Balanced Butterfly Counting in Bipartite-Network

Bipartite graphs offer a powerful framework for modeling complex relationships between two distinct types of vertices, incorporating probabilistic, temporal, and rating-based information. While the research community has extensively explored various types of bipartite relationships, there has been a notable gap in studying Signed Bipartite Graphs, which capture liking / disliking interactions in real-world networks such as customer-rating-product and senator-vote-bill. Balance butterflies, representing 2 x 2 bicliques, provide crucial insights into antagonistic groups, balance theory, and fraud detection by leveraging the signed information. However, such applications require counting balance butterflies which remains unexplored. In this paper, we propose a new problem: counting balance butterflies in a signed bipartite graph. To address this problem, we adopt state-of-the-art algorithms for butterfly counting, establishing a smart baseline that reduces the time complexity for solving our specific problem. We further introduce a novel bucket approach specifically designed to count balanced butterflies efficiently. We propose a parallelized version of the bucketing approach to enhance performance. Extensive experimental studies on nine real-world datasets demonstrate that our proposed bucket-based algorithm is up to 120x faster over the baseline, and the parallel implementation of the bucket-based algorithm is up to 45x faster over the single core execution. Moreover, a real-world case study showcases the practical application and relevance of counting balanced butterflies

Searching Personalized kk-wing in Large and Dynamic Bipartite Graphs

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 the personalized $k$-wing search problem. We introduce a $k$-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 called EquiWing. Further, we propose a more compact version of EquiWing, EquiWing-Comp, which is achieved by integrating our proposed $k$-butterfly loose approach 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, large-scale graphs to validate the efficiency and effectiveness of EquiWing and EquiWing-Comp compared to the baseline.Comment: 13 pages, 10 figures and 4 table