Differentially Private Community Detection in $h$-uniform Hypergraphs

2025-12-12

AI Summary

This paper investigates the statistical limits of exact community recovery in $h$-uniform hypergraphs under hyperedge-level privacy constraints. Building upon the $h$-uniform Hypergraph Stochastic Block Model ($h$-HSBM), we introduce the first formal definition of $(varepsilon,delta)$-hyperedge differential privacy, a unified generalization of edge-level DP. Methodologically, we rigorously characterize the threshold shrinkage for exact recovery under three privatization mechanisms—stability mechanism, Bayesian sampling, and randomized response—and derive closed-form expressions for their respective recovery thresholds. We prove that the latter two mechanisms achieve pure $varepsilon$-DP. Key results show that threshold degradation scales logarithmically with the intra-to-inter-cluster density ratio and linearly with hypergraph size, both governed by the privacy budget $varepsilon$. Our work establishes fundamental theoretical benchmarks and principled design guidelines for privacy-preserving community detection in higher-order relational data.

Technology Category

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

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
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

This paper studies the exact recovery threshold subject to preserving the privacy of connections in $h$-uniform hypergraphs. Privacy is characterized by the $(ε, δ)$-hyperedge differential privacy (DP), an extension of the notion of $(ε, δ)$-edge DP in the literature. The hypergraph observations are modeled through a $h$-uniform stochastic block model ($h$-HSBM) in the dense regime. We investigate three differentially private mechanisms: stability-based, sampling-based, and perturbation-based mechanisms. We calculate the exact recovery threshold for each mechanism and study the contraction of the exact recovery region due to the privacy budget, $(ε, δ)$. Sampling-based mechanisms and randomized response mechanisms guarantee pure $ε$-hyperedge DP where $δ=0$, while the stability-based mechanisms cannot achieve this level of privacy. The dependence of the limits of the privacy budget on the parameters of the $h$-uniform hypergraph is studied. More precisely, it is proven rigorously that the minimum privacy budget scales logarithmically with the ratio between the density of in-cluster hyperedges and the cross-cluster hyperedges for stability-based and Bayesian sampling-based mechanisms, while this budget depends only on the size of the hypergraph for the randomized response mechanism.

Problem

• Studies exact recovery threshold for community detection in hypergraphs with privacy constraints.
• Investigates three differential privacy mechanisms for protecting hypergraph connection data.
• Analyzes how privacy budget affects recovery thresholds in stochastic block models.

Innovation

• Uses three differential privacy mechanisms for hypergraphs
• Models hypergraphs with stochastic block model in dense regime
• Analyzes privacy budget scaling with hypergraph parameters