  Training deep graph neural networks (GNNs) poses a challenging task, as the
performance of GNNs may suffer from the number of hidden message-passing
layers. The literature has focused on the proposals of {over-smoothing} and
{under-reaching} to explain the performance deterioration of deep GNNs. In this
paper, we propose a new explanation for such deteriorated performance
phenomenon, {mis-simplification}, that is, mistakenly simplifying graphs by
preventing self-loops and forcing edges to be unweighted. We show that such
simplifying can reduce the potential of message-passing layers to capture the
structural information of graphs. In view of this, we propose a new framework,
edge enhanced graph neural network (EEGNN). EEGNN uses the structural
information extracted from the proposed Dirichlet mixture Poisson graph model
(DMPGM), a Bayesian nonparametric model for graphs, to improve the performance
of various deep message-passing GNNs. We propose a Markov chain Monte Carlo
inference framework for DMPGM. Experiments over different datasets show that
our method achieves considerable performance increase compared to baselines.
