  While graph neural networks (GNNs) are widely used for node and graph
representation learning tasks, the reliability of GNN uncertainty estimates
under distribution shifts remains relatively under-explored. Indeed, while
post-hoc calibration strategies can be used to improve in-distribution
calibration, they need not also improve calibration under distribution shift.
However, techniques which produce GNNs with better intrinsic uncertainty
estimates are particularly valuable, as they can always be combined with
post-hoc strategies later. Therefore, in this work, we propose G-$\Delta$UQ, a
novel training framework designed to improve intrinsic GNN uncertainty
estimates. Our framework adapts the principle of stochastic data centering to
graph data through novel graph anchoring strategies, and is able to support
partially stochastic GNNs. While, the prevalent wisdom is that fully stochastic
networks are necessary to obtain reliable estimates, we find that the
functional diversity induced by our anchoring strategies when sampling
hypotheses renders this unnecessary and allows us to support G-$\Delta$UQ on
pretrained models. Indeed, through extensive evaluation under covariate,
concept and graph size shifts, we show that G-$\Delta$UQ leads to better
calibrated GNNs for node and graph classification. Further, it also improves
performance on the uncertainty-based tasks of out-of-distribution detection and
generalization gap estimation. Overall, our work provides insights into
uncertainty estimation for GNNs, and demonstrates the utility of G-$\Delta$UQ
in obtaining reliable estimates.
