  Graph neural networks (GNNs) have been shown promising in improving the
efficiency of learning communication policies by leveraging their permutation
properties. Nonetheless, existing works design GNNs only for specific wireless
policies, lacking a systematical approach for modeling graph and selecting
structure. Based on the observation that the mismatched permutation property
from the policies and the information loss during the update of hidden
representations have large impact on the learning performance and efficiency,
in this paper we propose a unified framework to learn permutable wireless
policies with multidimensional GNNs. To avoid the information loss, the GNNs
update the hidden representations of hyper-edges. To exploit all possible
permutations of a policy, we provide a method to identify vertices in a graph.
We also investigate the permutability of wireless channels that affects the
sample efficiency, and show how to trade off the training, inference, and
designing complexities of GNNs. We take precoding in different systems as
examples to demonstrate how to apply the framework. Simulation results show
that the proposed GNNs can achieve close performance to numerical algorithms,
and require much fewer training samples and trainable parameters to achieve the
same learning performance as the commonly used convolutional neural networks.
