Graphs provide a powerful framework for modeling complex systems, but their structural variability makes analysis and classification challenging. To address this, we introduce GAUDI (Graph Autoencoder Uncovering Descriptive Information), a novel unsupervised geometric deep learning framework that captures both local details and global structure. GAUDI employs an innovative hourglass architecture with hierarchical pooling and upsampling layers, linked through skip connections to preserve essential connectivity information throughout the encoding-decoding process. By mapping different realizations of a system - generated from the same underlying parameters - into a continuous, structured latent space, GAUDI disentangles invariant process-level features from stochastic noise. We demonstrate its power across multiple applications, including modeling small-world networks, characterizing protein assemblies from super-resolution microscopy, analyzing collective motion in the Vicsek model, and capturing age-related changes in brain connectivity. This approach not only improves the analysis of complex graphs but also provides new insights into emergent phenomena across diverse scientific domains.

翻译：图结构为复杂系统建模提供了强大的框架，但其结构的多变性使得分析和分类具有挑战性。为此，我们提出了GAUDI（Graph Autoencoder Uncovering Descriptive Information），一种新颖的无监督几何深度学习框架，能够同时捕捉局部细节与全局结构。GAUDI采用创新的沙漏架构，包含分层池化与上采样层，并通过跳跃连接相链接，以在整个编码-解码过程中保持关键的连通性信息。通过将系统（由相同底层参数生成）的不同实现映射到一个连续、结构化的潜在空间中，GAUDI能够将不变的过程级特征与随机噪声分离开来。我们在多个应用中展示了其强大能力，包括小世界网络建模、超分辨率显微镜下的蛋白质组装表征、Vicsek模型中的集体运动分析，以及大脑连接中与年龄相关变化的捕捉。该方法不仅改进了复杂图的分析，还为跨不同科学领域涌现现象的理解提供了新的见解。