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Privacy utility trade offs for parameter estimation in degree heterogeneous higher order networks

Bibhabasu Mandal, Sagnik Nandy

TL;DR

This work analyzes privacy-utility trade-offs in estimating β-model parameters from degree-based network summaries under differential privacy, extending from graphs to higher-order hypergraphs. It derives finite-sample minimax lower bounds under both local and central edge DP, showing that local DP incurs a cost roughly proportional to $1/(\\varepsilon^2 n^{r-1})$ while central DP adds a second-order term $\max\{1/n^{r-1}, 1/(\\varepsilon^2 n^{2(r-1)})\}$. The authors propose rate-optimal procedures: a locally private discrete Laplace mechanism for releasing r-degrees combined with a regularized MLE achieves the minimax rate up to logarithmic factors, and a centrally DP gradient-descent estimator achieves matching rates with explicit dependencies on $n$, $\\varepsilon$, and $\\delta$. Through simulations and a real Enron network analysis, the paper demonstrates that central DP yields smaller privacy costs than local DP and quantifies the cost of privacy on parameter estimation and link prediction, informing practical deployment of private degree-based analyses in higher-order networks.

Abstract

In sensitive applications involving relational datasets, protecting information about individual links from adversarial queries is of paramount importance. In many such settings, the available data are summarized solely through the degrees of the nodes in the network. We adopt the $β$ model, which is the prototypical statistical model adopted for this form of aggregated relational information, and study the problem of minimax-optimal parameter estimation under both local and central differential privacy constraints. We establish finite sample minimax lower bounds that characterize the precise dependence of the estimation risk on the network size and the privacy parameters, and we propose simple estimators that achieve these bounds up to constants and logarithmic factors under both local and central differential privacy frameworks. Our results provide the first comprehensive finite sample characterization of privacy utility trade offs for parameter estimation in $β$ models, addressing the classical graph case and extending the analysis to higher order hypergraph models. We further demonstrate the effectiveness of our methods through experiments on synthetic data and a real world communication network.

Privacy utility trade offs for parameter estimation in degree heterogeneous higher order networks

TL;DR

This work analyzes privacy-utility trade-offs in estimating β-model parameters from degree-based network summaries under differential privacy, extending from graphs to higher-order hypergraphs. It derives finite-sample minimax lower bounds under both local and central edge DP, showing that local DP incurs a cost roughly proportional to while central DP adds a second-order term . The authors propose rate-optimal procedures: a locally private discrete Laplace mechanism for releasing r-degrees combined with a regularized MLE achieves the minimax rate up to logarithmic factors, and a centrally DP gradient-descent estimator achieves matching rates with explicit dependencies on , , and . Through simulations and a real Enron network analysis, the paper demonstrates that central DP yields smaller privacy costs than local DP and quantifies the cost of privacy on parameter estimation and link prediction, informing practical deployment of private degree-based analyses in higher-order networks.

Abstract

In sensitive applications involving relational datasets, protecting information about individual links from adversarial queries is of paramount importance. In many such settings, the available data are summarized solely through the degrees of the nodes in the network. We adopt the model, which is the prototypical statistical model adopted for this form of aggregated relational information, and study the problem of minimax-optimal parameter estimation under both local and central differential privacy constraints. We establish finite sample minimax lower bounds that characterize the precise dependence of the estimation risk on the network size and the privacy parameters, and we propose simple estimators that achieve these bounds up to constants and logarithmic factors under both local and central differential privacy frameworks. Our results provide the first comprehensive finite sample characterization of privacy utility trade offs for parameter estimation in models, addressing the classical graph case and extending the analysis to higher order hypergraph models. We further demonstrate the effectiveness of our methods through experiments on synthetic data and a real world communication network.
Paper Structure (32 sections, 9 theorems, 95 equations, 4 figures, 2 tables)

This paper contains 32 sections, 9 theorems, 95 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Statistical model and problem formulation
  3. Privacy Framework
  4. Major Contributions
  5. Related Work
  6. Notation and paper organization
  7. Privacy--Utility Trade-offs
  8. Lower bounds under local differential privacy
  9. Lower bounds under central differential privacy
  10. Locally Private Release of Degrees via the Discrete Laplace Mechanism
  11. DP-GD Based Parameter Estimation under Central Differential Privacy
  12. Numerical Experiment: Cost of privacy in parameter estimation
  13. Privacy-preserving link prediction on the Enron email dataset
  14. Discussion and future directions
  15. Use of generative AI.
  16. ...and 17 more sections

Key Result

Theorem 2.1

For any $\varepsilon\in[0,\varepsilon_0]$, there exists a constant $c_0=c_0(r,M,\varepsilon_0)>0$ such that

Figures (4)

  • Figure 1: Local DP in hypergraph $\beta$-model
  • Figure 2: Central DP in hypergraph $\beta$-model
  • Figure 3: Log excess risk as a function of $\log_{10} n$ for $\varepsilon=0.01$.
  • Figure 4: Log excess risk as a function of $\log_{10}\varepsilon$ for $n=200$.

Theorems & Definitions (12)

  • Definition 1.1: Edge Differential Privacy
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Proposition A.1
  • ...and 2 more