  Beyond 5G wireless technology Cell-Free Massive MIMO (CFmMIMO) downlink
relies on carefully designed precoders and power control to attain uniformly
high rate coverage. Many such power control problems can be calculated via
second order cone programming (SOCP). In practice, several order of magnitude
faster numerical procedure is required because power control has to be rapidly
updated to adapt to changing channel conditions. We propose a Graph Neural
Network (GNN) based solution to replace SOCP. Specifically, we develop a GNN to
obtain downlink max-min power control for a CFmMIMO with maximum ratio
transmission (MRT) beamforming. We construct a graph representation of the
problem that properly captures the dominant dependence relationship between
access points (APs) and user equipments (UEs). We exploit a symmetry property,
called permutation equivariance, to attain training simplicity and efficiency.
Simulation results show the superiority of our approach in terms of
computational complexity, scalability and generalizability for different system
sizes and deployment scenarios.
