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.

Periodic group behaviors often exist in temporal interaction networks, such as monthly group meetings, quarterly animal migrations, and yearly birthday parties. In real life, these events are usually quasi-periodic, meaning that the time intervals between two adjacent events are nearly constant but not exactly constant. Most existing studies mainly focus on identifying exact periodic group behaviors, which may result in an incomplete detection of periodic patterns in temporal networks. To fill this gap, we focus on a quasi-periodic community mining problem, which aims to find the most representative cohesive sub graphs, including the quasi-periodic$k$-core and quasi-periodic k-clique. The number of quasi-periodic communities is much larger than that of periodic communities, since the number of quasi-periodic sub-sequences is larger than that of periodic sub-sequences in a given time sequence. To efficiently compute the quasi-periodic communities, we propose a novel two-stage framework. In the first stage, the framework checks whether the time sequence of each vertex contains quasi-periodic sub-sequences. To this end, we develop a new structure, the DAG oracle, which comprises a set of concise DAGs that enables rapid extraction of all quasi-periodic sub-sequences. Based on the DAG oracle, we can easily compute all quasi-periodic sub-sequences for every vertex. In the second stage, the framework computes local quasi-periodic subgraphs that contain the vertex, which allows for the application of existing community mining algorithms. Given the large number of these subgraphs, we propose several carefully -designed pruning rules to further reduce redundant computations. Extensive experiments on 5 real-life datasets demonstrate the efficiency and effectiveness of our proposed solutions.

Data warehouse and OLAP (Online Analytical Processing) are effective tools for decision support on traditional relational data and static multidimensional network data. However, many real-world multidimensional networks are often modeled as temporal multidimensional networks, where the edges in the network are associated with temporal information. Such temporal multidimensional networks typically cannot be handled by traditional data warehouse and OLAP techniques. To fill this gap, we propose a novel data warehouse model, named$\mathsf {Temporal{ }\; Graph{ }\; Cube}$, to support OLAP queries on temporal multidimensional networks. Through supporting OLAP queries in any time range, users can obtain summarized information of the network in the time range of interest, which cannot be derived by using traditional static graph OLAP techniques. We propose a segment-tree based indexing technique to speed up the OLAP queries, and also develop an index-updating technique to maintain the index when the temporal multidimensional network evolves over time. In addition, we also propose a novel concept called$\mathsf {similarity{ }\; of{ }\; snapshots}$which shows a strong correlation with the efficiency of indexing technique and can provide a good reference on the necessity of building the index. The results of extensive experiments on two large real-world datasets demonstrate the effectiveness and efficiency of the proposed method.