Detecting anomalies for dynamic graphs has drawn increasing attention due to their wide applications in social networks, e-commerce, and cybersecurity. Recent deep learning-based approaches have shown promising results over shallow methods. However, they fail to address two core challenges of anomaly detection in dynamic graphs: the lack of informative encoding for unattributed nodes and the difficulty of learning discriminate knowledge from coupled spatial-temporal dynamic graphs. To overcome these challenges, in this paper, we present a novel T ransformer-based A nomaly D etection framework for DY namic graphs ( TADDY ). Our framework constructs a comprehensive node encoding strategy to better represent each node’s structural and temporal roles in an evolving graphs stream. Meanwhile, TADDY captures informative representation from dynamic graphs with coupled spatial-temporal patterns via a dynamic graph transformer model. The extensive experimental results demonstrate that our proposed TADDY framework outperforms the state-of-the-art methods by a large margin on six real-world datasets.

Turnover numbers (kcat), which indicate an enzyme's catalytic efficiency, have a wide range of applications in fields including protein engineering and synthetic biology. Experimentally measuring the enzymes' kcat is always time-consuming. Recently, the prediction of kcat using deep learning models has mitigated this problem. However, the accuracy and robustness in kcat prediction still needs to be improved significantly, particularly when dealing with enzymes with low sequence similarity compared to those within the training dataset. Herein, we present DeepEnzyme, a cutting-edge deep learning model that combines the most recent Transformer and Graph Convolutional Network (GCN) to capture the information of both the sequence and 3D-structure of a protein. To improve the prediction accuracy, DeepEnzyme was trained by leveraging the integrated features from both sequences and 3D-structures. Consequently, DeepEnzyme exhibits remarkable robustness when processing enzymes with low sequence similarity compared to those in the training dataset by utilizing additional features from high-quality protein 3D-structures. DeepEnzyme also makes it possible to evaluate how point mutations affect the catalytic activity of the enzyme, which helps identify residue sites that are crucial for the catalytic function. In summary, DeepEnzyme represents a pioneering effort in predicting enzymes' kcat values with improved accuracy and robustness compared to previous algorithms. This advancement will significantly contribute to our comprehension of enzyme function and its evolutionary patterns across species.

Geometric properties of N random points distributed independently and uniformly on the unit sphere with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as N → ∞) for the expected moments of the radii of spherical caps associated with the facets of the convex hull of N random points on . We provide conjectures for the asymptotic distribution of the scaled radii of these spherical caps and the expected value of the largest of these radii (the covering radius). Numerical evidence is included to support these conjectures. Furthermore, utilizing the extreme law for pairwise angles of Cai et al., we derive precise asymptotics for the expected separation of random points on .