Learning representations on graphs is foundational for many downstream tasks, and its synergy with diffusion models has emerged as a promising direction. However, diffusion-based methods for heterogeneous graphs remain underexplored, confronting two principal challenges: (1) The presence of noise and structural heterogeneity in graphs makes it challenging to accurately capture semantic transitions among diverse relation types. (2) The isotropic Gaussian noise used in forward diffusion fails to reflect graphs' inherent semantics and structural anisotropy. To address these, we propose ARDiff, a novel framework that integrates residual diffusion with anisotropic noise for heterogeneous graph learning. Specifically, we propose a semantic residual diffusion mechanism that progressively refines node embeddings by orchestrating transitions from low-semantic (high-noise) to high-semantic (low-noise) relational contexts, thus enabling step-wise distillation of task-relevant information. In addition, to address the limitations of conventional diffusion, we introduce an anisotropic diffusion strategy: in the forward process, noise injection is oriented by structural and semantic priors; in the denoising step, a conditional diffusion mechanism is guided by a random walk encoding, enhancing both topological consistency and semantic alignment. Extensive evaluation on heterogeneous graph datasets demonstrates that ARDiff significantly surpasses current leading methods in link prediction and node classification, setting a new paradigm and benchmark in heterogeneous graph representation learning.

The sparse user-item interactions in recommender systems hinder the quality of embedding representations and degraded recommendation performance. Existing methods attempt to alleviate this sparsity issue by incorporating auxiliary information via item tags, but often neglect structured characteristics of embedding space, such as semantic distribution and logical relations. To this end, we propose VoRec, a novel framework that explores the spatial distribution of items and their associated tags to achieve accurate recommendations in hyperbolic space. Specifically, we employ the Voronoi diagram to partition hyperbolic space into logically related subspaces, based on tag distributions and relationships derived from existing tag taxonomies. In addition, we combine the Voronoi diagram with the Hyperbolic Graph Convolutional Network (HGCN) and exploit the respective advantages of the Poincaré and Lorentz models in hyperbolic space. Finally, we develop two types of Voronoi site update strategies, namely active and passive ones, to optimize the Voronoi diagram for recommendation tasks. The active strategy employs contrastive learning to guide updates to the Voronoi diagram, while the passive strategy adaptively freezes parameters based on information gain to regulate the update rate. Extensive experiments on four real-world benchmark datasets demonstrate that our proposed VoRec framework delivers substantial performance improvements, achieving an average 16.35% enhancement in Recall and NDCG metrics compared to state-of-the-art baselines. The model implementation is publicly available at: https://github.com/s35lay/VoRec.