  A theoretical performance analysis of the graph neural network (GNN) is
presented. For classification tasks, the neural network approach has the
advantage in terms of flexibility that it can be employed in a data-driven
manner, whereas Bayesian inference requires the assumption of a specific model.
A fundamental question is then whether GNN has a high accuracy in addition to
this flexibility. Moreover, whether the achieved performance is predominately a
result of the backpropagation or the architecture itself is a matter of
considerable interest. To gain a better insight into these questions, a
mean-field theory of a minimal GNN architecture is developed for the graph
partitioning problem. This demonstrates a good agreement with numerical
experiments.
