Local Differential Privacy-Preserving Spectral Clustering for General Graphs
Sayan Mukherjee, Vorapong Suppakitpaisarn
TL;DR
This work addresses the robustness of spectral clustering under local differential privacy for general graphs by modeling privacy with edge flipping. It introduces a formal main theorem showing that, for flip probability $p = O(\log n / n)$, the spectral clustering output changes only by $O(\eta(G)\cdot n)$ in the $d_{\text{size}}$ metric, where $\eta(G)$ is the graph's spectral robustness. A key contribution is the spectral-robustness parameter and a tight analysis that yields a privacy-budget lower bound of $\Theta(\log n)$ in the worst case, complemented by an instability result for larger noise levels. Empirically, the authors validate their theory on real networks, demonstrating strong clustering stability under modest edge perturbations and outlining limits as privacy budgets grow larger.
Abstract
Spectral clustering is a widely used algorithm to find clusters in networks. Several researchers have studied the stability of spectral clustering under local differential privacy with the additional assumption that the underlying networks are generated from the stochastic block model (SBM). However, we argue that this assumption is too restrictive since social networks do not originate from the SBM. Thus, we delve into an analysis for general graphs in this work. Our primary focus is the edge flipping method -- a common technique for protecting local differential privacy. We show that, when the edges of an $n$-vertex graph satisfying some reasonable well-clustering assumptions are flipped with a probability of $O(\log n/n)$, the clustering outcomes are largely consistent. Empirical tests further corroborate these theoretical findings. Conversely, although clustering outcomes have been stable for non-sparse and well-clustered graphs produced from the SBM, we show that in general, spectral clustering may yield highly erratic results on certain well-clustered graphs when the flipping probability is $ω(\log n/n)$. This indicates that the best privacy budget obtainable for general graphs is $Θ(\log n)$.