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Fully Dynamic Graph Algorithms with Edge Differential Privacy

Sofya Raskhodnikova, Teresa Anna Steiner

TL;DR

The paper advances the study of differential privacy in fully dynamic graphs under continual release by delivering the first private algorithms for several core graph statistics (including triangle count, connected components, maximum matching, degree histogram, and degree list) and by establishing tight upper and lower bounds for both event-level and item-level edge-DP. Central to the approach are a suite of techniques: a sequential embedding from batch problems, reductions from submatrix queries to triangle counting, a general lower-bound framework built on 2-edge gadgets, a transformation from degree-restricted to general DP, and a recomputing-with-noise scheme via a binary-tree mechanism for triangle counting. The results illuminate fundamental privacy-utility limits in the fully dynamic setting, showing exponential-in-time-horizon lower bounds for several problems and tight, near-optimal upper bounds in multiple regimes, while also connecting lower bounds across event-level, item-level, and output-determined paradigms. Together, these contributions provide a comprehensive toolkit for designing and analyzing private fully dynamic graph algorithms with continual outputs, highlighting both what is achievable and where intrinsic limits lie.

Abstract

We study differentially private algorithms for analyzing graphs in the challenging setting of continual release with fully dynamic updates, where edges are inserted and deleted over time, and the algorithm is required to update the solution at every time step. Previous work has presented differentially private algorithms for many graph problems that can handle insertions only or deletions only (called partially dynamic algorithms) and obtained some hardness results for the fully dynamic setting. The only algorithms in the latter setting were for the edge count, given by Fichtenberger, Henzinger, and Ost (ESA 21), and for releasing the values of all graph cuts, given by Fichtenberger, Henzinger, and Upadhyay (ICML 23). We provide the first differentially private and fully dynamic graph algorithms for several other fundamental graph statistics (including the triangle count, the number of connected components, the size of the maximum matching, and the degree histogram), analyze their error and show strong lower bounds on the error for all algorithms in this setting. We study two variants of edge differential privacy for fully dynamic graph algorithms: event-level and item-level. We give upper and lower bounds on the error of both event-level and item-level fully dynamic algorithms for several fundamental graph problems. No fully dynamic algorithms that are private at the item-level (the more stringent of the two notions) were known before. In the case of item-level privacy, for several problems, our algorithms match our lower bounds.

Fully Dynamic Graph Algorithms with Edge Differential Privacy

TL;DR

The paper advances the study of differential privacy in fully dynamic graphs under continual release by delivering the first private algorithms for several core graph statistics (including triangle count, connected components, maximum matching, degree histogram, and degree list) and by establishing tight upper and lower bounds for both event-level and item-level edge-DP. Central to the approach are a suite of techniques: a sequential embedding from batch problems, reductions from submatrix queries to triangle counting, a general lower-bound framework built on 2-edge gadgets, a transformation from degree-restricted to general DP, and a recomputing-with-noise scheme via a binary-tree mechanism for triangle counting. The results illuminate fundamental privacy-utility limits in the fully dynamic setting, showing exponential-in-time-horizon lower bounds for several problems and tight, near-optimal upper bounds in multiple regimes, while also connecting lower bounds across event-level, item-level, and output-determined paradigms. Together, these contributions provide a comprehensive toolkit for designing and analyzing private fully dynamic graph algorithms with continual outputs, highlighting both what is achievable and where intrinsic limits lie.

Abstract

We study differentially private algorithms for analyzing graphs in the challenging setting of continual release with fully dynamic updates, where edges are inserted and deleted over time, and the algorithm is required to update the solution at every time step. Previous work has presented differentially private algorithms for many graph problems that can handle insertions only or deletions only (called partially dynamic algorithms) and obtained some hardness results for the fully dynamic setting. The only algorithms in the latter setting were for the edge count, given by Fichtenberger, Henzinger, and Ost (ESA 21), and for releasing the values of all graph cuts, given by Fichtenberger, Henzinger, and Upadhyay (ICML 23). We provide the first differentially private and fully dynamic graph algorithms for several other fundamental graph statistics (including the triangle count, the number of connected components, the size of the maximum matching, and the degree histogram), analyze their error and show strong lower bounds on the error for all algorithms in this setting. We study two variants of edge differential privacy for fully dynamic graph algorithms: event-level and item-level. We give upper and lower bounds on the error of both event-level and item-level fully dynamic algorithms for several fundamental graph problems. No fully dynamic algorithms that are private at the item-level (the more stringent of the two notions) were known before. In the case of item-level privacy, for several problems, our algorithms match our lower bounds.
Paper Structure (28 sections, 33 theorems, 21 equations, 3 figures, 2 tables)

This paper contains 28 sections, 33 theorems, 21 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our results
  3. Our Techniques
  4. Additional Related Work
  5. Discussion and Open Questions
  6. Organization of Technical Sections
  7. Preliminaries
  8. Graph Sequences
  9. Continual release graph algorithms
  10. $(\alpha,\beta)$-accuracy for graph functions
  11. Problem Definitions
  12. Fully Dynamic Private Triangle Counting
  13. Lower Bounds for Event-Level TriangleCount via Submatrix Queries
  14. Event-Level Private Algorithm for Triangle Count
  15. Lower Bound for Item-Level TriangleCount
  16. ...and 13 more sections

Key Result

Lemma 2.5

Let $k\in\mathbb N$ and $\varepsilon>0$ and $f:\mathcal{U}^{*}\rightarrow \mathbb{R}^k$ be a function with $L_1$-sensitivity $\Delta_1$. The Laplace mechanism is defined as $\mathcal{A}(x)=f(x)+(Y_1,\dots,Y_k)$, where $Y_i \sim \mathrm{Lap}(\Delta_1/\varepsilon)$ are independent random variables for

Figures (3)

  • Figure 3.1: An example of the construction in the proof of \ref{['lem:red_triangles_submatrix']} for $n=4$ and $w=5$. The input matrix is $Y=\left(1010110000011110\right)$; the dashed lines show the edges inserted and then deleted for the query $(a,b)$ with $a^T=(0~0~1~0)$ and $b^T=(0~1~0~1)$. There are $5$ triangles between $x_3$, $v_4$, and $z_1,\dots,z_5$, and $(3,4)$ is the only pair $(i,j)$ satisfying $Y[ij]=a[i]=b[j]=1$.
  • Figure 3.2: An example of the construction in the proof of \ref{['lem:red_triangles_submatrix_boundedD']} for $n=4$, $B=2$ and $w=5$. The input matrix is $Y=\left(1010110000011110\right)$; the dashed lines show the edges inserted and then deleted for the query $(a,b)$ with $a^T=(0~0~1~0)$ and $b^T=(0~1~0~1)$.
  • Figure 5.1: 2-edge distinguishing gadgets for $f_{MM},f_{CC}$ and $-f_{\geq 1}$ of constant size and weight. Note that for $f_{MM}$, we have $e_1,e_2\notin E'$, and for $f_{CC}$ and $-f_{\geq 1}$, we have $e_1,e_2\in E'$.

Theorems & Definitions (80)

  • Definition 2.1: Neighboring datasets
  • Definition 2.2: Differential privacy DworkMNS06DworkKMMN06
  • Definition 2.3: Sensitivity
  • Definition 2.4: Laplace distribution
  • Lemma 2.5: Laplace Mechanism DworkMNS06
  • Definition 2.6: Normal Distribution
  • Lemma 2.7: Gaussian mechanism BlumDMN05BunS16
  • Lemma 2.8: Gaussian tail bound
  • Lemma 2.9: Simple composition DworkL09DworkRV10DworkKMMN06
  • Definition 2.10: Edge-neighboring
  • ...and 70 more