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Graph neural networks (GNNs) update the hidden representations of vertices(called Vertex-GNNs) or hidden representations of edges (called Edge-GNNs) byprocessing and pooling the information of neighboring vertices and edges andcombining to exploit topology information. When learning resource allocationpolicies, GNNs cannot perform well if their expressive power is weak, i.e., ifthey cannot differentiate all input features such as channel matrices. In thispaper, we analyze the expressive power of the Vertex-GNNs and Edge-GNNs forlearning three representative wireless policies: link scheduling, powercontrol, and precoding policies. We find that the expressive power of the GNNsdepends on the linearity and output dimensions of the processing andcombination functions. When linear processors are used, the Vertex-GNNs cannotdifferentiate all channel matrices due to the loss of channel information,while the Edge-GNNs can. When learning the precoding policy, even theVertex-GNNs with non-linear processors may not be with strong expressiveability due to the dimension compression. We proceed to provide necessaryconditions for the GNNs to well learn the precoding policy. Simulation resultsvalidate the analyses and show that the Edge-GNNs can achieve the sameperformance as the Vertex-GNNs with much lower training and inference time.

Learning Resource Allocation Policy: Vertex-GNN or Edge-GNN?