Graph neural networks (GNNs) have been extensively utilized in various fields due to their efficient and strong capability in processing graph-structured data. However, as the scale of real-world graphs continues to grow, processing large-scale graphs with GNNs has become increasingly storage-intensive and time-consuming. Graph condensation, which aims to synthesize a smaller yet informative substitute graph, has emerged as a promising solution. Most existing approaches, however, primarily focus on feature condensation while paying less attention to faithfully reconstructing structural information. Moreover, the condensed representations are typically dense, which not only limits storage and computational efficiency but also risks retaining noise and redundancy. To address these limitations, we propose DGC, a Discrete Graph Condensation framework, where numerous nodes in the original graph can be represented by the discrete codes in the codebook while maintaining the original structure information. Specifically, we leverage high-order proximity between nodes to capture the original graph's structure and transfer it to the condensed graph via low-rank approximation. Meanwhile, node features are discretized and compressed through a vector quantized variational autoencoder, effectively reducing redundancy and enhancing storage efficiency while retaining critical information. Extensive experiments on multiple real-world graphs demonstrate the effectiveness of our proposed model against nine competitive baseline methods across different condensation ratios.

In recent years, Message Passing Neural Networks (MPNNs) have emerged as a leading paradigm for learning over graph-structured data, owing to their simple architecture and excellent scalability. However, their expressive power is strictly limited by the one-dimensional Weisfeiler–Leman (1-WL) test. A common approach to enhance expressivity is designing models based on higher-order k -WL and k -Folklore WL ( k -FWL) tests, collectively referred to as tuple-based GNNs. Despite their greater expressive power, these methods face two major challenges: (1) The neighboring tuples in k -WL/FWL are generated by replacing any node in the graph, disregarding adjacency relationships and thus failing to capture structural dependencies between tuples. (2) The design of k -WL/FWL lacks flexibility, while increasing k enhances expressivity, it also results in an exponential surge in time and space complexity. To overcome these limitations, we propose Trail-WL, a more expressive variant of the WL test, along with its neural network counterpart, Trail-based Tuple Neural Network (TTNN). We redefine node tuples as collections of trail structures, providing a more intuitive representation of graph connectivity compared to prior tuple-based methods. By incorporating variable trail lengths and node distance constraints, our model adaptively balances expressivity and scalability while preserving essential structural features. Furthermore, we propose and formally prove three theoretical expressivity theorems for Trail-WL. Finally, we conduct comprehensive experiments on both synthetic and real-world datasets, demonstrating that our approach consistently matches or outperforms baseline methods, thereby substantiating its effectiveness.