On the Price of Differential Privacy for Spectral Clustering over Stochastic Block Models

TL;DR

This work addresses private spectral clustering for community detection in SBMs under edge differential privacy. It introduces three scalable mechanisms—Graph Perturbation, Subsampling Stability, and Noisy Power Iteration—each accompanied by theoretical guarantees that relate the privacy budget $\varepsilon$ and failure probability to recoverability of the true communities. A general DP lower bound is derived, and concrete bounds on the distance between private and true eigenvectors are provided, leading to overlap guarantees under privacy. The methods are validated on synthetic SBMs and the Political Blogs dataset, demonstrating the privacy–utility trade-offs and practical viability of scalable, privacy-preserving community detection.

Abstract

We investigate privacy-preserving spectral clustering for community detection within stochastic block models (SBMs). Specifically, we focus on edge differential privacy (DP) and propose private algorithms for community recovery. Our work explores the fundamental trade-offs between the privacy budget and the accurate recovery of community labels. Furthermore, we establish information-theoretic conditions that guarantee the accuracy of our methods, providing theoretical assurances for successful community recovery under edge DP.

Figures (5)

• Figure 1: Overlap rate vs $n$, for $\epsilon = 1$, $p = 0.25$, $q = 0.0025$, and $\delta = 10^{-6}$. A fair comparison is ensured by setting the same total failure probability $\delta_{\text{failure}} = 0.01$ for both mechanisms. For the Graph-perturbation mechanism, the confidence level $\eta$ is directly set to $\delta_{\text{failure}}$. For the Subsampling stability mechanism, $\eta$ is adjusted to satisfy the condition $3m\eta = \delta_{\text{failure}}$.
• Figure 2: Overlap vs $\epsilon$, when $n=200$, $p =0.2$, $q = 0.02$, $\delta=n^{-2}$.
• Figure 3: Overlap vs $\epsilon$, when $n=400$, $p =0.2$, $q = 0.02$, $\delta=n^{-2}$.
• Figure 4: Overlap vs $\epsilon$, when $n=800$, $p =0.2$, $q = 0.02$, $\delta=n^{-2}$.
• Figure 5: Overlap vs $\epsilon$ for the Political Blogs dataset.

Theorems & Definitions (35)

• Definition 1: Laplacian Matrix
• Definition 2: $(\beta, \eta)$-Accurate Recovery
• Definition 3: $(\epsilon, \delta)$-edge DP
• Lemma 3.1
• proof
• Theorem 3.1: Necessary Condition
• proof
• Definition 4: Randomized Response Perturbation Mechanism
• Lemma 3.2: Concentration of Laplacian Matrices le2017concentration
• Lemma 3.3: Concentration of Perturbed Laplacian Matrices via Randomized Response Mechanism