Efficient and privacy-preserving butterfly counting on encrypted bipartite graphs

Bipartite graphs have numerous real-world applications, with the butterfly motif serving as a key higher order structure that models cohesion within these graphs. Analyzing butterflies is crucial fora comprehensive understanding of networks, making butterfly counting a significant focus for researchers. In recent years, various efficient methods for exact butterfly counting, along with sampling-based approximate schemes, have been proposed for plaintext bipartite graphs. However, these methods often overlook data privacy concerns, which are critical in real-world scenarios such as doctor-patient and user-item relationships. Additionally, traditional encryption methods do not work due to the nature of graph structures. To tackle these challenges, we propose two schemes for exact butterfly counting on encrypted b ipartite graphs (EB-BFC), enabling butterfly counting for specific vertices or edges to protect privacy of butterfly counting. Firstly, we demonstrate how structured encryption techniques could be used to encrypt the bipartite graph and construct a secure index, resulting in the efficient, privacy-preserving scheme EB-BFC1. Secondly, to ensure vertex data privacy, we propose a butterfly counting scheme based on Private Set Intersection, EB-BFC2. Finally, we demonstrate the security and efficiency of our proposed schemes through theoretical proofs and experiments on real-world datasets.