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Relationship between Generalized Origin Destination and Flow Matrix – A Tutorial

By Kardi Teknomo, PhD.

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What is lacking from graph theory in general is the existence of moving agents. Moving agents create trajectories. Based on the ordinal graph trajectories that utilize the network, I have related the structure of a graph (e.g. adjacency matrix or path distance matrix) with the utilization of the graph (e.g. flow matrix or origin-destination matrix) by the agents.  In summary, the relationship between network structure and network utilization can be simplified into simple algebraic equation of element wise product of the matrices and matrix addition:

Where the utilization matrices are
= generalized origin-destination matrix
= flow matrix
= alternative route matrix
and the network structure is represented by
When there is no alternative route, the formula is even shorter

.

In this tutorial, I give 2 numerical examples of the key concepts presented in our paper.

This tutorial is a companion of our paper in ATR (Teknomo, K. and Fernandez, P., A theoretical foundation for the relationship between generalized origin–destination matrix and flow matrix based on ordinal graph trajectories, Journal of Advance Transportation Research DOI: 10.1002/atr.1214). It is recommended to read the full paper from Wiley. Personal copy of the published paper is available upon request.

The topic of this tutorial is as follow:
Relationship between Network Structure and Network Utilization
Numerical Examples
Weaving Section
General Network Graph
Matrix-Set

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Java Program Documentation (contribution of Michael F. Ybanez). The program is available for download.

Press release: Automatic Gathering Origin Destination Data

Preferable reference for this tutorial is

Teknomo, Kardi. (2013) Relationship between Generalized Origin Destination and Flow Matrix – A Tutorial
http://people.revoledu.com/kardi/research/trajectory/od/