Enumerating all the bipartite cohesive subgraphs in a bipartite graph has been studied extensively. However, for some applications, one is interested in finding bipartite cohesive subgraphs containing a specific vertex. In this paper, we study a new query-dependent bipartite cohesive subgraph search problem based on k-wing model. To address the problem, we propose two efficient and wing number conserving indexing schemes, EquiWing-Graph and a more compact index, EquiWing-Tree, which is achieved by using our proposed k-butterfly loose approach and discovered hierarchy properties. Moreover, we discover novel properties that help us localize the scope of the maintenance in our proposed indices at a lower cost for evolving bipartite graphs. Extensive experimental results evidence the efficiency and effectiveness of our proposed approaches.

There are extensive studies focusing on the application scenario that all the bipartite cohesive subgraphs need to be discovered in a bipartite graph. However, we observe that, for some applications, one is interested in finding bipartite cohesive subgraphs containing a specific vertex. In this paper, we study a new query-dependent bipartite cohesive subgraph search problem based on$k$-wing model, named as personalized$k$-wing search problem. We study the$k$-wing equivalence relationship to summarize the edges of a bipartite graph$G$into groups. Therefore, all the edges of$G$are segregated into different groups, i.e.$k$-wing equivalence class, forming an efficient and wing number conserving index calledEquiWing-Graph. Further, we propose a more compact index,EquiWing-Tree, which is achieved by using our proposed$k$-butterfly looseapproach and discovered hierarchy properties. These indices are used to expedite the personalized$k$-wing search with a non-repetitive access to$G$, which leads to linear algorithms for searching the personalized$k$-wing. Moreover, we conduct a thorough study on the maintenance of the proposed indices for evolving bipartite graphs. We discover novel properties that help us localize the scope of the maintenance at a low cost. By exploiting the discoveries, we propose novel algorithms for maintaining the two indices, which substantially reduces the cost of maintenance. We perform extensive experimental studies in real-world graphs to validate the efficiency and effectiveness ofEquiWing-GraphandEquiWing-Treecompared to the baseline.

Enumeration of all maximal bicliques in bipartite graphs is a well-studied fundamental problem. However, a wide range of applications need less overlapping bicliques with specific size constraints instead of all the maximal bicliques. In this paper, we study a new biclique problem, called the top-k t-biclique coverage problem. A t-biclique is a biclique with a size constraint$t$for one vertex set and the problem aims to find$k$t-bicliques maximising the coverage on the other vertex set. The top-k t-biclique coverage problem has novel applications such as finding top-k courses while maximising student engagement. We prove that this problem is NP-hard. A straightforward way to address the problem first needs to enumerate and store all t-bicliques and then greedily select$k$promising t-bicliques, leading an approximate guarantee on the coverage. However, it takes exponential space, which is impractical. We then apply a fast approximation scheme to solve this problem, which shaves the exponential space consumption by progressively updating top-k results during the t-biclique enumeration. Observing that the fast approximation algorithm takes too much time on updating the results due to the coverage is computed from scratch for each update, an online index is devised to address the drawback. Due the hardness of the problem, even the fast approximation algorithm cannot scale to large dataset. To devise a scalable solution, we then propose a heuristic algorithm running in polynomial time. Thanks for four carefully designed heuristic rules, the heuristic algorithm can find large coverage top-k t-bicliques extremely fast for large datasets. Apart from that, the heuristic result with large coverage can effectively prune unpromising enumerations in the fast greedy algorithm, which improves the efficiency of the fast approximation algorithm without compromising the approximation ratio. Extensive experiments are conducted on real datasets to justify the effectiveness and efficiency of the proposed algorithms.

Enumerating maximal bicliques in a bipartite graph is an important problem in data mining, with innumerable real-world applications across different domains such as web community, bioinformatics, etc. Although substantial research has been conducted on this problem, surprisingly, we find that pivot-based search space pruning, which is quite effective in clique enumeration, has not been exploited in biclique scenario. Therefore, in this paper, we explore the pivot-based pruning for biclique enumeration. We propose an algorithm for implementing the pivot-based pruning, powered by an effective index structure Containment Directed Acyclic Graph (CDAG). Meanwhile, existing literature indicates contradictory findings on the order of vertex selection in biclique enumeration. As such, we re-examine the problem and suggest an offline ordering of vertices which expedites the pivot pruning. We conduct an extensive performance study using real-world datasets from a wide range of domains. The experimental results demonstrate that our algorithm is more scalable and outperforms all the existing algorithms across all datasets and can achieve a significant speedup against the previous algorithms.