Topological Federated Clustering via Gravitational Potential Fields under Local Differential Privacy

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 $\epsilon$ 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 ($\epsilon < 1$), 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.