Ateneo de Manila, Philippines

Transcription

1 Ideal Flow Based on Random Walk on Directed Graph Ateneo de Manila, Philippines

2 Background Problem: how the traffic flow in a network should ideally be distributed? Current technique: use Wardrop s Principle: User Equilibrium: Nash equilibrium Social Equilibrium: minimum system travel time Ideal condition: the most efficient utilization of a network happens when the flow is distributed uniformly over space and time.

3 Ideal Traffic Flow Distribution Queuing Theory Results: the most efficient utilization of a network happens when the flow is distributed uniformly over space and time lead to Random Walk on Network

4 Motivation If we have such ideal traffic flow distribution, we may use the ideal traffic flow matrix as a guideline to manage the actual traffic flow (e.g. by optimizing signal timing on area wide network, or by providing intelligence traffic info) in such a way such that the actual flow will be transformed as close as possible to the ideal traffic flow matrix.

5 Simulation N agents move from random source nodes with no destinations. On each, the agents take random choice to the available choice of directed edge. The agents keep moving until time T-> is achieved. Simplified version: edge length = 1, agent speed = 1 (each agent jump from one node to the next node at 1 time step)

6 Trajectory Our interest: recorded trajectory data of the agents. Flow = sum of trajectories on each edge

7 Research Goal We investigate the connection between network properties and the results of trajectory analysis over any network. Specifically: Effect of adding a link on network to the relative flow distribution

8 Example: Simulation Setting N=200 agents, T=1000 time steps A=[ ; ; ; ; ]

9 Simulation Result Flow Distribution = Relative Flow Distribution = Flow Distribution / Total Flow Distribution =

10 Flow Ratio Relative Flow Distribution = Flow Ratio = Relative Flow Distribution / Minimum Relative Flow Distribution

11 Relative Nodes Distribution Node Number Nodes Distribution Relative Nodes Distribution Ratio of node distribution

12 Interesting Results Regardless the number of agents N or total simulation time T, when N*T are quite large to fill the network we have asymptotic values of relative flow ratio and relative node ratio Flow ratio and node ratio (number of agents visit on edges and nodes) depends on network structure and not depends on the simulation setting

13 Flow Ratio Manual computation For a small network, no need simulation: Set any node as origin. Set 100 flow into node origin When a node has only one in-edge and one outedge, the same amount of flow continue from inedge to out-edge When a node has more than one in-edge, sum all the flow from in-edges into the flow set in the node. When a node has more than one out-edge, distribute the flow equally among all out-edges. ratio of flow distribution is obtained by dividing the flow with the minimum flow.

14 Example

15 Interesting Results Flow ratio is a good indicator of the importance of an edge utilization on a network. This edge relative importance is based on the network structure rather than the utilization of the network. Thus, it is based on inherent properties of the graph structure. The relative flow distribution of random walk on network is asymptotically equal to manually traced distribution of flow over the network with uniform distribution. Nodes preserve the flow. All flow in into a node is equal to all flow out of the same node.

16 Linear Algebra Approach

17 Example A=[ ; ; ; ; ]

18 Example (cont d) Constraint: e 12 =e 13, or e 12 -e 13 =0 e 23 =e 24 =e 25, or e 23 -e 24 =0 and e 24 -e 25 =0 B

19 Example (cont d)

20 Dynamic Network

21 Interesting Results Adding a link is not always diffuse congestion. Unexpected result: Adding a link in certain link may cause congestion somewhere else far away from that added link. A link can be added to divert the congestion by providing more direct alternative route and that link may contribute to reduce congestion.

22 Potential Applications Given many choices about where to build new expressway, where the expressway should be connected? If the expressway is built, what would be the impact (+ & -) to other road network? If certain road are deleted due to disaster, demonstration, parade, what would be the impact to each other links in the network? Impact is not only connectivity but also priority to rebuild.

23 More Theoretical Results Ideal Flow maximizes network Entropy max H p log p s. t. p 1 p j k j 2 j j j1 j1 Premagic matrix is a square matrix where the vector row sum is equal to the transpose of vector column sum. Theorem: Ideal flow matrix is always Premagic matrix Node flow conservation (see proof on paper). k T T Fj j F

24 Example A flow F

25 Real World Example

26 Conclusions We propose an ideal flow based on random walk of multi agents in a directed network graph. We found out that the ideal flow is invariant from simulation number of agents and length of simulation. This implies that ideal flow matrix depends only on the network structure. We also prove our main theorem that that ideal flow matrix is always premagic matrix because premagic matrix characterizes flow conservation on nodes. The uniform distribution ideal flow maximizes network entropy. Adding edge on network may increase or decrease congestion level due to increase of importance level of the edge