  Graph-structured data are ubiquitous. However, graphs encode diverse types of
information and thus play different roles in data representation. In this
paper, we distinguish the \textit{representational} and the
\textit{correlational} roles played by the graphs in node-level prediction
tasks, and we investigate how Graph Neural Network (GNN) models can effectively
leverage both types of information. Conceptually, the representational
information provides guidance for the model to construct better node features;
while the correlational information indicates the correlation between node
outcomes conditional on node features. Through a simulation study, we find that
many popular GNN models are incapable of effectively utilizing the
correlational information. By leveraging the idea of the copula, a principled
way to describe the dependence among multivariate random variables, we offer a
general solution. The proposed Copula Graph Neural Network (CopulaGNN) can take
a wide range of GNN models as base models and utilize both representational and
correlational information stored in the graphs. Experimental results on two
types of regression tasks verify the effectiveness of the proposed method.
