Butterfly Counting over Bipartite Graphs with Local Differential Privacy

Butterfly counting on bipartite graphs has gained increasing attention in past decades. Inevitably, butterfly counts can reveal the presence of certain edges, posing a privacy risk in real applications. Edge local differential privacy (edge LDP), which requires each vertex to perturb its neighbors locally, has been applied to protect edge privacy in graphs. This paper, for the first time, investigates butterfly counting on bipartite graphs with edge LDP. Although a straightforward approach that allows each vertex to perturb its incident edges locally to construct a noisy graph and perform butterfly counting preserves edge LDP, it often results in severe over-counting and significant bias since the resulting noisy graph is generally much denser than the input graph. To obtain unbiased butterfly counts, we propose a multiple-round interaction algorithm to allow the vertices to download the noisy graph and compute local motif counts. Moreover, to avoid adding substantial noise to satisfy edge LDP, we further propose the Download-free Butterfly. Estimation (DBE) algorithm, which captures motif transformation probabilities and relies on motif counts from the noisy graph to yield unbiased butterfly estimates. DBE significantly enhances accuracy via reduced communication between vertices and the data curator. Extensive experiments on 14 datasets validate the effectiveness and efficiency of our proposed techniques.