Locally Differentially Private Graph Clustering via the Power Iteration Method

TL;DR

This work tackles graph clustering under edge local differential privacy (LDP) by introducing an interactive power-iteration clustering framework. It employs a leading-eigenvector elimination strategy via $\tilde{W}=W-\frac{1}{n}J$ and a lazy random-walk update to stabilize convergence, enabling a constant privacy budget under graphs with minimum degree $\tilde{\Omega}(\sqrt{n})$ and well-defined cluster structure. The proposed Private Power Iteration Clustering (PPI-C) runs in $O(n\log n)$ time with $O(n)$ memory and achieves $\epsilon$-edge LDP, outperforming spectral clustering on randomized-response graphs in both SBM and Reddit-network experiments. The approach extends the applicability of LDP graph clustering to general, well-clustered graphs beyond stochastic-block-model assumptions and offers substantial practical gains in privacy-utility trade-offs for large-scale networks.

Abstract

We propose a locally differentially private graph clustering algorithm. Previous works have explored this problem, including approaches that apply spectral clustering to graphs generated via the randomized response algorithm. However, these methods only achieve accurate results when the privacy budget is in $Ω(\log n)$, which is unsuitable for many practical applications. In response, we present an interactive algorithm based on the power iteration method. Given that the noise introduced by the largest eigenvector constant can be significant, we incorporate a technique to eliminate this constant. As a result, our algorithm attains local differential privacy with a constant privacy budget when the graph is well-clustered and has a minimum degree of $\tildeΩ(\sqrt{n})$. In contrast, while randomized response has been shown to produce accurate results under the same minimum degree condition, it is limited to graphs generated from the stochastic block model. We perform experiments to demonstrate that our method outperforms spectral clustering applied to randomized response results.

Figures (7)

• Figure 1: Comparison of the normalized discrepancy between our algorithm and the randomized response-based algorithm on the graphs generated from the stochastic block model. The results shown in Figures \ref{['subfig:density1']} and \ref{['subfig:density15']} represent the differences in the normalized discrepancies between the two algorithms.
• Figure 2: The normalized discrepancies of our algorithm for the graph extracted from the Reddit graph
• Figure 3: Power iteration on $\mathrm{BSBM}(1000, 1000, 1000, 1000, 0.5, 0.2)$ for lazy random walk matrices $W_\alpha$ with $\alpha\in\{0, 0.1,\ldots, 0.9\}$.
• Figure 4: (Left): Heatmap of average $d_{\mathrm{norm}}(\text{NonElim})-d_{\mathrm{norm}}(\text{Ours})$ over $20$ SBMs with $n_1=n_2=1000$, with varying probabilities $p,q\in\{0.05,0.1,\ldots, 0.95\}$, and privacy budget $\epsilon=2.0$. (Right): Discrepancy with increasing $\epsilon$ for 20 SBMs with $p=0.3, q=0.2$.
• Figure 5: The normalized discrepancies of our algorithm for the degree-corrected stochastic block model following a power law in the degree distribution

Theorems & Definitions (32)

• Definition 2.1: $\epsilon$-Edge LDP Query
• Definition 2.2: $\epsilon$-edge LDP Algorithm qin2017generating
• Definition 2.3: Edge Local Laplacian Query hillebrand2023unbiased
• Proposition B.1
• proof
• Claim B.2
• Lemma B.3
• Claim B.4
• Lemma B.5
• Claim B.6