • Using Land use model trips produced and attracted are calculated for each cell
    • Cells are assigned to network nodes using voronoi diagram
    • Trips produced and attracted are calculated for a network node using voronoi diagram
travel demand model contd
Travel Demand Model(Contd.)
  • Trip Distribution
    • Calculates trips between network nodes
    • Gravity model

Where,

trs is trips from origin node r to destination node s,

pr is trips produced from node r,

qsis trips attracted to node s,

drsis cost of travel between nodes r and s along shortest path

travel demand model contd14
Travel Demand Model(Contd.)
  • Traffic Assignment
    • Path with least cost of traveling
    • Cost of traveling a link is

Where, la is length of link a

va is speed of link a

 is value of time

o is tax rate

1, 2 are coefficients

travel demand model contd15
Travel Demand Model(Contd.)
  • Traffic Assignment ( Contd.)
    • Dijkstras Algorithm
    • Flow on a link is

Where,

Krsis a set of links along the shortest path from node r to node s,

a,rs = 1 if a  Krs, 0 otherwise

revenue model
Revenue Model
  • Toll is the only source of revenue
  • Annual revenue generated by a link is total toll paid by the travelers

Where, coefficients are same as coefficients used in traveling cost function

  • A central revenue handling agent can be modeled
cost model
Cost Model
  • Assuming only one type of cost
  • Cost of a link is

Where, c is cost rate,

1, 2, 3 are coefficients.

  • Introducing more cost functions makes the model more complicated and probably more realistic
network investment model
Network Investment Model
  • A link based model
  • Speed of a link improves if revenue is more than cost of maintenance, drops otherwise

Where, vat is speed of link a at time step t,

          • is speed reduction coefficient.
  • No revenue sharing between links: Revenue from a link is used in its own investment
examples
Examples
  • Base case
    • Network - speed ~ U(1, 1)
    • Land use ~ U(10, 10)
    • Travel cost da { = 1.0, o = 1.0, 1 = 1.0, 2 = 0.0}
    • Cost model {c =365, 1 = 1.0, 2 = 0.75, 3 = 0.75}
    • Improvement model {  = 1.0}
    • Speeds on links running in opposite direction between same nodes are averaged
    • Symmetric route assignment
slide20

Initial network

Equilibrium network state after 8 iterations

Slow

Fast

Figure 4 Equilibrium speed distribution for the base case on a 10x10 grid network

Base Case

slide21

Base Case

Initial network

Equilibrium network state after 9 iterations

Slow

Fast

Figure 5 Equilibrium speed distribution for the base case on a 15x15 grid network

case 2 same as base case but initial speeds u 1 5

Initial network

Equilibrium network state after 8 iterations

Slow

Fast

Figure 6 Equilibrium speed distribution for case 2 on a 10x10 grid network

Case 2: Same as base case but initial speeds ~ U(1, 5)
case 2 same as base case but initial speeds u 1 523

Initial network

Equilibrium network state after 8 iterations

Slow

Fast

Figure 7 Equilibrium speed distribution for case 2 on a 15x15 grid network

Case 2: Same as base case but initial speeds ~ U(1, 5)
case 3 base case with a downtown

Initial network

Equilibrium network state after 7 iterations

Slow

Fast

Figure 8 Equilibrium speed distribution for case 3 on a 10x10 grid network

Case 3: Base case with a downtown
case 3 base case with a downtown25

Initial network

Equilibrium network state after 7 iterations

Slow

Fast

Figure 9 Equilibrium speed distribution for case 3 on a 15x15 grid network

Case 3: Base case with a downtown
conclusions
Conclusions
  • Succeeded in growing transportation networks
  • Sufficiency of simple link based revenue and investment rules in mimicking the network structure
  • Hierarchical structure of transportation networks is a property not entirely a design
  • Future work will help in better understanding of network dynamics
future work
Future Work
  • Trip distribution - An agent based combined trip distribution and traffic assignment
  • Traffic assignment - Deterministic and Stochastic user equilibrium
  • Revenue sharing between links
  • Modeling construction of new links
  • Maintenance and construction costs
  • Spatial dependent revenue and investment models