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Publishing Below-Threshold Triangle Counts under Local Weight Differential Privacy

Kevin Pfisterer, Quentin Hillebrand, Vorapong Suppakitpaisarn

TL;DR

The paper tackles counting below-threshold triangles in weighted graphs under local weight differential privacy, a setting where topology is public but edge weights are private. It introduces a two-round mechanism: first, privatized incident weights are published to form a noisy graph, then nodes count locally using a triangle-ownership assignment function $\rho$ to reduce variance, employing either a biased or an unbiased estimator. A key contribution is the development of a scalable, $O(d^2\log^3 d)$-time algorithm to compute $eta$-smooth sensitivity, enabling tighter noise calibration via the smooth sensitivity framework, along with a pre-computation step that reduces covariance and improves accuracy. Empirical results on real networks show substantial gains over naive baselines, with the unbiased estimator plus smooth sensitivity delivering strong performance on graphs with many triangles and the variance-minimizing assignment further enhancing scalability. The work provides a practical foundation for privacy-preserving subgraph statistics in weighted networks and suggests natural extensions to broader subgraph counts.

Abstract

We propose an algorithm for counting below-threshold triangles in weighted graphs under local weight differential privacy. While prior work has largely focused on unweighted graphs, edge weights are intrinsic to many real-world networks. We consider the setting in which the graph topology is publicly known and privacy is required only for the contribution of an individual to incident edge weights, capturing practical scenarios such as road and telecommunication networks. Our method uses two rounds of communication. In the first round, each node releases privatized information about its incident edge weights under local weight differential privacy. In the second round, nodes locally count below-threshold triangles using this privatized information; we introduce both biased and unbiased variants of the estimator. We further develop two refinements: (i) a pre-computation step that reduces covariance and thus lowers expected error, and (ii) an efficient procedure for computing smooth sensitivity, which substantially reduces running time relative to a straightforward implementation. Finally, we present experimental results that quantify the trade-offs between the biased and unbiased variants and demonstrate the effectiveness of the proposed improvements.

Publishing Below-Threshold Triangle Counts under Local Weight Differential Privacy

TL;DR

The paper tackles counting below-threshold triangles in weighted graphs under local weight differential privacy, a setting where topology is public but edge weights are private. It introduces a two-round mechanism: first, privatized incident weights are published to form a noisy graph, then nodes count locally using a triangle-ownership assignment function to reduce variance, employing either a biased or an unbiased estimator. A key contribution is the development of a scalable, -time algorithm to compute -smooth sensitivity, enabling tighter noise calibration via the smooth sensitivity framework, along with a pre-computation step that reduces covariance and improves accuracy. Empirical results on real networks show substantial gains over naive baselines, with the unbiased estimator plus smooth sensitivity delivering strong performance on graphs with many triangles and the variance-minimizing assignment further enhancing scalability. The work provides a practical foundation for privacy-preserving subgraph statistics in weighted networks and suggests natural extensions to broader subgraph counts.

Abstract

We propose an algorithm for counting below-threshold triangles in weighted graphs under local weight differential privacy. While prior work has largely focused on unweighted graphs, edge weights are intrinsic to many real-world networks. We consider the setting in which the graph topology is publicly known and privacy is required only for the contribution of an individual to incident edge weights, capturing practical scenarios such as road and telecommunication networks. Our method uses two rounds of communication. In the first round, each node releases privatized information about its incident edge weights under local weight differential privacy. In the second round, nodes locally count below-threshold triangles using this privatized information; we introduce both biased and unbiased variants of the estimator. We further develop two refinements: (i) a pre-computation step that reduces covariance and thus lowers expected error, and (ii) an efficient procedure for computing smooth sensitivity, which substantially reduces running time relative to a straightforward implementation. Finally, we present experimental results that quantify the trade-offs between the biased and unbiased variants and demonstrate the effectiveness of the proposed improvements.
Paper Structure (42 sections, 40 theorems, 96 equations, 4 figures, 1 table, 4 algorithms)

This paper contains 42 sections, 40 theorems, 96 equations, 4 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. We design an assignment that minimizes the variance of the triangle count in Section \ref{['sec:mincov']}
  4. We develop a scalable algorithm to compute the smooth sensitivity in Section \ref{['sec:smooth']}.
  5. Preliminaries
  6. Problem definition
  7. Local weight differential privacy
  8. Two-step algorithm for publishing below-threshold triangle counts
  9. Noisy releases and task assignments
  10. Local counting
  11. Publishing local counts
  12. Privacy of our mechanism
  13. Communication costs
  14. Precision of the biased estimator
  15. Precision of the unbiased estimator
  16. ...and 27 more sections

Key Result

Theorem 1

The global sensitivity of $f'_v$ is given by where $GS(g^T_v) = 1$ when using the biased estimator and $GS(g^T_v) = 1 + 2\frac{p}{(1-p)^2}$ when using the unbiased estimator.

Figures (4)

  • Figure 1: Visualization of $C'_4$ instance introducing covariance
  • Figure 2: Representation of smooth sensitivity as a problem of shifting values to a target
  • Figure 3: Relative error across all methods under different settings for the input graph ML-Tele-278: (a) as a function of the privacy parameter $\varepsilon$, (b) as a function of the number of below-threshold triangles $|\Delta|$, (c) as a function of the threshold $\lambda$.
  • Figure 4: Relative error for varying $\lambda$ on the input graph GMWCS

Theorems & Definitions (88)

  • Definition 1
  • Definition 2: Neighboring weight vectors
  • Definition 3: Local weight differential privacy brito2023global
  • Definition 4: Discrete Laplace query ghosh2009universally
  • Definition 5: Global sensitivity dwork
  • Definition 6: Laplace query dwork
  • Definition 7: Biased estimator
  • Definition 8: Unbiased estimator
  • Theorem 1: Global sensitivity of $f_v'$
  • proof
  • ...and 78 more