Designing Heterogeneous GNNs with Desired Permutation Properties for Wireless Resource Allocation

Graph neural networks (GNNs) have been designed for learning a variety of wireless policies, i.e., the mappings from environment parameters to decision variables, thanks to their superior performance, and the potential in enabling scalability and size generalizability. These merits are rooted in leveraging permutation prior, i.e., satisfying the permutation property of the policy to be learned (referred to as desired permutation property). Many wireless policies are with complicated permutation properties. To satisfy these properties, heterogeneous GNNs (HetGNNs) should be used to learn such policies. There are two critical factors that enable a HetGNN to satisfy a desired permutation property: constructing an appropriate heterogeneous graph and judiciously designing the architecture of the HetGNN. However, both the graph and the HetGNN are designed heuristically so far. In this paper, we strive to provide a systematic approach for the design to satisfy the desired permutation property. We first propose a method for constructing a graph for a policy, where the edges and their types are defined for the sake of satisfying complicated permutation properties. Then, we provide and prove three sufficient conditions to design a HetGNN such that it can satisfy the desired permutation property when learning over an appropriate graph. These conditions suggest a method of designing the HetGNN with desired permutation property by sharing the processing, combining, and pooling functions according to the types of vertices and edges of the graph. We take power allocation and hybrid precoding policies as examples for demonstrating how to apply the proposed methods and validating the impact of the permutation prior by simulations.