Clustering non-independent and identically distributed (non-IID) data under local differential privacy (LDP) in federated settings presents a critical challenge: preserving privacy while maintaining accuracy without iterative communication. Existing one-shot methods rely on unstable pairwise centroid distances or neighborhood rankings, degrading severely under strong LDP noise and data heterogeneity. We present Gravitational Federated Clustering (GFC), a novel approach to privacy-preserving federated clustering that overcomes the limitations of distance-based methods under varying LDP. Addressing the critical challenge of clustering non-IID data with diverse privacy guarantees, GFC transforms privatized client centroids into a global gravitational potential field where true cluster centers emerge as topologically persistent singularities. Our framework introduces two key innovations: (1) a client-side compactness-aware perturbation mechanism that encodes local cluster geometry as "mass" values, and (2) a server-side topological aggregation phase that extracts stable centroids through persistent homology analysis of the potential field's superlevel sets. Theoretically, we establish a closed-form bound between the privacy budget $ε$ and centroid estimation error, proving the potential field's Lipschitz smoothing properties exponentially suppress noise in high-density regions. Empirically, GFC outperforms state-of-the-art methods on ten benchmarks, especially under strong LDP constraints ($ε< 1$), while maintaining comparable performance at lower privacy budgets. By reformulating federated clustering as a topological persistence problem in a synthetic physics-inspired space, GFC achieves unprecedented privacy-accuracy trade-offs without iterative communication, providing a new perspective for privacy-preserving distributed learning.

翻译：在联邦学习环境中，对非独立同分布（non-IID）数据进行局部差分隐私（LDP）下的聚类面临一个关键挑战：如何在避免迭代通信的同时，既保护隐私又保持准确性。现有的一步式方法依赖于不稳定的成对质心距离或邻域排序，在强LDP噪声和数据异构性下性能严重下降。我们提出引力联邦聚类（GFC），这是一种新颖的隐私保护联邦聚类方法，克服了基于距离的方法在不同LDP下的局限性。针对具有多样化隐私保证的非IID数据聚类的关键挑战，GFC将私有化的客户端质心转化为全局引力势场，其中真实的聚类中心作为拓扑持久的奇点显现。我们的框架引入两个关键创新：（1）一种客户端侧紧凑性感知扰动机制，将局部聚类几何编码为“质量”值；（2）一种服务器侧拓扑聚合阶段，通过势场超水平集的持久同调分析提取稳定质心。理论上，我们建立了隐私预算$ε$与质心估计误差之间的闭式界，证明了势场的Lipschitz平滑特性在高密度区域指数级抑制噪声。实证上，GFC在十个基准测试中优于最先进的方法，特别是在强LDP约束下（$ε< 1$），同时在较低隐私预算下保持相当性能。通过将联邦聚类重新表述为合成物理启发空间中的拓扑持久性问题，GFC实现了前所未有的隐私-准确性权衡，无需迭代通信，为隐私保护分布式学习提供了新视角。