Graph Convolutional Network (GCN) is an important method for learning graph representations of nodes. For large-scale graphs, the GCN could meet with the neighborhood expansion phenomenon, which makes the model complexity high and the training time long. An efficient solution is to adopt graph sampling techniques, such as node sampling and random walk sampling. However, the existing sampling methods still suffer from aggregating too many neighbor nodes and ignoring node feature information. Therefore, in this paper, we propose a new subgraph sampling method, namely, Similarity-Aware Random Walk (SARW), for GCN with large-scale graphs. A novel similarity index between two adjacent nodes is proposed, describing the relationship of nodes with their neighbors. Then, we design a sampling probability expression between adjacent nodes using node feature information, degree information, neighbor set information, etc. Moreover, we prove the unbiasedness of the SARW-based GCN model for node representations. The simplified version of SARW (SSARW) has a much smaller variance, which indicates the effectiveness of our subgraph sampling method in large-scale graphs for GCN learning. Experiments on six datasets show our method achieves superior performance over the state-of-the-art graph sampling approaches for the large-scale graph node classification task.

Storing information in DNA molecules is of great interest because of its advantages in longevity, high storage density, and low maintenance cost. A key step in the DNA storage pipeline is to efficiently cluster the retrieved DNA sequences according to their similarities. Levenshtein distance is the most suitable metric on the similarity between two DNA sequences, but it is inferior in terms of computational complexity and less compatible with mature clustering algorithms. In this work, we propose a novel deep squared Euclidean embedding for DNA sequences using Siamese neural network, squared Euclidean embedding, and chi-squared regression. The Levenshtein distance is approximated by the squared Euclidean distance between the embedding vectors, which is fast calculated and clustering algorithm friendly. The proposed approach is analyzed theoretically and experimentally. The results show that the proposed embedding is efficient and robust.

Different from deep neural networks for non-graph data classification, graph neural networks (GNNs) leverage the information exchange between nodes (or samples) when representing nodes. The category distribution shows an imbalance or even a highly-skewed trend on nearly all existing benchmark GNN data sets. The imbalanced distribution will cause misclassification of nodes in the minority classes, and even cause the classification performance on the entire data set to decrease. This study explores the effects of the imbalance problem on the performances of GNNs and proposes new methodologies to solve it. First, a node-level index, namely, the label difference index ($LDI$), is defined to quantitatively analyze the relationship between imbalance and misclassification. The less samples in a class, the higher the value of its average$LDI$; the higher the$LDI$of a sample, the more likely the sample will be misclassified. We define a new loss and propose four new methods based on$LDI$. Experimental results indicate that the classification accuracies of the three among our proposed four new methods are better in both transductive and inductive settings. The$LDI$can be applied to other GNNs.

In this paper, we solve the existence problem of telescopers for rational functions in three discrete variables. We reduce the problem to that of deciding the summability of bivariate rational functions, a problem which has recently been solved. This existence criteria is used, for example, for detecting the termination of Zeilberger's algorithm to the function classes studied in this paper.