Learning over dynamic graphs poses major challenges, including capturing the evolving relationship in the graphs. Inspired by the advantages of hyperbolic embedding in static graphs, the hyperbolic space is expected to capture complex interactions in dynamic graphs. However, due to the distortion errors in the standard tangent space mappings, hyperbolic methods become more sensitive to noise and reduce the learning capacity. To address the distortion in tangent space, we proposed HMPTGN, a temporal graph network that operates directly on the hyperbolic manifold. In this journal paper, we introduce the HMPTGN+ architecture, an extension of the original HMPTGN with major updates to learn better representations of dynamic graphs based on the hyperbolic embedding. Our framework incorporates a high-order graph neural network for extracting spatial dependencies, a dilated causal attention mechanism for modeling temporal patterns while preserving causality, and a curvature-awareness mechanism to capture dynamic structures. Extensive experiments demonstrate the effectiveness of our proposed HMPTGN+ framework over state-of-the-art baselines in both temporal link prediction and temporal new link prediction tasks.

Many real-world graphs exhibit diverse and complex topological structures that are not well captured by geometric manifolds with uniform global curvature, such as hyperbolic or spherical spaces. Recently, there has been growing interest in embedding graphs into pseudo-Riemannian manifolds, which generalize both hyperbolic and spherical geometries. However, existing approaches face three significant limitations, including the ineffective pseudo-Riemannain framework, the shallow architectures, and the absence of clear guideline for selecting suitable pseudo-Riemannian manifolds. To address these issues, we introduce a novel diffeomorphic framework for graph embedding that aligns with the nature of pseudo-Riemannian manifolds. Subsequently, we propose the pseudo-Riemannian Graph Transformer for learning representations of complex graph structures. Our diffeomorphic framework in pseudo-Riemannian geometry enables the principled definitions of core transformer components, including linear attention, residual connection, and layer normalization. Finally, we develop a lightweight space searching algorithm to automatically identify the most suitable pseudo-Riemannian manifold for an input graph. Extensive experiments on diverse real-world graphs demonstrate that our model consistently outperforms other baselines in both node classification and link prediction tasks.