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

2025-04-24

AI Summary

This paper addresses the problem of recovering users’ latent positions from a random dot product graph (RDPG) under ε-edge local differential privacy (LDP), ensuring that edge relationships between any two nodes remain provably indistinguishable to the data curator. We first characterize the systematic geometric distortion induced by LDP perturbation on the underlying latent geometry, then propose a correction-based statistical inference framework that achieves consistent recovery of latent positions under a generalized RDPG model; we further establish that its convergence rate nearly attains the minimax lower bound under the LDP constraint. Extending the framework to higher-order geometric-topological structures, we enable robust reconstruction of persistence diagrams via persistent homology. Our core contribution is the first theoretical linkage between LDP and geometric graph recovery—yielding a privacy-preserving graph learning solution that is both statistically optimal and topologically interpretable.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Recover latent information from graphs under local differential privacy
• Adjust inference for privatized graphs to correct geometric distortion
• Achieve near-optimal recovery of geometric and topological latent features

Innovation

• Adjusts inference for privatized graph recovery
• Uses geometric distortion for latent positions
• Extends privacy to richer model classes