  There is increasing evidence suggesting neural networks' sensitivity to
distribution shifts, so that research on out-of-distribution (OOD)
generalization comes into the spotlight. Nonetheless, current endeavors mostly
focus on Euclidean data, and its formulation for graph-structured data is not
clear and remains under-explored, given two-fold fundamental challenges: 1) the
inter-connection among nodes in one graph, which induces non-IID generation of
data points even under the same environment, and 2) the structural information
in the input graph, which is also informative for prediction. In this paper, we
formulate the OOD problem on graphs and develop a new invariant learning
approach, Explore-to-Extrapolate Risk Minimization (EERM), that facilitates
graph neural networks to leverage invariance principles for prediction. EERM
resorts to multiple context explorers (specified as graph structure editers in
our case) that are adversarially trained to maximize the variance of risks from
multiple virtual environments. Such a design enables the model to extrapolate
from a single observed environment which is the common case for node-level
prediction. We prove the validity of our method by theoretically showing its
guarantee of a valid OOD solution and further demonstrate its power on various
real-world datasets for handling distribution shifts from artificial spurious
features, cross-domain transfers and dynamic graph evolution.
