  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.
