The distance-generalized core, also called ( k , h )-core, is defined as the maximal subgraph in which every vertex has at least k vertices at distance no longer than h. Compared with k -core, ( k , h )-core can identify more fine-grained subgraphs and, hence, is more useful for the applications such as network analysis and graph coloring. The state-of-the-art algorithms for ( k , h )-core decomposition are peeling algorithms, which iteratively delete the vertex with the minimum h -degree (i.e., the least number of neighbors within h hops). However, they suffer from some limitations, such as low parallelism and incapability of supporting dynamic graphs. To address these limitations, in this paper, we revisit the problem of ( k , h )-core decomposition. First, we introduce two novel concepts of pairwise h-attainability index and n-order H-index based on an insightful observation. Then, we thoroughly analyze the properties of n -order H-index and propose a parallelizable local algorithm for ( k , h )-core decomposition. Moreover, several optimizations are presented to accelerate the local algorithm. Furthermore, we extend the proposed local algorithm to address the ( k , h )-core maintenance problem for dynamic graphs. Experimental studies on real-world graphs show that, compared to the best existing solution, our proposed algorithms can reduce the ( k , h )-core decomposition time by 1--3 orders of magnitude and save the maintenance cost by 1--2 orders of magnitude.

Learning to rank from logged user feedback, such as clicks or purchases, is a central component of many real-world information systems. Different from human-annotated relevance labels, the user feedback is always noisy and biased. Many existing learning to rank methods infer the underlying relevance of query–item pairs based on different assumptions of examination, and still optimize a relevance based objective. Such methods rely heavily on the correct estimation of examination, which is often difficult to achieve in practice. In this work, we propose a general framework U-rank+ for learning to rank with logged user feedback from the perspective of graph matching. We systematically analyze the biases in user feedback, including examination bias and selection bias. Then, we take both biases into consideration for unbiased utility estimation that directly based on user feedback, instead of relevance. In order to maximize the estimated utility in an efficient manner, we design two different solvers based on Sinkhorn and LambdaLoss for U-rank+ . The former is based on a standard graph matching algorithm, and the latter is inspired by the traditional method of learning to rank. Both of the algorithms have good theoretical properties to optimize the unbiased utility objective while the latter is proved to be empirically more effective and efficient in practice. Our framework U-rank+ can deal with a general utility function and can be used in a widespread of applications including web search, recommendation, and online advertising. Semi-synthetic experiments on three benchmark learning to rank datasets demonstrate the effectiveness of U-rank+ . Furthermore, our proposed framework has been deployed on two different scenarios of a mainstream App store, where the online A/B testing shows that U-rank+ achieves an average improvement of 19.2% on click-through rate and 20.8% improvement on conversion rate in recommendation scenario, and 5.12% on platform revenue in online advertising scenario over the production baselines.