In sensitive applications involving relational datasets, protecting information about individual links from adversarial queries is of paramount importance. In many such settings, the available data are summarized solely through the degrees of the nodes in the network. We adopt the $β$ model, which is the prototypical statistical model adopted for this form of aggregated relational information, and study the problem of minimax-optimal parameter estimation under both local and central differential privacy constraints. We establish finite sample minimax lower bounds that characterize the precise dependence of the estimation risk on the network size and the privacy parameters, and we propose simple estimators that achieve these bounds up to constants and logarithmic factors under both local and central differential privacy frameworks. Our results provide the first comprehensive finite sample characterization of privacy utility trade offs for parameter estimation in $β$ models, addressing the classical graph case and extending the analysis to higher order hypergraph models. We further demonstrate the effectiveness of our methods through experiments on synthetic data and a real world communication network.

翻译：在涉及关系数据集的敏感应用中，保护个体链接信息免受对抗性查询至关重要。在此类场景中，可用数据通常仅通过网络中节点的度信息进行汇总。本文采用$β$模型——作为此类聚合关系信息的典型统计模型，研究在本地与中心化差分隐私约束下的极小极大最优参数估计问题。我们建立了有限样本极小极大下界，精确刻画了估计风险对网络规模与隐私参数的依赖关系，并提出了在本地与中心化差分隐私框架下均能达到该下界（至常数与对数因子）的简洁估计量。我们的研究首次为$β$模型中的参数估计提供了完整的有限样本隐私-效用权衡表征，涵盖了经典图模型情形，并将分析拓展至高阶超图模型。通过合成数据与真实世界通信网络的实验，我们进一步验证了所提方法的有效性。