This article studies the batch processing of truss maintenance in large graphs. Trussness is a widely used index in graph analytics for cohesive subgraph mining. It is defined on edges to reflect the closeness of vertices connected by the edges. The trussness maintenance problem, i.e., updating trussness after edge insertions/deletions and avoiding recomputation, was proposed by Cohen (2008) with the assumption that real graphs are continuously evolving in nature and their dynamicity usually only affects few edges. Different from the existing work that mainly focuses on simple single edge/vertex insertion/deletion, we propose batch processing algorithms for truss maintenance with multiedge insertions/deletions. By presenting an edge structure called triangle disjoint set, whose insertion/deletion can make each edge change its trussness by at most 1, we tackle the difficulty in quantifying the trussness changes of the edges in batch processing. More specifically, we propose two indices, namely sustain support and pivotal support, to help measure whether the edges have the potential to change their trussness, such that the search range of the potential edges is greatly reduced. The extensive experiments on real-world graphs illustrate that compared with the single-edge processing approach, our batch processing algorithms can significantly improve the processing efficiency, and the improvement becomes more obvious when more edges are inserted/deleted.

Exploring the hierarchical structure of graphs presents notable advantages for graph analysis, revealing insights ranging from individual vertex behavior to community distribution and overall graph stability. This paper studies hierarchical structures within hypergraphs, where a hyperedge can connect multiple vertices. We observed that directly extending hierarchical frameworks from pairwise graphs to hypergraphs overlooks high-order interactions and can result in either high computational complexity or sparse hierarchy structure. To address this challenge, we introduce a dual-layer hypergraph hierarchy consisting of a primary hierarchy and a secondary hierarchy, enabling the construction of a refined hypergraph hierarchy in linear time. The dual-layer hierarchy establishes a global hierarchy based on vertex cohesion, utilizing vertex-induced subhypergraphs, and a local hierarchy based on hyperedge containment, employing edge-induced subhypergraphs. The combination of global and local hierarchy mitigates the homogeneity and sparsity issues inherent in single-layer hierarchies, allowing more effective modeling of high-order interactions. Furthermore, we propose an efficient hierarchical construction algorithm by leveraging a novel hyperedge-based disjoint set to identify connected subhypergraphs. Additionally, to optimize the local hierarchy further and prevent the emergence of excessively redundant levels, we introduce a compact local hierarchy by defining a restricted subgraph metric to eliminate redundancy caused by large-sized hyperedges. Empirical studies on real-world hypergraphs demonstrate the effectiveness of our approach.

This article studies the core maintenance problem in dynamic graphs. The core number is a fundamental index reflecting the cohesiveness of a graph, which is widely used in large-scale graph analytics. The core maintenance problem requires updating the core numbers of vertices after a set of edges and vertices are inserted into or deleted from the graph. Previous works focus on the scenario of single-edge updates and process the edges one by one when multiple edges are inserted/deleted. We initiate the studies of processing multiple edges concurrently to improve the efficiency of core maintenance. Specifically, we discover a structure of inserted/deleted edges, superior edge set , which can be processed together to greatly reduce unnecessarily repeated visits of vertices in the procedure of sequential edge processing. Based on the structure of the superior edge set, efficient algorithms are then devised for incremental and decremental core maintenance, respectively. Compared with single-edge processing algorithms, our algorithms show a significant speedup in the processing time. Furthermore, our algorithms admit parallel implementations, which can further improve the update efficiency. We also conduct extensive experiments on different types of real-world, temporal, and synthetic data sets, and the results illustrate that the proposed algorithms exhibit good efficiency, stability, and scalability.