  Safe deployment of graph neural networks (GNNs) under distribution shift
requires models to provide accurate confidence indicators (CI). However, while
it is well-known in computer vision that CI quality diminishes under
distribution shift, this behavior remains understudied for GNNs. Hence, we
begin with a case study on CI calibration under controlled structural and
feature distribution shifts and demonstrate that increased expressivity or
model size do not always lead to improved CI performance. Consequently, we
instead advocate for the use of epistemic uncertainty quantification (UQ)
methods to modulate CIs. To this end, we propose G-$\Delta$UQ, a new single
model UQ method that extends the recently proposed stochastic centering
framework to support structured data and partial stochasticity. Evaluated
across covariate, concept, and graph size shifts, G-$\Delta$UQ not only
outperforms several popular UQ methods in obtaining calibrated CIs, but also
outperforms alternatives when CIs are used for generalization gap prediction or
OOD detection. Overall, our work not only introduces a new, flexible GNN UQ
method, but also provides novel insights into GNN CIs on safety-critical tasks.
