Accelerating D-Core Maintenance over Dynamic Directed Graphs

Given a directed graph $G$ and two non-negative integers $k$ and $l$, a D-core, or ($k$, l)-core, is the maximal subgraph $H\subseteq G$ where each vertex in $H$ has an in-degree and out-degree not smaller than $k$ and $I$, respectively. D-cores have found extensive applications, such as social network analysis, fraud detection, and graph visualization. In these applications, graphs are highly dynamic and frequently updated with the insertions and deletions of vertices and edges, making it costly to recompute the D-cores from scratch to handle the updates. In the literature, the peeling-based algorithm has been proposed to handle D-core maintenance. However, the peeling-based method suffers from efficiency issues, e.g., it may degenerate into recomputing all the D-cores and is inefficient for batch updates due to sequential processing. To address these limitations, we introduce novel algorithms for incrementally maintaining D-cores in dynamic graphs. We begin by presenting the theoretical findings to identify the D-cores that should be updated. By leveraging these theoretical analysis results, we propose a local-search-based algorithm with optimizations to handle single-edge insertions and deletions. We further propose an H-index-based algorithm for scenarios involving batch updates. Several novel edge-grouping strategies are proposed to improve the efficiency of the H-index-based algorithm. Extensive empirical evaluations over both real-world and synthetic networks demonstrate that our proposed algorithms are up to 5 orders of magnitude faster than the peeling-based method.