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Tighter Bounds for Local Differentially Private Core Decomposition and Densest Subgraph

Monika Henzinger, A. R. Sricharan, Leqi Zhu

TL;DR

This work advances privacy-preserving graph analysis by establishing tight additive-error lower bounds for core decomposition in centralized models and 1-round local models, and by delivering local mechanisms with near-optimal accuracy. The authors introduce continual counting-based peeling to implement private core computations, enabling memoryless and memoryful local algorithms that achieve exact or near-exact core decompositions with additive errors polynomial in $\log n$ and $\log \Delta$. They also show that these core-decomposition results translate to densest-subgraph guarantees via reductions, yielding improved local-private densest-subgraph mechanisms. Finally, the paper delineates memory versus nonmemory trade-offs and lays groundwork for future separations between local memory regimes. Overall, the results tighten the known privacy-accuracy trade-offs in the local model and provide practical, near-optimal tools for privacy-preserving graph analysis.

Abstract

Computing the core decomposition of a graph is a fundamental problem that has recently been studied in the differentially private setting, motivated by practical applications in data mining. In particular, Dhulipala et al. [FOCS 2022] gave the first mechanism for approximate core decomposition in the challenging and practically relevant setting of local differential privacy. One of the main open problems left by their work is whether the accuracy, i.e., the approximation ratio and additive error, of their mechanism can be improved. We show the first lower bounds on the additive error of approximate and exact core decomposition mechanisms in the centralized and local model of differential privacy, respectively. We also give mechanisms for exact and approximate core decomposition in the local model, with almost matching additive error bounds. Our mechanisms are based on a black-box application of continual counting. They also yield improved mechanisms for the approximate densest subgraph problem in the local model.

Tighter Bounds for Local Differentially Private Core Decomposition and Densest Subgraph

TL;DR

This work advances privacy-preserving graph analysis by establishing tight additive-error lower bounds for core decomposition in centralized models and 1-round local models, and by delivering local mechanisms with near-optimal accuracy. The authors introduce continual counting-based peeling to implement private core computations, enabling memoryless and memoryful local algorithms that achieve exact or near-exact core decompositions with additive errors polynomial in and . They also show that these core-decomposition results translate to densest-subgraph guarantees via reductions, yielding improved local-private densest-subgraph mechanisms. Finally, the paper delineates memory versus nonmemory trade-offs and lays groundwork for future separations between local memory regimes. Overall, the results tighten the known privacy-accuracy trade-offs in the local model and provide practical, near-optimal tools for privacy-preserving graph analysis.

Abstract

Computing the core decomposition of a graph is a fundamental problem that has recently been studied in the differentially private setting, motivated by practical applications in data mining. In particular, Dhulipala et al. [FOCS 2022] gave the first mechanism for approximate core decomposition in the challenging and practically relevant setting of local differential privacy. One of the main open problems left by their work is whether the accuracy, i.e., the approximation ratio and additive error, of their mechanism can be improved. We show the first lower bounds on the additive error of approximate and exact core decomposition mechanisms in the centralized and local model of differential privacy, respectively. We also give mechanisms for exact and approximate core decomposition in the local model, with almost matching additive error bounds. Our mechanisms are based on a black-box application of continual counting. They also yield improved mechanisms for the approximate densest subgraph problem in the local model.
Paper Structure (17 sections, 26 theorems, 10 equations, 1 figure, 1 table, 4 algorithms)

This paper contains 17 sections, 26 theorems, 10 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Technical overview
  4. Preliminaries
  5. Core decomposition via continual counting
  6. Privacy
  7. Accuracy
  8. Memoryless modification
  9. Further applications
  10. Lower bounds for core decomposition
  11. Formal definition of differential privacy in the local model
  12. Memoryless continual counting
  13. Further applications of continual counting
  14. Approximate core decomposition
  15. Approximate densest subgraph
  16. ...and 2 more sections

Key Result

Theorem 1

[DNPR10dpcoCSS10bintreemech] Let $\epsilon > 0$ be a constant. There is an $\epsilon$-differentially private adaptive counting mechanism such that, with probability at least $1 - \beta$, at any single time step $t \in [T]$, the error is bounded by

Figures (1)

  • Figure :

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Theorem 4
  • ...and 24 more