Mining dense subgraphs in bipartite graphs is a fundamental problem in graph analysis. However, existing cohesive models for dense subgraph mining often encounter practical limitations. For example, biclique requires that any vertex from one set connects all vertices from the other set, leading to an NP-hard computational problem, while \((\alpha ,\beta )\) -core requires that any vertex from one set connects at least a certain number \((\alpha \) or \(\beta )\) of vertices from the other set, potentially extracting vertices with weak internal relationships. To overcome these limitations, we propose a novel model called \((\alpha ,\beta )\) -butterfly. The \((\alpha ,\beta )\) -butterfly computation makes an optimal balance between preserving strong internal relationships between vertices and ensuring computational efficiency. The \((\alpha ,\beta )\) -butterfly employs butterfly number rather than degree as the metric, thereby preserving strong internal relationships between the extracted vertices. We propose a bloom-based structure called BC (Blooms-Cluster) which records the butterfly relationship between vertices. Based on BC, we devise an efficient approach by directly finding and updating affected vertices on BC instead of visiting the neighbors of removed vertices on the original graph, enabling the computation to achieve a high computational efficiency. We further propose a BC-based index, which supports efficient computation of satisfactory results in response to changes in parameter values. We theoretically prove that BC can correctly maintain the butterfly relationship between vertices to ensure the correctness of our algorithms. Also, we conduct extensive experiments on 9 real-world datasets. The experimental results demonstrate that our algorithms can efficiently compute \((\alpha ,\beta )\) -butterfly.
