  Recent studies have significantly improved the prediction accuracy of travel
demand using graph neural networks. However, these studies largely ignored
uncertainty that inevitably exists in travel demand prediction. To fill this
gap, this study proposes a framework of probabilistic graph neural networks
(Prob-GNN) to quantify the spatiotemporal uncertainty of travel demand. This
Prob-GNN framework is substantiated by deterministic and probabilistic
assumptions, and empirically applied to the task of predicting the transit and
ridesharing demand in Chicago. We found that the probabilistic assumptions
(e.g. distribution tail, support) have a greater impact on uncertainty
prediction than the deterministic ones (e.g. deep modules, depth). Among the
family of Prob-GNNs, the GNNs with truncated Gaussian and Laplace distributions
achieve the highest performance in transit and ridesharing data. Even under
significant domain shifts, Prob-GNNs can predict the ridership uncertainty in a
stable manner, when the models are trained on pre-COVID data and tested across
multiple periods during and after the COVID-19 pandemic. Prob-GNNs also reveal
the spatiotemporal pattern of uncertainty, which is concentrated on the
afternoon peak hours and the areas with large travel volumes. Overall, our
findings highlight the importance of incorporating randomness into deep
learning for spatiotemporal ridership prediction. Future research should
continue to investigate versatile probabilistic assumptions to capture
behavioral randomness, and further develop methods to quantify uncertainty to
build resilient cities.
