Under an internet background involving artificial intelligence and big data—unstructured, materialized, network graph-structured data, such as social networks, knowledge graphs, and compound molecules, have gradually entered into various specific business scenarios. One problem that urgently needs to be solved in the industry involves how to perform feature extractions, transformations, and operations in graph-structured data to solve downstream tasks, such as node classifications and graph classifications in actual business scenarios. Therefore, this paper proposes a gated recursion-based graph neural network (GR-GNN) algorithm to solve tasks such as node depth-dependent feature extractions and node classifications for graph-structured data. The GRU neural network unit was used to complete the node classification task and, thereby, construct the GR-GNN model. In order to verify the accuracy, effectiveness, and superiority of the algorithm on the open datasets Cora, CiteseerX, and PubMed, the algorithm was used to compare the operation results with the classical graph neural network baseline algorithms GCN, GAT, and GraphSAGE, respectively. The experimental results show that, on the validation set, the accuracy and target loss of the GR-GNN algorithm are better than or equal to other baseline algorithms; in terms of algorithm convergence speed, the performance of the GR-GNN algorithm is comparable to that of the GCN algorithm, which is higher than other algorithms. The research results show that the GR-GNN algorithm proposed in this paper has high accuracy and computational efficiency, and very wide application significance.

Single-index models are common dimensionreduction models in statistics. In practical applications, the link function may havesingularities, including jumps at some unknown positions and structural changes of therelated process. Detection of such singularities is important for estimating the coefficientsand understanding the structural changes. This article proposes a jump detection methodbased on the refined minimum average conditional variance estimation and zero-crossingproperties of second-order derivative of a function. Using the detected jumps, we givesemiparametric jump-detection-based estimators for the parameter vector and the linkfunction. Then the selection of procedure parameters is discussed. Under some mild conditions,we establish large sample properties of the jump detection procedure and the proposedestimators. Numerical studies and a real data analysis are conducted to assess the finitesample property of the proposed methods.