Important communities are densely connected subgraphs containing vertices with high importance values, which have received wide attention recently. However, existing methods, predominantly based on the$k$-core model, suffer from limitations such as rigid degree constraints and suboptimal density, often failing to capture highly important vertices. To address these limitations, we propose a new community model based on pseudoarboricity that guarantees near-optimal density while preserving important vertices. Further, we introduce a novel problem of Psudoarboricity-based Skyline Important Community (PSIC), which uniquely treats density and importance as independent attributes. To efficiently address PSIC, we first devise a basic algorithm climbStairs, which iteratively refines communities by peeling vertices with low importance. To boost efficiency, we develop an advanced algorithm DivAndCon, which employs a recursive divide-and-conquer strategy combined with weight-based and pseudoarboricity-based pruning techniques, significantly reducing the search space. For massive graphs with billions of edges, inspired by a recursive division tree, we develop several parallel algorithms utilizing thread-pool and free-synchronization mechanism. Finally, we conduct extensive experiments on 10 real-world networks, and the results demonstrate the superiority of our solutions in terms of effectiveness, efficiency, and scalability.

The densest subgraph (DS) search over a directed graph focuses on finding the subgraph with the highest density among all subgraphs. This problem has raised numerous applications, such as fraud detection and community detection. The state-of-the-art DS algorithms have prohibitively high costs or poor approximation ratios, making them unsuitable for practical applications. To address these dilemmas, in this paper, we propose a novel model called integral densest subgraph (IDS). We show that IDS can serve as a near-DS model that has a tight floor relationship with the density of the DS. To compute IDS, we first propose a novel flow network named (α,β)-dense network, based on which we design an exact network-flow algorithm GetIDS with O(p • log |V| • |E| 1.5 ) time complexity, where p is typically a small constant in real-world graphs. Additionally, we propose several non-trivial pruning techniques to further improve the efficiency. Subsequently, we propose a novel (2 + ε)-approximation algorithm MultiCore with near-linear time complexity, providing a good approximation guarantee with high efficiency. Finally, our extensive experiments on 10 real-world graphs demonstrate the effectiveness of the proposed IDS model, and the high efficiency and scalability of the proposed solutions.

Mining dense subgraphs in a bipartite graph is a fundamental task in bipartite graph analysis, with numerous applications in community detection, fraud detection, and e-commerce recommendation. Existing dense subgraph models, such as biclique, k -biplex, k -bitruss, and (α,β)-core, often face challenges due to their high computational complexity or limitations in effectively capturing the density of the graph. To overcome these issues, in this paper, we propose a new dense subgraph model for bipartite graphs, namely (α,β)-dense subgraph, designed to capture the density structure inherent in bipartite graphs. We show that all (α,β)-dense subgraphs are nested within each other, forming a hierarchical density decomposition of the bipartite graph. To efficiently compute the (α,β)-dense subgraph, we develop a novel network flow algorithm with a carefully-designed core pruning technique. The time complexity of our algorithm is O(|E|+|E(R)| 1.5 ), where |E| denotes the number of edges and |E(R)| is the number of edges of the pruned graph, often significantly smaller than |E|. Armed with this algorithm, we also propose a novel and efficient divide-and-conquer algorithm to compute the entire density decomposition of the bipartite graph within O(p ⋅ log d max ⋅ |E| 1.5 ) time, where p is typically a small constant in real-world bipartite graphs and d max is the maximum degree. Extensive experiments and case studies on 11 real-world datasets demonstrate the effectiveness of our (α,β)-dense subgraph model and the high efficiency and scalability of our proposed algorithms.

The arboricity a ( G ) of a graph G is defined as the minimum number of edge-disjoint forests that the edge set of G can be partitioned into. It is a fundamental metric and has been widely used in many graph analysis applications. However, computing a ( G ) is typically a challenging task. To address this, an easier-to-compute alternative called pseudoarboricity was proposed. Pseudoarboricity has been shown to be closely connected to many important measures in graphs, including the arboricity and the densest subgraph density ρ ( G ). Computing the exact pseudoarboricity can be achieved by employing a parametric max-flow algorithm, but it becomes computationally expensive for large graphs. Existing 2-approximation algorithms, while more efficient, often lack satisfactory approximation accuracy. To overcome these limitations, we propose two new approximation algorithms with theoretical guarantees to approximate the pseudoarboricity. We show that our approximation algorithms can significantly reduce the number of times the max-flow algorithm is invoked, greatly improving its efficiency for exact pseudoarboricity computation. In addition, we also study the pseudoarboricity maintenance problem in dynamic graphs. We propose two novel and efficient algorithms for maintaining the pseudoarboricity when the graph is updated by edge insertions or deletions. Furthermore, we develop two incremental pseudoarboricity maintenance algorithms specifically designed for insertion-only scenarios. We conduct extensive experiments on 195 real-world graphs, and the results demonstrate the high efficiency and scalability of the proposed algorithms in computing pseudoarboricity for both 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.

Densest subgraph search, aiming to identify a subgraph with maximum edge density, faces limitations as the edge density inadequately reflects biases towards a given vertex set R. To address this, the R -subgraph density was introduced, refining the doubled edge density by penalizing vertices in a subgraph but not in R , using the degree as a penalty factor. This advancement leads to the Anchored Densest Subgraph (ADS) search problem, which finds the subgraph Š with the highest R -subgraph density for a given set R. Nonetheless, current algorithms for ADS search face significant inefficiencies in handling large-scale graphs or the sizable R set. Furthermore, these algorithms require re-computing the ADS whenever the graph is updated, complicating the efficient maintenance within dynamic graphs. To tackle these challenges, we propose the concept of integer R -subgraph density and study the problem of finding a subgraph S * ⊆ V with the highest integer R -subgraph density. We reveal that the R -subgraph density of S* provides an additive approximation to that of ADS with a difference of less than 1, and hence S * is termed the Approximate Anchored Densest Subgraph (AADS). For searching the AADS, we present an efficient global algorithm incorporating the re-orientation network flow technique and binary search, operating in a time polynomial to the graph's size. Additionally, we propose a novel local algorithm using shortest-path-based methods for the max-flow computation from s to t around R , markedly boosting performance in scenarios with larger R sets. For dynamic graphs, both basic and improved algorithms are developed to efficiently maintain the AADS when an edge is updated. Extensive experiments and a case study demonstrate the efficiency, scalability, and effectiveness of our solutions.