Hyperbolic neural networks (HNNs) are emerging as a promising tool for representing data embedded in non-Euclidean geometries, yet their adoption has been hindered by challenges related to stability and robustness. In this work, we conduct a rigorous Lipschitz analysis for HNNs and propose using Lipschitz regularization as a novel strategy to enhance their robustness. Our comprehensive investigation spans both the Poincaré ball model and the hyperboloid model, establishing Lipschitz bounds for HNN layers. Importantly, our analysis provides detailed insights into the behavior of the Lipschitz bounds as they relate to feature norms, particularly distinguishing between scenarios where features have unit norms and those with large norms. Further, we study regularization using the derived Lipschitz bounds. Our empirical validations demonstrate consistent improvements in HNN robustness against noisy perturbations.

The recent rapid growth in mobile data traffic entails a pressing demand for improving the throughput of the underlying wireless communication networks. Network node deployment has been considered as an effective approach for throughput enhancement which, however, often leads to highly nontrivial nonconvex optimizations. Although convex-approximation-based solutions are considered in the literature, their approximation to the actual throughput may be loose and sometimes lead to unsatisfactory performance. With this consideration, in this article, we propose a novel graph neural network (GNN) method for the network node deployment problem. Specifically, we fit a GNN to the network throughput and use the gradients of this GNN to iteratively update the locations of the network nodes. Besides, we show that an expressive GNN has the capacity to approximate both the function value and the gradients of a multivariate permutation-invariant function, as a theoretic support to the proposed method. To further improve the throughput, we also study a hybrid node deployment method based on this approach. To train the desired GNN, we adopt a policy gradient algorithm to create datasets containing good training samples. Numerical experiments show that the proposed methods produce competitive results compared with the baselines.