Designing Heterogeneous GNNs With Desired Permutation Properties for Wireless Resource Allocation

Graph neural networks (GNNs) have been designed for learning a variety of resource allocation policies, i.e., the mappings from environment parameters to allocated resources, thanks to their superior performance, and the potential in enabling size generalizability and scalability. 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 have complex permutation properties, and heterogeneous GNNs (HetGNNs) ought to be used to satisfy these properties. Two key factors enable a HetGNN to satisfy desired permutation property: a heterogeneous graph and the architecture of the HetGNN, both are usually designed heuristically for a specific problem. In this paper, we strive to provide a systematic and general approach for the design to satisfy the desired permutation property. We first propose a method for constructing a graph to learn a policy, where the edges and their types are defined for the sake of satisfying complex permutation properties. Then, we provide three sufficient conditions with rigorous proof for ensuring a HetGNN to satisfy the desired permutation property when learning over an appropriate graph. These conditions suggest a method for designing the HetGNN with desired permutation property by tying its processing, combining, and pooling functions based on the types of vertices and edges of the graph. We focus on two representative examples – power allocation in multi-cell-multi-user systems and hybrid precoding in multi-user multi-antenna systems – for demonstrating how to apply the proposed methods and validating the impact of exploiting permutation prior through simulations.