The core numbers of vertices in a graph are one of the most well-studied cohesive subgraph models because of the linear running time. In practice, many data graphs are dynamic graphs that are continuously changing by inserting or removing edges. The core numbers are updated in dynamic graphs with edge insertions and deletions, which is called core maintenance. When a burst of a large number of inserted or removed edges come in, we have to handle these edges on time to keep up with the data stream. There are two main sequential algorithms for core maintenance, \textsc{Traversal} and \textsc{Order}. It is proved that the \textsc{Order} algorithm significantly outperforms the \alg{Traversal} algorithm over all tested graphs with up to 2,083 times speedups. To the best of our knowledge, all existing parallel approaches are based on the \alg{Traversal} algorithm; also, their parallelism exists only for affected vertices with different core numbers, which will reduce to sequential when all vertices have the same core numbers. In this paper, we propose a new parallel core maintenance algorithm based on the \alg{Order} algorithm. Importantly, our new approach always has parallelism, even for the graphs where all vertices have the same core numbers. Extensive experiments are conducted over real-world, temporal, and synthetic graphs on a 64-core machine. The results show that for inserting and removing 100,000 edges using 16-worker, our method achieves up to 289x and 10x times speedups compared with the most efficient existing method, respectively.

翻译：图中的顶点核数是研究最充分的凝聚子图模型之一，因其具有线性运行时间。在实际应用中，许多数据图是动态图，通过插入或删除边不断变化。核数在具有边插入和删除的动态图中更新，这称为核维护。当大量插入或删除边突发到来时，我们必须及时处理这些边以跟上数据流。目前有两种主要的核维护顺序算法：\textsc{Travel}和\textsc{Order}。已证明，\textsc{Order}算法在所有测试图上显著优于\textsc{Traversal}算法，加速比最高达2083倍。据我们所知，所有现有的并行方法均基于\textsc{Traversal}算法；此外，它们的并行性仅存在于具有不同核数的受影响顶点之间，当所有顶点具有相同核数时会退化为顺序执行。本文提出了一种基于\textsc{Order}算法的新型并行核维护算法。重要的是，我们的新方法始终具有并行性，即使对于所有顶点核数相同的图也是如此。我们在64核机器上对真实图、时间图和合成图进行了大量实验。结果表明，在使用16个工作进程插入和删除10万条边时，与现有最有效方法相比，我们的方法分别实现了最高289倍和10倍的加速。