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## Modeling Transportation Network Dynamics as a Complex System

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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? • How can this improved knowledge help in planning transportation networks?**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?**• 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**• 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**Network Layer**Land Use Layer Figure 1: Splitting the system into network layer and land use layer**Models Required**• Land use and population model • Travel demand model • Revenue model • Cost model • Network investment model**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**• 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**• 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.)**• 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(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(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**• 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**• 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**• 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**• 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**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**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**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)**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)**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**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**• 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**• 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