A cumulative gravity model for inter-urban spatial interaction at different scales

The classical gravity model assumes the knowledge of origin-destination (OD) flows between places. While this is reasonable for some modes of transportation such as air travel, it presents challenges to many other applications such as road travel where the observed flow on road segments, such as annual average daily traffic (AADT), may involve traffic for multiple OD pairs. This article introduces a methodology to calibrate a cumulative gravity model on a scale-specific road network. The model infers the traffic on OD pairs from the knowledge of link flows and we document its effectiveness in predicting the bidirectional volume of traffic on highways in the United States. Census data and state AADT were used to calibrate both the noncumulative and cumulative gravity models in three different regions in the western US: Arizona only, 10 western US states, and 16 western US states. The road networks were built by defining nodes as population centers of >10,000 inhabitants for Arizona and >500,000 for the other two networks. A systematic method to remove junction nodes that do not satisfy population threshold requirements but are necessary to maintain network connectivity is presented. The cumulative gravity model performed better (with a R-squared value of 0.93 in Arizona) than the standard model, but the improvement, based on two goodness-of-fit metrics (Common Part of Commuters and least squares) was above 2% only in the Arizona network. Removal of commercial traffic from the data further improved the model's calibration in the 10-state network. A thresholding method that connects the cumulative to the standard gravity model reveals that most road trips in Arizona are within the 150-mile range and that this distance increases to 500 miles in both western networks. Potential future applications of the present work are discussed.