Efficient Algorithms for Density Decomposition on Large Static and Dynamic Graphs

Locally-densest subgraph (LDS) decomposition is a fundamental decomposition in graph analysis that finds numerous applications in various domains, including community detection, fraud detection, graph querying, and graph visualization. However, the LDS decomposition is computationally challenging for both static and dynamic graphs. Furthermore, the LDS decomposition often produces an excessive number of dense subgraph layers, leading to the unnecessary partition of tightly-connected subgraphs. To address these limitations, an alternative concept called density decomposition was proposed, which can generate a more reasonable number of dense subgraph layers. However, the state-of-the-art algorithm for density decomposition requires O(m(2)) time (m is the number of edges of the graph), which is very costly for large graphs. In this paper, we conduct an in-depth investigation of density decomposition and propose efficient algorithms for computing it on both static and dynamic graphs. First, we establish a novel relationship between density decomposition and LDS decomposition. Second, based on these relationships, we propose novel algorithms to compute the density decomposition on static graphs with carefully designed network flow and divide-and-conquer techniques. Our proposed static algorithms significantly reduce the time complexity to O(m(3/2) log p) (p is often a very small constant in real-world graphs). Third, for dynamic graphs, we develop three dynamic algorithms with efficient O(m) time complexity. Extensive experiments on several large real-world graphs demonstrate the high efficiency, scalability, and effectiveness of the proposed algorithms.