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Signal Recovery from Random Dot-Product Graphs Under Local Differential Privacy

Siddharth Vishwanath, Jonathan Hehir

TL;DR

This work addresses recovering latent positions in generalized random dot-product graphs under $\varepsilon$-edge local differential privacy. It shows that the edgeFlip privacy mechanism induces a tractable geometric distortion captured by an affine map $\varphi_\varepsilon$, enabling a privacy-aware inference pipeline based on a privacy-adjusted spectral embedding. The authors establish a minimax lower bound for latent-position estimation under $\varepsilon$-edge LDP and demonstrate that their embedding achieves near-minimax rates in dense regimes, up to a $\sqrt{\log n}$ factor; they also extend the analysis to topological inference using persistence diagrams, showing consistent recovery of underlying geometry and topology. The results generalize private community-detection insights to the richer GRDPG framework and provide practical tools for geometry- and topology-based analysis of privatized networks.

Abstract

We consider the problem of recovering latent information from graphs under $\varepsilon$-edge local differential privacy where the presence of relationships/edges between two users/vertices remains confidential, even from the data curator. For the class of generalized random dot-product graphs, we show that a standard local differential privacy mechanism induces a specific geometric distortion in the latent positions. Leveraging this insight, we show that consistent recovery of the latent positions is achievable by appropriately adjusting the statistical inference procedure for the privatized graph. Furthermore, we prove that our procedure is nearly minimax-optimal under local edge differential privacy constraints. Lastly, we show that this framework allows for consistent recovery of geometric and topological information underlying the latent positions, as encoded in their persistence diagrams. Our results extend previous work from the private community detection literature to a substantially richer class of models and inferential tasks.

Signal Recovery from Random Dot-Product Graphs Under Local Differential Privacy

TL;DR

This work addresses recovering latent positions in generalized random dot-product graphs under -edge local differential privacy. It shows that the edgeFlip privacy mechanism induces a tractable geometric distortion captured by an affine map , enabling a privacy-aware inference pipeline based on a privacy-adjusted spectral embedding. The authors establish a minimax lower bound for latent-position estimation under -edge LDP and demonstrate that their embedding achieves near-minimax rates in dense regimes, up to a factor; they also extend the analysis to topological inference using persistence diagrams, showing consistent recovery of underlying geometry and topology. The results generalize private community-detection insights to the richer GRDPG framework and provide practical tools for geometry- and topology-based analysis of privatized networks.

Abstract

We consider the problem of recovering latent information from graphs under -edge local differential privacy where the presence of relationships/edges between two users/vertices remains confidential, even from the data curator. For the class of generalized random dot-product graphs, we show that a standard local differential privacy mechanism induces a specific geometric distortion in the latent positions. Leveraging this insight, we show that consistent recovery of the latent positions is achievable by appropriately adjusting the statistical inference procedure for the privatized graph. Furthermore, we prove that our procedure is nearly minimax-optimal under local edge differential privacy constraints. Lastly, we show that this framework allows for consistent recovery of geometric and topological information underlying the latent positions, as encoded in their persistence diagrams. Our results extend previous work from the private community detection literature to a substantially richer class of models and inferential tasks.

Paper Structure

This paper contains 23 sections, 16 theorems, 149 equations, 10 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Differential Privacy for Graphs
  4. Spectral Embedding and GRDPGs
  5. Main Results
  6. Closure of GRDPGs under edgeFlip
  7. Minimax rates for GRDPGs
  8. Privacy-Adjusted Spectral Embedding
  9. Topological inference under edgeFlip
  10. Experiments
  11. Conclusion
  12. Spectral alignment matrices
  13. Auxiliary Results
  14. Properties of P and $\Delta$
  15. Upper bounds for GRDPG Estimation
  16. ...and 8 more sections

Key Result

Theorem 3.1

Suppose $(\Av, \Xv) \sim \mathcal{G}(\pr, \rho_n; p, q)$, and for $\varepsilon\ge 0$, let $\mathcal{M}_\varepsilon(\Av)$ denote the $\varepsilon$-edge LDP graph under edgeFlip. Let $\varphi_\varepsilon: \R^{d} \to \R^{d+1}$ be the map given by $\varphi_\varepsilon(\xv) = \te \oplus \se\rho_n^{1/2}\x Then, $(\mathcal{M}_\varepsilon(\Av), \varphi_\varepsilon(\textup{X})) \sim \mathcal{G}\pa{\varphi_

Figures (10)

  • Figure 1: Hierarchy of network models.
  • Figure 2: Illustration of \ref{['prop:closure']}. Spectral embedding of $\mathcal{M}_\varepsilon(\Av)$ after edgeFlip when $\textup{X}\!\sim\!\pr\!=\!\!\textup{Unif}(\bbS^1)$ and $\rho_n\!\equiv\!1$.
  • Figure 3: Illustration of edgeFlip for $\textsf{SBM}(n; \gamma, \rho_n)$
  • Figure 4: Illustration of the bounds in \ref{['thm:convergence-rate']}. Heatmap of $d_{2,\infty}({\widecheck{\textup{X}}}, \textup{X})$ error for $\varepsilon$ vs. $n$ in \ref{['exp:convergence']}.
  • Figure 5: Spectral embedding followed by UMAP for the OpenFlights network in \ref{['exp:flights']}.
  • ...and 5 more figures

Theorems & Definitions (31)

  • Definition 2.1: $\varepsilon$-edge DP
  • Definition 2.2: $\varepsilon$-edge LDP
  • Definition 2.3: edgeFlip
  • Definition 2.4
  • Definition 2.5: GRDPG
  • Remark 2.1
  • Theorem 3.1
  • Corollary 3.1
  • Proposition 3.1: paraphrased; see \ref{['sec:composition']}
  • Definition 3.1: $d_{2,\infty}$ metric
  • ...and 21 more