Differentially Private Community Detection in $h$-uniform Hypergraphs

TL;DR

This paper investigates the thresholds for exact community detection in $h$-uniform hypergraphs under differential privacy constraints, analyzing how privacy budgets affect the ability to recover community structure.

Contribution

It introduces and compares three differentially private mechanisms for hypergraph community detection, providing exact recovery thresholds and analyzing privacy-utility trade-offs.

Findings

Sampling-based and randomized response mechanisms achieve pure $\e$-hyperedge DP.

Stability-based mechanisms cannot achieve pure $\e$-DP.

Privacy budget scales logarithmically with hyperedge density ratio or depends on hypergraph size.

Abstract

This paper studies the exact recovery threshold subject to preserving the privacy of connections in $h$-uniform hypergraphs. Privacy is characterized by the $(\epsilon, \delta)$-hyperedge differential privacy (DP), an extension of the notion of $(\epsilon, \delta)$-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, $(\epsilon, \delta)$. Sampling-based mechanisms and randomized response mechanisms guarantee pure $\epsilon$-hyperedge DP where $\delta=0$, 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.