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Private Synthetic Graph Generation and Fused Gromov-Wasserstein Distance

Leoni Carla Wirth, Gholamali Aminian, Gesine Reinert

TL;DR

This work tackles private synthetic graph generation from complex data by introducing PSGG, which jointly outputs a true attributed graph and an ε-DP synthetic graph under a random connection model. It leverages TV-PSMM to produce a DP attribute measure hat{ν} and uses the fused Gromov-Wasserstein distance to quantify utility between structured graph distributions. The authors provide two rigorous utility guarantees: (i) an accuracy bound in FGW distance between the true and private graphs, and (ii) a distributional bound between the respective graph-generating processes, with rates that depend on the ambient dimension d and privacy level ε. The framework accommodates extensions to directed/weighted graphs and (ε,δ)-DP, highlighting practical implications for privacy-preserving graph data sharing and method development.

Abstract

Networks are popular for representing complex data. In particular, differentially private synthetic networks are much in demand for method and algorithm development. The network generator should be easy to implement and should come with theoretical guarantees. Here we start with complex data as input and jointly provide a network representation as well as a synthetic network generator. Using a random connection model, we devise an effective algorithmic approach for generating attributed synthetic graphs which is $ε$-differentially private at the vertex level, while preserving utility under an appropriate notion of distance which we develop. We provide theoretical guarantees for the accuracy of the private synthetic graphs using the fused Gromov-Wasserstein distance, which extends the Wasserstein metric to structured data. Our method draws inspiration from the PSMM method of \citet{he2023}.

Private Synthetic Graph Generation and Fused Gromov-Wasserstein Distance

TL;DR

This work tackles private synthetic graph generation from complex data by introducing PSGG, which jointly outputs a true attributed graph and an ε-DP synthetic graph under a random connection model. It leverages TV-PSMM to produce a DP attribute measure hat{ν} and uses the fused Gromov-Wasserstein distance to quantify utility between structured graph distributions. The authors provide two rigorous utility guarantees: (i) an accuracy bound in FGW distance between the true and private graphs, and (ii) a distributional bound between the respective graph-generating processes, with rates that depend on the ambient dimension d and privacy level ε. The framework accommodates extensions to directed/weighted graphs and (ε,δ)-DP, highlighting practical implications for privacy-preserving graph data sharing and method development.

Abstract

Networks are popular for representing complex data. In particular, differentially private synthetic networks are much in demand for method and algorithm development. The network generator should be easy to implement and should come with theoretical guarantees. Here we start with complex data as input and jointly provide a network representation as well as a synthetic network generator. Using a random connection model, we devise an effective algorithmic approach for generating attributed synthetic graphs which is -differentially private at the vertex level, while preserving utility under an appropriate notion of distance which we develop. We provide theoretical guarantees for the accuracy of the private synthetic graphs using the fused Gromov-Wasserstein distance, which extends the Wasserstein metric to structured data. Our method draws inspiration from the PSMM method of \citet{he2023}.

Paper Structure

This paper contains 18 sections, 7 theorems, 65 equations, 1 figure, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential Privacy
  4. Fused Gromov-Wasserstein distance
  5. Total variation distance
  6. An $\epsilon$- differential private synthetic data algorithm
  7. Private synthetic graph generation
  8. Graph model
  9. The private synthetic graph generator
  10. Theoretical guarantees on utility
  11. Accuracy of Algorithm 3
  12. Bounds on the difference between the graph distributions
  13. Choice of parameters
  14. Conclusion and Future Works
  15. More Related Works
  16. ...and 3 more sections

Key Result

Proposition 2.1

If for all integers $k$ in the support of $P_\Lambda$ and for $a \in \{-1, 1\},$ then the Algorithm alg: PSMM is $\epsilon$-differential private.

Figures (1)

  • Figure 1: Graphs of Chung-Lu type generated with Algorithm \ref{['alg: graph generation']} on $\Omega = [0,1]$ with $n=1000$, $m = \lceil\sqrt{\epsilon n}\rceil$ and $a=b=100$. Graph 1 shows a realization of the "true" graph with $\mathcal{X} = \{0,\ldots,0,1,\ldots,1\}$. Graph 2-4 are realizations of the private synthetic graph generated with Laplacian noise $\text{Lap}_{\mathbb{Z}\xspace}(1/\epsilon)$ for $\epsilon\in\{1,0.1,0.01\}$. The vertex color visualizes the attribute, where small attributes correspond to dark colors.

Theorems & Definitions (13)

  • Proposition 2.1
  • Example 2.2
  • Proposition 2.3
  • Theorem 3.1
  • Theorem 4.1
  • Corollary 4.2
  • Remark 4.3
  • Theorem 4.4
  • Corollary 4.5
  • proof
  • ...and 3 more