  Optimizing power control in multi-cell cellular networks with deep learning
enables such a non-convex problem to be implemented in real-time. When channels
are time-varying, the deep neural networks (DNNs) need to be re-trained
frequently, which calls for low training complexity. To reduce the number of
training samples and the size of DNN required to achieve good performance, a
promising approach is to embed the DNNs with priori knowledge. Since cellular
networks can be modelled as a graph, it is natural to employ graph neural
networks (GNNs) for learning, which exhibit permutation invariance (PI) and
equivalence (PE) properties. Unlike the homogeneous GNNs that have been used
for wireless problems, whose outputs are invariant or equivalent to arbitrary
permutations of vertexes, heterogeneous GNNs (HetGNNs), which are more
appropriate to model cellular networks, are only invariant or equivalent to
some permutations. If the PI or PE properties of the HetGNN do not match the
property of the task to be learned, the performance degrades dramatically. In
this paper, we show that the power control policy has a combination of
different PI and PE properties, and existing HetGNN does not satisfy these
properties. We then design a parameter sharing scheme for HetGNN such that the
learned relationship satisfies the desired properties. Simulation results show
that the sample complexity and the size of designed GNN for learning the
optimal power control policy in multi-user multi-cell networks are much lower
than the existing DNNs, when achieving the same sum rate loss from the
numerically obtained solutions.
