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Near-Optimal Differentially Private k-Core Decomposition

Laxman Dhulipala, George Z. Li, Quanquan C. Liu

TL;DR

The paper tackles the problem of computing the k-core decomposition under $\varepsilon$-edge differential privacy, improving prior results by obtaining a private algorithm with no multiplicative error and $O(\log n / \varepsilon)$ additive error. Central to the approach is the Multidimensional AboveThreshold (MAT), a generalized sparse vector technique that handles multidimensional threshold queries and supports threshold-based peeling in graph algorithms, with both central and local privacy realizations. The authors present a private variant of the classical peeling algorithm, achieve near-linear time, and extend the framework to related problems such as densest subgraph and low out-degree ordering, including improved guarantees in the LEDP and centralized settings. The results yield near-optimal privacy-utility tradeoffs and strengthen the bridge between distributed/parallel graph algorithms and privately released graph analytics, with practical implications for private graph analytics on large-scale networks.

Abstract

Recent work by Dhulipala et al. \cite{DLRSSY22} initiated the study of the $k$-core decomposition problem under differential privacy via a connection between low round/depth distributed/parallel graph algorithms and private algorithms with small error bounds. They showed that one can output differentially private approximate $k$-core numbers, while only incurring a multiplicative error of $(2 +η)$ (for any constant $η>0$) and additive error of $\poly(\log(n))/\eps$. In this paper, we revisit this problem. Our main result is an $\eps$-edge differentially private algorithm for $k$-core decomposition which outputs the core numbers with no multiplicative error and $O(\text{log}(n)/\eps)$ additive error. This improves upon previous work by a factor of 2 in the multiplicative error, while giving near-optimal additive error. Our result relies on a novel generalized form of the sparse vector technique, which is especially well-suited for threshold-based graph algorithms; thus, we further strengthen the connection between distributed/parallel graph algorithms and differentially private algorithms.

Near-Optimal Differentially Private k-Core Decomposition

TL;DR

The paper tackles the problem of computing the k-core decomposition under -edge differential privacy, improving prior results by obtaining a private algorithm with no multiplicative error and additive error. Central to the approach is the Multidimensional AboveThreshold (MAT), a generalized sparse vector technique that handles multidimensional threshold queries and supports threshold-based peeling in graph algorithms, with both central and local privacy realizations. The authors present a private variant of the classical peeling algorithm, achieve near-linear time, and extend the framework to related problems such as densest subgraph and low out-degree ordering, including improved guarantees in the LEDP and centralized settings. The results yield near-optimal privacy-utility tradeoffs and strengthen the bridge between distributed/parallel graph algorithms and privately released graph analytics, with practical implications for private graph analytics on large-scale networks.

Abstract

Recent work by Dhulipala et al. \cite{DLRSSY22} initiated the study of the -core decomposition problem under differential privacy via a connection between low round/depth distributed/parallel graph algorithms and private algorithms with small error bounds. They showed that one can output differentially private approximate -core numbers, while only incurring a multiplicative error of (for any constant ) and additive error of . In this paper, we revisit this problem. Our main result is an -edge differentially private algorithm for -core decomposition which outputs the core numbers with no multiplicative error and additive error. This improves upon previous work by a factor of 2 in the multiplicative error, while giving near-optimal additive error. Our result relies on a novel generalized form of the sparse vector technique, which is especially well-suited for threshold-based graph algorithms; thus, we further strengthen the connection between distributed/parallel graph algorithms and differentially private algorithms.
Paper Structure (18 sections, 17 theorems, 15 equations, 8 algorithms)

This paper contains 18 sections, 17 theorems, 15 equations, 8 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Concurrent, Independent Work.
  5. Differential Privacy Background
  6. Local Edge Differential Privacy
  7. Generalized Sparse Vector Technique
  8. Local Privacy Guarantees
  9. Differentially Private $k$-Core Decomposition
  10. A Variation of the Classical Algorithm
  11. Private Implementation of the Algorithm
  12. Efficient Implementation of the Algorithm
  13. An Improved Analysis of DLRSSY22
  14. Related Problems
  15. Differentially Private Densest Subgraph
  16. ...and 3 more sections

Key Result

Theorem 1.3

There is an algorithm which gives $(1, O(\frac{\log(n)}{\varepsilon}))$-approximate core numbers that is $\varepsilon$-(local) edge differentially private and guarantees the approximation with high probability.

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8: $(\phi, \zeta)$-Approximate Low Out-Degree Ordering
  • Theorem 1.9
  • Definition 2.1
  • ...and 26 more