Modeling Transportation Network Dynamics as a Complex System

1. Modeling Transportation Network Dynamics as a Complex System • David Levinson • Bhanu Yerra

2. 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?

3. 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

4. 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

5. 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

6. Network Layer Land Use Layer Figure 1: Splitting the system into network layer and land use layer

7. Models Required • Land use and population model • Travel demand model • Revenue model • Cost model • Network investment model

8. Figure 2: Overview of modeling process

9. 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

10. 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

11. 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

12. Figure 3: An example representation of voronoi diagram

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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)

23. 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)

24. 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

25. 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

26. 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

27. 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

28. Demo