Graph analytics attract much attention from both research and industry communities. Due to the linear time complexity, the $k$-core decomposition is widely used in many real-world applications such as biology, social networks, community detection, ecology, and information spreading. In many such applications, the data graphs continuously change over time. The changes correspond to edge insertion and removal. Instead of recomputing the $k$-core, which is time-consuming, we study how to maintain the $k$-core efficiently. That is, when inserting or deleting an edge, we need to identify the affected vertices by searching for more vertices. The state-of-the-art order-based method maintains an order, the so-called $k$-order, among all vertices, which can significantly reduce the searching space. However, this order-based method is complicated for understanding and implementation, and its correctness is not formally discussed. In this work, we propose a simplified order-based approach by introducing the classical Order Data Structure to maintain the $k$-order, which significantly improves the worst-case time complexity for both edge insertion and removal algorithms. Also, our simplified method is intuitive to understand and implement; it is easy to argue the correctness formally. Additionally, we discuss a simplified batch insertion approach. The experiments evaluate our simplified method over 12 real and synthetic graphs with billions of vertices. Compared with the existing method, our simplified approach achieves high speedups up to 7.7x and 9.7x for edge insertion and removal, respectively.

翻译：图表分析吸引了研究和产业界的注意力。 由于时间的线性复杂性, 美元核心分解在生物学、 社交网络、 社区检测、 生态和信息传播等许多真实世界的应用中被广泛使用。 在许多这样的应用中, 数据图表会随着时间的变化而不断变化。 这些变化与边缘插入和删除相对应。 这种基于秩序的方法对于理解和执行来说比较复杂,而且没有正式讨论这种方法的正确性。 在这项工作中, 我们建议采用基于秩序的简化方法, 采用经典秩序数据结构来维持美元水平的分流, 从而大大改进基于秩序的分流。 基于秩序的状态方法在生物学、 社会网络、 社区检测、 以及信息传播信息传播等方面都维持着秩序。 最糟糕的分流速度, 使用最简化的分流方法, 使用最简化的分流方法, 并使用更简化的分流法, 使用更简化的分流法, 使用更精确的分解法, 使用更精确的分解法, 使用更简化的分解法, 进行更精确的分解。