Hurricane in Bipartite Graphs: The Lethal Nodes of Butterflies

Abstract

Bipartite graphs are widely used when modeling the relationships between two different types of entities, such as purchase relationships. In a bipartite graph, the number of butterflies, i.e., 2 × 2 biclique, is a fundamental metric for analyzing the structures and properties of bipartite graphs. Considering the deletion of critical nodes may affect the stability of bipartite graphs, we propose the butterfly minimization problem, where the attacker aims to maximize the number of butterflies removed from the graph by deleting b nodes. We prove the problem is NP-hard, and the objective function is monotonic and submodular. We adopt a greedy algorithm to solve the problem with 1 − 1/e approximation ratio. To scale for large graphs, novel methods are developed to reduce the searching space. Experiments over real-world bipartite graphs are conducted to demonstrate the advantages of proposed techniques.